A Profile Likelihood Analysis of the Constrained MSSM with Genetic Algorithms
Yashar Akrami, Pat Scott, Joakim Edsjö, Jan Conrad, Lars Bergström
PPreprint typeset in JHEP style - HYPER VERSION
A Profile Likelihood Analysis of the ConstrainedMSSM with Genetic Algorithms
Yashar Akrami, Pat Scott, Joakim Edsj¨o, Jan Conrad and Lars Bergstr¨om
Oskar Klein Centre for Cosmoparticle PhysicsDepartment of Physics, Stockholm UniversityAlbaNova, SE-10691 Stockholm, SwedenE-mails: yashar, pat, edsjo, conrad, [email protected]
Abstract:
The Constrained Minimal Supersymmetric Standard Model (CMSSM) is oneof the simplest and most widely-studied supersymmetric extensions to the standard modelof particle physics. Nevertheless, current data do not sufficiently constrain the modelparameters in a way completely independent of priors, statistical measures and scanningtechniques. We present a new technique for scanning supersymmetric parameter spaces,optimised for frequentist profile likelihood analyses and based on Genetic Algorithms. Weapply this technique to the CMSSM, taking into account existing collider and cosmologicaldata in our global fit. We compare our method to the
MultiNest algorithm, an efficientBayesian technique, paying particular attention to the best-fit points and implications forparticle masses at the LHC and dark matter searches. Our global best-fit point lies in thefocus point region. We find many high-likelihood points in both the stau co-annihilation andfocus point regions, including a previously neglected section of the co-annihilation regionat large m . We show that there are many high-likelihood points in the CMSSM parameterspace commonly missed by existing scanning techniques, especially at high masses. Thishas a significant influence on the derived confidence regions for parameters and observables,and can dramatically change the entire statistical inference of such scans. Keywords:
Supersymmetry Phenomenology, Supersymmetric Standard Model,Cosmology of Theories beyond the SM. a r X i v : . [ h e p - ph ] M a r ontents
1. Introduction 12. Model and analysis 6
3. Results and discussion 15
4. Summary and conclusions 34
1. Introduction
New physics beyond the Standard Model (SM) is broadly conjectured to appear at TeVenergy scales. Particular attention has been paid to supersymmetric (SUSY) extensions ofthe SM, widely hoped to show up at the Large Hadron Collider (LHC). One of the strongestmotivations for physics at the new scale is the absence of any SM mechanism for protectingthe Higgs mass against radiative corrections; this is known as the hierarchy or fine-tuningproblem [1]. Softly-broken weak-scale supersymmetry (for an introduction, see Ref. 2)provides a natural solution to this problem via the cancellation of quadratic divergences inradiative corrections to the Higgs mass. The natural connection between SUSY and grandunified theories (GUTs) also offers extensive scope for achieving gauge-coupling unificationin this framework [3]. Supersymmetry has even turned out to be a natural component ofmany string theories, so it may be worth incorporating into extensions of the SM anyway(though in these models it is not at all necessary for SUSY to be detectable at low energies).Another major theoretical motivation for supersymmetry is that most weak-scale ver-sions contain a viable dark matter (DM) candidate [4]. Its stability is typically achievedvia a conserved discrete symmetry ( R -parity) which arises naturally in some GUTs, and– 1 –akes the lightest supersymmetric particle (LSP) stable. Its ‘darkness’ is achieved byhaving the LSP be a neutral particle, such as the lightest neutralino or sneutrino. Theseare both weakly-interacting massive particles (WIMPs), making them prime dark mattercandidates [4]. The sneutrino is strongly constrained due to its large nuclear-scatteringcross-section, but the neutralino remains arguably the leading candidate for DM.Describing the low-energy behaviour of a supersymmetric model typically requiresadding many new parameters to the SM. This makes phenomenological analyses highlycomplicated. Even upgrading the SM to its most minimal SUSY form, the Minimal Su-persymmetric Standard Model (MSSM; for a recent review, see Ref. 5), introduces morethan a hundred free parameters. All but one of these come from the soft terms in theSUSY-breaking sector. Fortunately, extensive regions of the full MSSM parameter spaceare ruled out phenomenologically, as generic values of many of the new parameters allowflavour changing neutral currents (FCNCs) or CP violation at levels excluded by experi-ment.One might be able to relate many of these seemingly-free parameters theoretically, dra-matically reducing their number. This requires specification of either the underlying SUSY-breaking mechanism itself, or a mediation mechanism by which SUSY-breaking would beconducted from some undiscovered particle sector to the known particle spectrum and itsSUSY counterparts. Several mediation mechanisms (for recent reviews, see e.g. Refs. 5and 6) have been proposed which relate the MSSM parameters in very different ways, butso far no clear preference has been established for one mechanism over another. For acomparison of some mediation models using current data, see Ref. 7. Gravity-mediatedSUSY breaking, based on supergravity unification, naturally leads to the suppression ofmany of the dangerous FCNC and CP-violating terms. Its simplest version is known asminimal supergravity (mSUGRA) [8, 9].An alternative approach is to directly specify a phenomenological MSSM reductionat low energy. Here one sets troublesome CP-violating and FCNC-generating terms tozero by hand, and further reduces the number of parameters by assuming high degrees ofsymmetry in e.g. mass and mixing matrices.A hybrid approach is to construct a phenomenological GUT-scale model, broadly moti-vated by the connection between SUSY and GUTs. Here one imposes boundary conditionsat the GUT scale ( ∼ GeV) and then explores the low-energy phenomenology by meansof the Renormalisation Group Equations (RGEs). One of the most popular schemes is theConstrained MSSM (CMSSM) [10], which incorporates the phenomenologically-interestingparts of mSUGRA. The CMSSM includes four continuous parameters: the ratio of the twoHiggs vacuum expectation values (tan β ), and the GUT-scale values of the SUSY-breakingscalar, gaugino and trilinear mass parameters ( m , m / and A ). The sign of µ (theMSSM Higgs/higgsino mass parameter) makes for one additional discrete parameter; itsmagnitude is determined by requiring that SUSY-breaking induces electroweak symmetry-breaking. Despite greatly curbing the range of possible phenomenological consequences,the small number of parameters in the CMSSM has made it a tractable way to explorebasic low-energy SUSY phenomenology.Before drawing conclusions about model selection or parameter values from experimen-– 2 –al data, one must choose a statistical framework to work in. There are two very differentfundamental interpretations of probability, resulting in two approaches to statistics (fora detailed discussion, see e.g. Ref. 11). Frequentism deals with relative frequencies, in-terpreting probability as the fraction of times an outcome would occur if a measurementwere repeated an infinite number of times. Bayesianism deals with subjective probabili-ties assigned to different hypotheses, interpreting probability as a measure of the degreeof belief that a hypothesis is true. The former is used for assigning statistical errors tomeasurements, whilst the latter can be used to quantify both statistical and systematicuncertainties. In the Bayesian approach one is interested in the probability of a set ofmodel parameters given some data, whereas in the frequentist approach the only quantityone can discuss is the probability of some dataset given a specific set of model parameters,i.e. a likelihood function.In a frequentist framework, one simply maps a model’s likelihood as a function of themodel parameters. The point with the highest likelihood is the best fit, and uncertain-ties upon the parameter values can be given by e.g. iso-likelihood contours in the modelparameter space. To obtain joint confidence intervals on a subset of parameters, the fulllikelihood is reduced to a lower-dimensional function by maximising it along the unwanteddirections in the parameter space. This is the profile likelihood procedure [12, and refer-ences therein]. In the Bayesian picture [13], probabilities are directly assigned to differentvolumes in the parameter space. One must therefore also consider the state of subjectiveknowledge about the relative probabilities of different parameter values, independent ofthe actual data; this is a prior. In this case, the statistical measure is not the likelihooditself, but a prior-weighted likelihood known as the posterior probability density function(PDF). Because this posterior is nothing but the joint PDF of all the parameters, con-straints on a subset of model parameters are obtained by marginalising (i.e. integrating)it over the unwanted parameters. This marginalised posterior is then the Bayesian coun-terpart to the profile likelihood. Bayesians report the posterior mean as the most-favouredpoint (given by the expectation values of the parameters according to the marginalisedposterior), with uncertainties defined by surfaces containing set percentages of the totalmarginalised posterior, or ‘probability mass’.One practically interesting consequence of including priors is that they are a powerfultool for estimating how robust a fit is. If the posterior is strongly dependent on the prior,the data are not sufficient to constrain the model parameters. It has been shown thatthe prior still plays a large role in Bayesian inferences in the CMSSM [14]. If an actualdetection occurs at the LHC, this dependency should disappear [44].Clearly the results of frequentist and Bayesian inferences will not coincide in general.This is especially true if the model likelihood has a complex dependence on the parameters(i.e. not just a simple Gaussian form), and if insufficient experimental data is available.Note that this is true even if the prior is taken to be flat; a flat prior in one parameterbasis is certainly not flat in every such basis. In the case of large sample limits bothapproaches give similar results, because the likelihood and posterior PDF both becomealmost Gaussian; this is why both are commonly used in scientific data analysis.The first CMSSM parameter scans were performed on fixed grids in parameter space [15,– 3 –6]. Predictions of e.g. the relic density of the neutralino as a cold dark matter candidateor the Higgs/superpartner masses were computed for each point on the grid, and comparedwith experimental data. In these earliest papers, points for which the predicted quantitieswere within an arbitrary confidence level (e.g. 1 σ , 2 σ ) were deemed “good”. Because allaccepted points are considered equivalently good, this method provides no way to deter-mine points’ relative goodnesses-of-fit, and precludes any deeper statistical interpretationof results.The first attempts at statistical interpretation were to perform (frequentist) χ analyseswith grid scans [17]. Limited random scans were also done in some of these cases. Despitesome advantages of grid scans, their main drawback is that the number of points sampled inan N -dimensional space with k points for each parameter grows as k N , making the methodextremely inefficient. This is even true for spaces of moderate dimension like the CMSSM.The lack of efficiency is mainly due to the complexity and highly non-linear nature of themapping of the CMSSM parameters to physical observables; many important features ofthe parameter space can be missed by not using a high enough grid resolution.Another class of techniques that has become popular in SUSY analyses is based on moresophisticated scanning algorithms. These are techniques designed around the Bayesianrequirement that a probability surface be mapped in such a way that the density of theresultant points is proportional to the actual probability. However, the points they returncan also be used in frequentist analyses. Foremost amongst these techniques is the MarkovChain Monte Carlo (MCMC) method [18–37], which has also been widely used in otherbranches of science, in particular cosmological data analysis [38]. The MCMC methodprovides a greatly improved scanning efficiency in comparison to traditional grid searches,scaling as kN instead of k N for an N -dimensional parameter space. More recently, theframework of nested sampling [39] has come to prominence, particularly via the publicly-available implementation MultiNest [40]. A handful of recent papers have explored theCMSSM parameter space or its observables using this technique [14, 32, 41–46], as wellas the higher-dimensional spaces of the Constrained Next-to-MSSM (CNMSSM) [47] andphenomenological MSSM (pMSSM) [48].
MultiNest was also the technique of choice in thesupersymmetry-breaking study of Ref. 7. A CMSSM scan with
MultiNest takes roughly afactor of ∼
200 less computational effort than a full MCMC scan, whilst results obtainedwith both algorithms are identical (up to numerical noise) [14].Besides improved computational efficiency, MCMCs and nested sampling offer otherconvenient features for both frequentist and Bayesian analyses. In a fully-defined statisti-cal framework, all significant sources of uncertainty can be included, including theoreticaluncertainties and our imperfect knowledge of the relevant SM parameters. These can beintroduced as additional ‘nuisance’ parameters in the scans, and resultant profile likeli-hoods and posterior PDFs profiled/marginalised over them. In a similar way, one canprofile and marginalise over all parameters at once to make straightforward statisticalinferences about any arbitrary function of the model parameters, like neutralino annihila-tion fluxes [27, 32, 46] or cross-sections [45]. The profiling/marginalisation takes almostno additional computational effort: profiling simply requires finding the sample with thehighest likelihood in a list, whereas marginalisation, given the design of MCMCs and nested– 4 –ampling, merely requires tallying the number of samples in the list. Finally, it is straight-forward to take into account all priors when using these techniques for Bayesian analyses.Although the prior-dependence of Bayesian inference can be useful for determiningthe robustness of a fit, it may be considered undesirable when trying to draw concreteconclusions from the fitting procedure. This is because the prior is a subjective quantity,and most researchers intuitively prefer their conclusions not to depend on subjective as-sessments. In this case, the natural preference would be to rely on a profile likelihoodanalysis rather than one based on the posterior PDF. The question then becomes: howdoes one effectively and efficiently explore a parameter space like the CMSSM, with itsmany finely-tuned regions, in the context of the profile likelihood?As a first attempt to answer this question, the profile likelihood of the CMSSM wasrecently mapped with MCMCs [25] and
MultiNest [14]. Despite the improved efficiency ofthese Bayesian methods with respect to grid searches by several orders of magnitude, theyare not optimised to look for isolated points with large likelihoods. They are thus verylikely to entirely miss high-likelihood regions occupying very tiny volumes in the parameterspace. Such regions might have a strong impact on the final results of the profile likelihoodscan , since the profile likelihood is normalised to the best-fit point and places all regions,no matter how small, on an equal footing. It appears that in the case of the CMSSM thereare many such fine-tuned regions. This is seen in e.g. CMSSM profile likelihood maps withdifferent MultiNest scanning priors [14]. Given that the profile likelihood is independentof the prior by definition, these results demonstrate that many high-likelihood regions aremissed when using a scanning algorithm optimised for Bayesian statistics. In order to makevalid statistical inferences in the context of the profile likelihood, the first (and perhapsmost crucial) step is to correctly locate the best-fit points. Setting confidence limits anddescribing other statistical characteristics of the parameter space makes sense only if thisfirst step is performed correctly.If one wishes to work confidently in a frequentist framework, some alternative scanningmethod is clearly needed. The method should be optimised for calculating the profilelikelihood, rather than the Bayesian evidence or posterior. Even if the results obtainedwith such a method turn out to be consistent with those of MCMCs and nested sampling,the exercise would greatly increase the utility and trustworthiness of those techniques.In this paper, we employ a particular class of optimisation techniques known as GeneticAlgorithms (GAs) to scan the CMSSM parameter space, performing a global frequentistfit to current collider and cosmological data. GAs and other evolutionary algorithms havenot yet been widely used in high energy physics or cosmology; to our knowledge, theironly prior use in SUSY phenomenology [49] has been for purposes completely different toours . There are two main reasons GAs should perform well in profile likelihood scans.Firstly, the sole design purpose of GAs is to maximise or minimise a function. This isexactly what is required by the profile likelihood; in the absence of any need to marginalise(i.e. integrate) over dimensions in the parameter space, it is more important that a search Missing fine-tuned regions could even modify the posterior PDF if they are numerous or large enough. See e.g. Refs. 50–52 for their use in high energy physics, Ref. 53 for uses in cosmology and generalrelativity, and Ref. 54 for applications to nuclear physics. – 5 –lgorithm finds the best-fit peaks than accurately maps the likelihood surface at lowerelevations. Secondly, the ability of GAs to probe global extrema excels most clearly overother techniques when the parameter space is very large, complex or poorly understood;this is precisely the situation for SUSY models. We focus exclusively on the CMSSM asour test-bed model, but the algorithms could easily be employed in higher-dimensionalSUSY parameter spaces without considerable change in the scanning efficiency (as theyscale as kN for an N -dimensional parameter space). We compare our profile likelihoodresults directly with those of the MultiNest scanning algorithm. This means that we alsocompare indirectly with MCMC scans, because
MultiNest and MCMCs give essentiallyidentical results [14]. We find that GAs uncover many better-fit points than previous
MultiNest scans.This paper proceeds as follows: in Sec. 2 we briefly review the parameters of theCMSSM, the predicted observables, constraints and bounds on them from collider andcosmological observations. We then introduce GAs as our specific scanning technique ofchoice. We present and discuss results in Sec. 3, comparing those obtained with GAs tothose produced by
MultiNest . We include the best-fit points and the highest-likelihoodregions of the parameter space, as well as the implications for particle discovery at theLHC and in direct and indirect dark matter searches. We draw conclusions and commenton future prospects in Sec. 4.
2. Model and analysis
Our goal is to compare the results of a profile likelihood analysis of the CMSSM performedwith GAs to those obtained using Bayesian scanning techniques, in particular
MultiNest .For this reason, we work with the same set of free parameters and ranges, the same observ-ables (measurable physical quantities predicted by the model) and the same constraints(the observed values of observables, as well as physicality requirements) as in Ref. 14. Wealso perform the theoretical calculations and construct the full likelihood function of themodel based on these variables and data in the same way as in Ref. 14. We limit ourselvesto just a brief review of these quantities and constraints here. For a detailed discussion,the reader is referred to previous papers [14, 21, 24].
As pointed out in Sec. 1, there are four new continuous parameters m / , m , A and tan β ,which are the main free parameters of the model to be fit to the data. There is also a newdiscrete parameter, the sign of µ , which we fix to be positive. The choice of positive µ is motivated largely by constraints on the CMSSM from the anomalous magneticmoment of the muon δa SUSY µ . The branching fraction BR ( B → X s γ ) actually prefers negative µ (see, forexample, Ref. 41 and the references therein). µ > δa SUSY µ and experimental data. We employ the samefixed sign in the present work for consistency, although it is of course possible to leave this a free discreteparameter in any global fit. – 6 –e add four additional nuisance parameters to the set of free parameters in our scans.These are the SM parameters with the largest uncertainties and strongest impacts uponCMSSM predictions: m t , the pole top quark mass, m b ( m b ) MS , the bottom quark mass eval-uated at m b , α em ( m Z ) MS , the electromagnetic coupling constant evaluated at the Z -bosonpole mass m Z , and α s ( m Z ) MS , the strong coupling constant, also evaluated at m Z . Theselast three quantities are computed in the modified minimal subtraction renormalisationscheme M S .The set of CMSSM and SM parameters together constitute an 8-dimensional parameterspace Θ = ( m , m / , A , tan β, m t , m b ( m b ) MS , α em ( m Z ) MS , α s ( m Z ) MS ) , (2.1)to be scanned and constrained according to the available experimental data. The rangesover which we scan the parameters are m , m / ∈ (50 GeV , A ∈ ( − , β ∈ (2 , m t ∈ [167 . , . m b ( m b ) MS ∈ [3 .
92 GeV , .
48 GeV],1 /α em ( m Z ) MS ∈ [127 . , . α s ( m Z ) MS ∈ [0 . , . In order to compare the predictions of each point in the parameter space with data, onehas to first derive some quantities which are experimentally measurable. For a globalfit, one needs to take into account all existing (and upcoming) data, such as collider andcosmological observations, and direct and indirect dark matter detection experiments. Thisis indeed the ultimate goal in any attempt to constrain a specific theoretical model like theCMSSM. Because our main goal in this paper is to assess how powerful GAs are comparedto conventional methods (in particular the
MultiNest algorithm), we restrict our analysis tothe same set of observables and constraints as in the comparison paper [14]. These includethe collider limits on Higgs and superpartner masses, electroweak precision measurements, B -physics quantities and the cosmologically-measured abundance of dark matter. Thesequantities and constraints are given in Table 1.The observables are of three types: • The SM nuisance parameters. Although they are considered free parameters of themodel along with the CMSSM parameters, these are rather well-constrained by thedata, and therefore used in constructing the full likelihood function. • Observables for which positive measurements have been made. These are the W -boson pole mass ( m W ), the effective leptonic weak mixing angle (sin θ eff ), anomalousmagnetic moment of the muon ( δa SUSY µ ), branching fraction BR ( B → X s γ ), B s – B s mass difference (∆ M B s ), branching fraction BR ( B u → τ ν ), and dark matter relicdensity (Ω χ h ) assuming the neutralino is the only constituent of dark matter. • Observables for which at the moment only (upper or lower) limits exist, i.e. thebranching fraction BR ( B s → µ + µ − ), the lightest MSSM Higgs boson mass m h (as-suming its coupling to the Z -boson is SM-like), and the superpartner masses.Our sources for these experimental data are indicated in the table. Note that thereare no theoretical uncertainties associated with the SM parameters, because they are si-multaneously observables and free input parameters. For details about these quantities,– 7 – bservable Mean value Uncertainties Reference(standard deviations)experimental theoreticalSM nuisance parameters m t . . m b ( m b ) MS .
20 GeV 0 .
07 GeV - [56] α s ( m Z ) MS . .
002 - [56]1 /α em ( m Z ) MS .
955 0 .
03 - [57]measured m W .
398 GeV 25 MeV 15 MeV [58]sin θ eff . × − × − [58] δa SUSY µ × BR ( B → X s γ ) × M B s .
77 ps − .
12 ps − . − [61] BR ( B u → τ ν ) × .
32 0 .
49 0 .
38 [60]Ω χ h . χ h [62]limits only (95% CL) BR ( B s → µ + µ − ) < . × −
14% [63] m h > . ζ h f ( m h ) (see Ref. 21) negligible [64] m ˜ χ >
50 GeV 5% [65] m ˜ χ ± > . > . m ˜ e R >
100 GeV ( >
73 GeV) 5% [66] ([67, 68]) m ˜ µ R >
95 GeV ( >
73 GeV) 5% [66] ([67, 68]) m ˜ τ >
87 GeV ( >
73 GeV) 5% [66] ([67, 68]) m ˜ ν >
94 GeV ( >
43 GeV) 5% [69] ([67, 68]) m ˜ t >
95 GeV ( >
65 GeV) 5% [66] ([67, 68]) m ˜ b >
95 GeV ( >
59 GeV) 5% [66] ([67, 68]) m ˜ q >
375 GeV 5% [56] m ˜ g >
289 GeV 5% [56]
Table 1:
List of all physical observables used in the analysis. These are: collider limits on Higgs andsuperpartner masses, electroweak precision measurements, B -physics quantities and the dark matter relicdensity. For the sake of comparability, these are the same quantities and values as used in Ref. 14. Theupper sub-table gives measurements of SM nuisance parameters. The central panel consists of observablesfor which a positive measurement has been made, and the lower panel shows observables for which onlylimits exist at the moment. The numbers in parenthesis are more conservative bounds applicable only underspecific conditions. For details and arguments, see Refs. 14, 21 and 24. Table adapted mostly from Ref. 42. experimental values and errors, particularly the reasoning behind theoretical uncertainties,see Refs. 14, 21 and 24.In order to calculate all observables and likelihoods for different points in the CMSSMparameter space, we have used SuperBayeS 1.35 [70], a publicly available package whichcombines
SoftSusy [71],
DarkSusy [72],
FeynHiggs [73],
Bdecay and
MicrOMEGAs [74] in astatistically consistent way. The public version of the package offers three different scanningalgorithms: MCMCs,
MultiNest , and fixed-grid scanning. We have modified the code toalso include GAs. Other global-fit packages are also available:
Fittino [36], for MCMCscans of the CMSSM,
SFitter [28], for MCMC scans of the CMSSM and also weak-scale– 8 –SSM, and
Gfitter [75], for Standard Model fits to electroweak precision data (SUSY fitswill soon be included as well). Amongst other search strategies,
Gfitter can make use ofGAs.In
SuperBayeS , the likelihoods of observables for which positive measurements exist(indicated in the upper and central panels of Table 1), are modeled by a multi-dimensionalGaussian function. The variance of this Gaussian is given by the sum of the experimen-tal and theoretical variances associated with each observable; the corresponding standarddeviations are shown in Table 1. For observables where only upper or lower limits ex-ist (indicated in the lower panel of Table 1), a smeared step-function likelihood is used,constructed taking into account estimated theoretical errors in calculating the predictedvalues of the observables. For details on the exact mathematical forms of these likelihoodfunctions, see Ref. 21.In addition to experimental constraints from collider and cosmological observations,one must also ensure that each point is physically self-consistent; those that are not shouldbe discarded or assigned a very low likelihood value. Unphysical points are ones whereno self-consistent solutions to the RGEs exist, the conditions of electroweak symmetry-breaking are not satisfied, one or more masses become tachyonic, or the theoretical assump-tion that the neutralino is the LSP is violated. This is done in
SuperBayeS by assigningan extremely small (almost zero) likelihood to the points that do not fulfil the physicalityconditions.
GAs [76–78] are a class of adaptive heuristic search techniques that draw on the evolu-tionary ideas of natural selection and survival of the fittest to solve optimisation problems.According to these principles, individuals in a breeding population which are better adaptedto their environment generate more offspring than others.GAs were invented in early 1970s, primarily by John Holland and colleagues for solvingoptimisation problems [79], although the idea of evolutionary computing had been intro-duced as early as the 1950s. Holland also introduced a formal mathematical framework,known as
Holland’s Schema Theorem , which is commonly considered to be the theoreti-cal explanation for the success of GAs. Now, about thirty years after their invention, GAshave amply demonstrated their practical usefulness (and robustness) in a variety of complexoptimisation problems in computational science, economics, medicine and engineering [77].The idea is very simple: in general, solving a problem means nothing more thanfinding the one solution most compatible with the conditions of the problem amongst manycandidate solutions, in an efficient way. In most cases, the quality of different candidatesolutions can be formulated in terms of a mathematical ‘fitness function’, to be maximisedby some algorithm employed to solve the problem. With a GA, one repeatedly modifiesa population of individual candidate solutions in such a way that after several iterations,the fittest point in the population evolves towards an optimal solution to the problem.These iterative modifications are designed to imitate the reproductive behaviour ofliving organisms. At each stage, some individuals are selected randomly or semi-randomlyfrom the current population to be parents. These parents produce children, which become– 9 –he next generation of candidate solutions. If parents with larger fitness values have morechance to recombine and produce children, over successive generations the best individualin the population should approach an optimal solution.Because GAs are based only on fitness values at each point, and are insensitive to thefunction’s gradients, they can also be applied to problems for which the fitness functionhas many discontinuities, is stochastic, highly non-linear or non-differentiable, or possessesany other special features which make the optimisation process extremely difficult. GAsare generally prescribed when the traditional optimisation algorithms either fail entirely orgive substandard results. These properties make GAs ideal for our particular problem, asthe CMSSM has exactly those unruly properties.In what follows, we describe the general algorithmic strategy employed in a GA, fol-lowed by our particular implementation for a profile likelihood analysis of the CMSSM.
Denote the full model likelihood by L (Θ), where Θ is the set of free parameters introducedin Eq. 2.1. This function, as a natural proxy for the goodness-of-fit given a fixed numberof degrees of freedom, indicates how fit each particular Θ, or individual, is. It is thus agood choice for the genetic fitness function. We now want to find a specific individual, sayΘ max , for which the fitness function L (Θ) is globally maximised.Consider a population of I candidate individuals Θ i ( i = 1 , ..., I ). Denote this entirepopulation by P . This population is operated on K times by a semi-random geneticoperator G to produce a series of new populations P k , where ( k = 1 , ..., K ) and P k = G P k − . The i th individual in the k th generation is Θ ki . For the general fitness of individualsto improve from one generation to the next, G must clearly depend upon L (Θ).The search must be initialised with some starting population P , which is then evolvedunder the action of G until some convergence criterion T is met. At this stage, the algo-rithm returns its best estimate of Θ max as the individual Θ where L (Θ ki ) is maximised.If G and T have been chosen appropriately, this should occur at k = K , i.e. in the lastgeneration. This algorithm can be summarised as follows: initialisation: P := { Θ i } , ∀ i ∈ [1 , I ] k := 0 reproduction loop: do while not T k := k + 1 generating new population through genetic operators: P k := G P k − end do reading the best-fit point: Θ max := Θ ml where L (Θ ml ) = max {L (Θ ki ) } , ∀ i ∈ [1 , I ] , ∀ k ∈ [1 , K ] Three main properties define a GA. Firstly, G operates on a population of points ratherthan a single individual. This makes GAs rather different from conventional Monte Carlo– 10 –earch techniques such as the MCMC, though the nested sampling algorithm does also acton a population of points. The parallelism of a GA means that if appropriate measuresof population diversity are incorporated into the algorithm, local maxima in the likelihoodsurface can be dealt with quite effectively; if a population is required to maintain a certainlevel of diversity, concentrations of individuals clustering too strongly around local maximawill be avoided by the remaining members of the population. This parallelism increasesthe convergence rate of the algorithm remarkably.Secondly, G does not operate directly on the real values of the parameters Θ i (thephenotype), but acts on their encoded versions (the chromosomes, or genotype). Dependingon the problem, individuals can be encoded as a string of binary or decimal digits, or evenmore complex data structures. G then acts on the chromosomes in the current generationbased only on their fitnesses, i.e. the likelihood function in our case. No further informationis required for the GA to work; this means discontinuities in the likelihood function or itsderivatives have virtually no affect on the performance of the algorithm.Finally, the transition rules used in G are probabilistic, not deterministic. The con-stituent genetic operators contained within G are the key elements of the algorithm, andhow they act on different populations defines different types of GAs. In our case G = RMCS , (2.2)where S is the selection procedure (how parents are selected for breeding from a source pop-ulation), C is the crossover (how offspring are to inherit properties from their parents), M isthe mutation process (random changes made to the properties of newly-created offspring),and R is the reproduction scheme used to place offspring into a broader population. C and M are stochastic processes defined at the genotype level, whereas S and R are phenotype-level processes, semi-random and essentially deterministic in nature, respectively.The randomised operators of GAs are strongly distinct from simple random walks.This is because every new generation of individuals inherits some desirable characteristicsfrom the present generation. Crossover (or recombination) rules play a crucial role indetermining how the parents create the children of the next generation. The children arenot copied directly to the next population; mutation rules specify that a certain degreeof random modification should be applied to the newly-produced offsprings’ chromosomes.Mutation is very important at this stage, since it is often the only mechanism preventing thealgorithm from getting stuck in local maxima; its strength is typically linked dynamicallyto some measure of population diversity.The reproduction loop is terminated whenever T is fulfilled. In our case, whenever apredefined number of generations N have been produced (i.e. K ≡ N ). The terminationparameter (i.e. the number of generations N ) and the chosen termination criterion itselfdepend upon the particular problem at hand and how accurate a solution is required. Thefittest individual in the final population is then accepted as the best-fit solution to theproblem. If one is also interested in mapping the likelihood function in the vicinity ofthe best-fit points, so as to be able to plot e.g. 1 and 2 σ confidence regions for differentCMSSM parameters, then it is useful to also retain all the individuals produced during theiterations of the algorithm. – 11 – .2.2 Our specific implementation Although any algorithm with the basic features described above ensures progressive im-provement over successive generations, to guarantee absolute maximisation one usuallyneeds to implement additional strategies and techniques, depending on the particular prob-lem at hand. To implement a GA for analysing the CMSSM parameter space, we havetaken advantage of the public GA package
PIKAIA [80, 81]. Here we briefly describe theoptions and ingredients from
PIKAIA 1.2 that we used in our GA implementation; themajority of these were the default choices. • Fitness function:
A natural fitness function to choose is the log-likelihood ofthe model, ln L (Θ). This function is however always negative (or zero). Since PIKAIA internally seeks to maximise a function, and this function must be positive definite, wechose the inverse chi-square (i.e. χ ) as the simplest appropriate fitness function. Exceptfor the way we adjust the mutation rate for different generational iterations (see below), allthe other genetic operators implemented in our algorithm are functions only of the rankingof individuals in the population; the actual values of the fitness function at these points donot matter as long as the ranking is preserved. • Encoding:
We encode individuals in the population (i.e. points in the CMSSMparameter space) using a decimal alphabet. That is, a string of base 10 integers, suchthat every normalised parameter θ i is encoded into a string d d ...d n d , where the d i ∈ [0 , m × n d = 8 × • Initialisation and population size:
We use completely random points in theparameter space for the initial population. We choose a population size of n p = 100 (thetypical number usually used in GAs), keeping it fixed throughout the entire evolutionaryprocess. • Selection:
In order to select parents in any given iteration, a stochastic mechanismis used. The probability of an individual to be selected for breeding is determined basedon its fitness in the following way: first, we assign to each individual Θ i a rank r i based onits fitness f i such that r = 1 corresponds to the fittest individual and r = n p to the mostunfit. Then a ranking fitness f (cid:48) i is defined in terms of this rank: f (cid:48) i = n p − r i + 1 . The sum of all ranking fitness values in the population is computed as F = n p (cid:88) i =1 f (cid:48) i , and n p running sums are defined as S j = j (cid:88) i =1 f (cid:48) i , j = 1 , ..., n p . – 12 –bviously S j +1 ≥ S j (since f (cid:48) i are all positive), and S n p = F . As the next step, a randomnumber R ∈ [0 , F ] is generated and the element S j is located for which S j − ≤ R < S j .The corresponding individual is one of the parents selected for breeding; the other oneis also chosen in the same manner. This selection procedure is called the Roulette WheelAlgorithm (see Ref. 80 and references therein for more on this procedure and the motivationfor using the ranking fitness in place of the true fitness). • Crossover:
When a pair of parent chromosomes have been chosen, a pair of offspringare produced via the crossover operator C . We use two different types of operators for thispurpose, namely, uniform one-point and two-point crossovers (see Ref. 82 for a compre-hensive review of existing crossover operators). Table 2 illustrates how these two processeswork. In our case, parents are encoded as 40-digit strings. The one-point crossover beginsby randomly selecting a cutting point along the chromosomes’ length, and dividing eachparent string into two sub-strings. This is done by generating a random integer K ∈ [1 , K , K ∈ [1 , uniform one-point crossoverinitial parent chromosomes selecting a random cutting point |
51 4394...05 | swapping the sub-strings |
70 4394...05 | final offspring uniform two-point crossoverinitial parent chromosomes selecting two random cutting points | | | | swapping the sub-strings | | | | final offspring Table 2:
Schematic description of the uniform one-point and two-point crossover operators employed inour analysis.
In our algorithm, for each pair of parent chromosomes, either one-point or two-pointcrossover is chosen with equal probability. This combination of the one-point and two-pointcrossovers is proposed to avoid the so-called “end-point bias” problem produced by usingonly the one-point crossover. This happens when, for example, a combination of two sub-strings situated at opposite ends of a parent string are advantageous when decoded backto the phenotypic level (in the sense that they give a high fitness), but only when they areexpressed simultaneously . Such a feature is impossible to pass on to offspring when using auniform one-point crossover, as cutting the string at any single point always destroys thiscombination. This is much less of a problem for sub-strings located more centrally in theparent string (see Ref. 80 again for more details on this issue).– 13 –nce two parents have been selected for breeding, the crossover operation is appliedonly with a preset probability. This probability is usually taken to be large but not 100%.We use 85% in our analysis, meaning that there is a 15% chance that any given breedingpair will be passed on intact to the next stage, where they will be affected by mutation. • Mutation:
We employ the so-called uniform one-point mutation operator. Differ-ent genes in the offspring’s chromosomes (i.e. decimal digits in the 40-digit strings) arereplaced with a predefined probability (the ‘mutation rate’), by a random integer in theinterval [0 , a priori . If too large, it can destroy a potentially excellent offspring and,in the extreme case, make the whole algorithm behave effectively as an entirely randomsearch. If too small, it can endanger the variability in the population and cause the wholepopulation to become trapped in local maxima; a large enough mutation rate is often theonly mechanism to escape premature convergence at local maxima. Therefore, instead ofusing a fixed mutation rate we allow it to vary dynamically throughout the run, such thatthe degree of ‘biodiversity’ is monitored and the mutation rate is adjusted accordingly.When the majority of individuals in a population (as estimated by the median individual)are very similar to the best individual, the population is probably clustered around a localmaximum and the mutation rate should increase. The converse is also true: a high degreeof diversity indicates that the mutation rate should be kept low. The degree of clusteringand the subsequent amount of adjustment in our algorithm are assessed based on the dif-ference between the actual fitness values of the best and median points. This is done bydefining the quantity ∆ f = ( f r =1 − f r = n p / ) / ( f r =1 + f r = n p / ) in which f r =1 and f r = n p / correspond to the fitnesses of the best and median individuals, respectively. The mutationrate is then increased (decreased) by a fixed multiplicative factor whenever ∆ f is smaller(larger) than a preset lower (upper) bound. We choose the lower and upper critical valuesof ∆ f to be 0 .
05 and 0 .
25 respectively, and the multiplicative factor to be 1 .
5. We boundthe mutation rate to lie between the typical values of 0 . .
25. We use an initialmutation rate of 0 . • Reproduction plans:
After selecting parents and producing offspring by acting onthem with the crossover and mutation operators, the newly bred individuals must somehowbe incorporated into the population. The strategy which controls this process is called areproduction plan. Although there are several advanced reproduction plans on the market,we utilise the simplest off-the-shelf option: full generational replacement. This means thatthe whole parent population is replaced by the newly-created children in each iteration, allin a single step. • Elitism:
There is always a possibility that the fittest individual is not passed onto the next generation, since it may be destroyed under the action of the crossover andmutation operations on the parents. To guarantee survival of this individual, we usean elitism feature in our reproduction plan. Under full generational replacement, elitismsimply means that the fittest individual in the offspring population is replaced by the fittestparent, if the latter has a higher fitness value. • Termination and number of generations:
There are several termination criteria– 14 –ne could use for a GA. Rather than evolving the population generation after generationuntil some tolerance criterion is met, we perform the evolution over a fixed and prede-termined number of generations. This is because the former strategy is claimed to bepotentially dangerous when approaching a new problem, in view of the usual convergencetrends exhibited by GA-based optimisers (see Ref. 80 for more details). In our analysis,we used 10 separate runs of the algorithm with different initial random seeds, and a fixednumber of likelihood evaluations ( ∼ × ) for each. We then compiled all points into asingle list, and used it to map the profile likelihood of the CMSSM.
3. Results and discussion
We now present and analyse the results of our global profile likelihood fit of the CMSSMusing GAs. We compare results with a similar global fit using the state-of-the-art Bayesianalgorithm
MultiNest [40]. In Sec. 3.1 we give the best-fit points and high-likelihood regionsin the CMSSM parameter space. In Sec. 3.2 we discuss and compare implications of bothmethods for the detection of supersymmetric and Higgs particles at the LHC. Sec. 3.3 isdevoted to an investigation of dark matter in the CMSSM and the prospects for its directand indirect detection. Throughout this section we compare our GA profile likelihoodresults mainly with those of the
MultiNest algorithm implemented with linear (flat) priors.The reasons for this are outlined in Sec. 3.4, along with a comparison of the two scanningtechniques in terms of the computational speed and convergence.
The χ at our best-fit point is 9 .
35. This is surprisingly better than the values of 13 . .
90 found by
MultiNest with linear and logarithmic priors, respectively, for the sameproblem. This improvement in the best fit can in principle have a drastic impact on thestatistical inference drawn about the model parameters.To demonstrate these effects, let us start with two-dimensional (2D) plots for theprincipal parameters of the CMSSM, i.e. m , m / , A and tan β , shown in Figs. 1 and2. These figures show 2D profile likelihood maps. In the first figure, the full likelihoodis maximised over all free (CMSSM plus SM nuisance) parameters except m and m / .Similar diagrams are shown in Fig. 2, but now for the 2D profile likelihoods in terms ofthe parameters A and tan β .Figs. 1a and 2a show the 2D profile likelihood maps obtained by taking into account allthe sample points in the parameter space resulting from the GA scan. The inner and outercontours indicate 68 .
3% (1 σ ) and 95 .
4% (2 σ ) confidence regions based on the GA best-fitpoint of χ = 9 .
35. That is, points with χ ≤ .
65 fall into the 1 σ region, and points with χ ≤ .
52 fall into the 2 σ region. Similar plots are presented for the MultiNest results inFigs. 1d and 2d, where the 1 σ and 2 σ contours are drawn based on the MultiNest best-fit of χ = 13 .
51 (1 and 2 σ regions are given by χ ≤ .
81 and χ ≤ .
68, respectively). Thesepanels reflect the outcomes of each scanning algorithm in the absence of any informationfrom the other. The sample points have been divided into 75 ×
75 bins in all plots and nosmoothing is applied. – 15 – a) Akrami, Scott, Edsjö, Conrad & Bergström (2010)
GA points + GA levels ) G e V ( m )GeV(m (b) Akrami, Scott, Edsjö, Conrad & Bergström (2010)
MN points + GA levels ) G e V ( m )GeV(m (c) Akrami, Scott, Edsjö, Conrad & Bergström (2010)
GA points + MN levels ) G e V ( m )GeV(m (d) Akrami, Scott, Edsjö, Conrad & Bergström (2010)
MN points + MN levels ) G e V ( m )GeV(m Figure 1:
Two-dimensional profile likelihoods in the m - m / plane for CMSSM scans with GAs (a) and MultiNest (d). These two panels show the statistically-consistent results of each scan. The inner and outercontours represent 68 .
3% (1 σ ) and 95 .
4% (2 σ ) confidence regions respectively, for each scan. The dottedcircle, square and triangle show respectively the GA global best-fit point with a χ of 9 .
35 (located in thefocus point region), the GA best-fit point in the stau co-annihilation region ( χ = 11 . MultiNest ( χ = 13 . MultiNest sample points as in (d), but with iso-likelihood contour levelsdrawn as in (a), based on the GA best-fit likelihood value. Panel (c) shows the same GA sample pointsas in (a), but with iso-likelihood contours as in (d), based on the
MultiNest best-fit likelihood value. Thesample points have been divided into 75 ×
75 bins in all plots. Here we see that the GA uncovers a largenumber of points with much higher likelihoods than
MultiNest , across large sections of the m - m / plane. It is important to realise that although both the 1 and 2 σ confidence regions in the GAresults seem to be rather smaller in size than the corresponding regions in the MultiNest scan(especially the 1 σ region), this is by no means an indication that MultiNest has found morehigh-likelihood points in the parameter space. The situation is in fact the exact opposite.– 16 – a) Akrami, Scott, Edsjö, Conrad & Bergström (2010)
GA points + GA levels (cid:2) tan ) G e V ( A (b) Akrami, Scott, Edsjö, Conrad & Bergström (2010)
MN points + GA levels (cid:2) tan ) G e V ( A (c) Akrami, Scott, Edsjö, Conrad & Bergström (2010)
GA points + MN levels (cid:2) tan ) G e V ( A (d) Akrami, Scott, Edsjö, Conrad & Bergström (2010)
MN points + MN levels (cid:2) tan ) G e V ( A Figure 2:
As in Fig. 1, but in the A -tan β plane. This is clear when we recall that the best-fit likelihood values are very different in thetwo cases giving rise to two completely different sets of iso-likelihood contours. To makethis point clear, suppose for the sake of argument that the GA best-fit likelihood value isindeed the absolute global maximum we were looking for. If so, we can now check howwell the
MultiNest algorithm has sampled the parameter space, by looking at Figs. 1b and2b. These show how many of the
MultiNest samples are located in the correct confidenceregions set by the absolute maximum; the contours are drawn based on this best-fit valuerather than the one found by
MultiNest itself. The plots show that
MultiNest has discoveredno points in the 1 σ region and only a small fraction of the 2 σ region. In particular, it isinteresting to notice that the MultiNest best-fit point, i.e. the one with χ = 13 .
51 (markedas dotted triangles in Figs. 1b,d and 2b,d) now sits in the 2 σ region. These all come fromthe fact that only points with χ ≤ .
65 and χ ≤ .
52 fall in the 1 σ and 2 σ regions,– 17 –espectively, and there are not many points found by MultiNest with such low χ s. Thesame statement holds for the log-prior MultiNest best-fit point with χ = 11 . χ ) which were missed (or skipped)by MultiNest . This indicates that the use of
MultiNest scans in the context of the frequentistprofile likelihood is rather questionable. This is not really surprising, given that
MultiNest isdesigned to sample the Bayesian posterior PDF, not map the profile likelihood.We can also use the resultant GA samples in a different way to clarify this result. InFigs. 1c and 2c, the GA samples are plotted with the same contours as in Figs. 1d and2d, i.e. based on the
MultiNest best-fit likelihood value. Compared to 1d and 2d, we seethat there are many high-likelihood points in the
MultiNest σ region found by the GA andmissed by MultiNest . In the sense of the profile likelihood, it appears that
MultiNest hasconverged prematurely; we see a much larger and more uniform pseudo-1 σ region with theGA data. Here we see that most of the region labeled as being within the 2 σ confidencelevel in the MultiNest scan is actually part of its 1 σ confidence region.Our results confirm the complexity of the CMSSM parameter space, showing thatmuch care should be taken in making any statistical statement about it. This is especiallytrue when using a frequentist approach, as this complication plays a crucial role in thefinal conclusions. It is of course true that the convergence criterion for MultiNest is definedon the basis of the Bayesian evidence, and the algorithm may have (indeed, probablyhas) converged properly in this context. The point we want to emphasise is that even if
MultiNest is converged for a Bayesian posterior PDF analysis of the model, this convergenceis far from acceptable for a profile likelihood analysis. The same is also very likely to betrue of other less sophisticated Bayesian methods, such as the MCMC; this is the case atleast for MCMC scans performed with the same physics and likelihood calculations as inour analysis (since MCMCs and
MultiNest give almost identical results in this case [14]).The aforementioned comparison does also suggest, however, that even the Bayesianposterior PDF obtained from
MultiNest and MCMC scans might not yet be quite properlymapped. This is because in principle, a significant amount of probability mass could becontained in the regions found by the GA but missed or skipped by
MultiNest . Given theabsence of any definition of a measure on the parameter space in profile likelihood analysessuch as the one we perform, we are unfortunately not in a position to make any conclusivestatement about the actual contribution of these regions to the Bayesian probability mass.Nevertheless, the difference in size between the blue regions in Figs. 1c and 1d is intriguing.We discuss these convergence questions further in Sec. 3.4.Our GA scan has found high-likelihood points in many of the CMSSM regions knownto be consistent with data, in particular the relic abundance of dark matter [83]. Theseinclude the stau (˜ τ ) co-annihilation (COA) region [84] usually at small m where thelightest stau is close in mass to the neutralino, the focus point (FP) region [85] at large m where a large Higgsino component causes neutralino annihilation into gauge bosonpairs, and the light Higgs boson funnel region [15, 86] at small m / . We have not foundany high-likelihood points in the stop (˜ t ) co-annihilation region [87–89] at large negative A , where the lightest stop is close in mass to the neutralino. This could be interpreted as– 18 –onfirmation of the claim that this region, although compatible with the WMAP constrainton the relic density of dark matter, is highly disfavoured when other observables are alsotaken into account [19].It is important to make the point that although our method does find some points inthe funnel region, it does not spread out very well around those points to map the wholeregion. Finding this very fine-tuned region is a known challenge for any scanning strategy,including nested sampling. The failure of the GA to map other points in the funnel regioncan be understood. We believe that this behaviour is caused by the specific crossoverscheme employed in our analysis (i.e. one/two-point crossover), and could probably becured by using a more advanced algorithm. Alternatively, a different parameterisation ofthe model, such as a logarithmic scaling of the mass parameters (or equivalently, geneticencoding in terms of the logarithms of these parameters), would probably find the funnelregion much more effectively (in the same way as it does when MultiNest is implementedwith logarithmic priors). In any case, it is important to realise that these types of regionsare findable by our method (although not very well), even without taking into account anyad hoc change in the model parameterisation (or choosing a non-linear prior such as thelogarithmic one in the Bayesian language). See Sec. 3.4 for more discussions about thepriors and parameterisation.Returning to the best-fit points, it is visible from the plots that the global best-fitpoint is located in the FP region (dotted circles in Figs. 1a,c and 2a,c). This has a veryinteresting phenomenological implication, which we return to later in this section when wediscuss contributions to the total likelihood of the best-fit model from different observables.For comparison, we have also marked the best-fit point located in the COA region, whichhas χ = 11 .
34 (dotted squares in Figs. 1a,c and 2a,c). This point is situated inside the1 σ confidence level contour (Figs. 1a and 2a) and is well-favoured by our analysis. It isinteresting to notice that the χ for this point, although worse than the global best-fit χ ,is still better than the best value found by the MultiNest scan, even when implemented witha log prior ( χ = 11 . MultiNest scans with logarithmic priors are usually considered a good wayto probe low-mass regions such as the COA. Our algorithm, even working with effectivelylinear priors (because the genome featured a linear encoding to the parameters), appearsto have found a better point in this region as well.As another exhibition of the consequences of our results compared to the Bayesiannested sampling technique, it is interesting to look at the 1D profile likelihoods for theCMSSM parameters (Fig. 3). The horizontal axes in the plots indicate the CMSSM pa-rameters and the vertical axes show the corresponding profile likelihoods, normalised tothe best-fit GA value (i.e. with χ = 9 . Multi-Nest
1D profile likelihoods respectively. We sorted points into 50 bins for these plots. Wehave also included the global (FP) and COA best-fit points in the plots, indicated by solidand dashed red lines, respectively.The very different results in Fig. 3 from the two different algorithms are yet anotherconfirmation that existing sampling techniques are probably still not sufficiently reliablefor the exploration and mapping of SUSY likelihoods, at least in a frequentist framework.– 19 – krami, Scott, Edsjö, Conrad & Bergström (2010) (cid:2) )GeV(m m ax P L / P L Akrami, Scott, Edsjö, Conrad & Bergström (2010) )GeV(m m ax P L / P L Akrami, Scott, Edsjö, Conrad & Bergström (2010) )GeV(A m ax P L / P L Akrami, Scott, Edsjö, Conrad & Bergström (2010) (cid:2) tan m ax P L / P L Figure 3:
One-dimensional profile likelihoods (
P L ) of CMSSM parameters, normalised to the global GAbest fit (
P L max ). The green and grey bars show results from GA and
MultiNest scans, respectively. Solidand dashed red lines represent the GA global best-fit point ( χ = 9 .
35, located in the focus point region)and the GA best-fit point in the stau co-annihilation region ( χ = 11 . These plots show that by employing a different scanning algorithm, it is quite possible tofind many important new points in the parameter space. This can in principle affect thewhole inference about the model, especially when we are interested not only in drawing ageneral statement about the high-likelihood regions, but also in performing a more detailedexploration of the model likelihood around the best-fit points. It also shows that the GAtechnology we made use of in this paper seems a better choice for frequentist analyses thanconventional tools, which are typically optimised for Bayesian searches; we have found– 20 – odel (+nuisance) parametersGA global BFP (located in FP region) GA COA BFP m . . m / . . A . . β . . m t . . m b ( m b ) MS .
19 GeV 4 .
20 GeV α s ( m Z ) MS . . /α em ( m Z ) MS .
955 127 . m W .
366 GeV 80 .
371 GeVsin θ eff . . δa SUSY µ × BR ( B → X s γ ) × M B s .
37 ps − . − BR ( B u → τ ν ) × .
32 1 . χ h BR ( B s → µ + µ − ) 4 . × − . × − Table 3:
Parameter and observable values at the best-fit points (BFPs) found using Genetic Algorithms.These quantities are shown for both the global best-fit point (located in the focus point (FP) region) andthe best-fit point in the stau co-annihilation (COA) region. Higgs and sparticle masses will be given inTable 5, when talking about implications for the LHC. higher likelihood values for almost all the regions within the interesting range of modelparameters. Whilst this certainly favours this technique over others, it should also serveas a warning. We can by no means be sure that the GA has actually found the trueglobal best-fit point. Clearly this concern should be taken much more seriously when oneis dealing with more complicated models than the CMSSM, with more parameters andmore complex theoretical structures.Listed in Tables 3 and 4 are all properties of the two GA best-fit points. The upperpart of the first table gives values of the CMSSM principal parameters and SM nuisanceparameters, whereas the lower part shows all physical observables employed in our calcu-lation of model likelihood. These are the same quantities as given in Table 1 except forthe Higgs and sparticle masses, which will be presented in the upcoming section on impli-cations for the LHC. To make the differences between the properties of these two “good”points more clear, in the second table we have indicated the individual contributions fromdifferent observables to the total χ at each point. These quantities are also given for MultiNest best-fit points found using flat and logarithmic priors.One interesting fact seen in Table 4 is the apparent tension between δa SUSY µ and theother observables, in particular BR ( B → X s γ ). This has been widely discussed in thepast [14, 31, 37, 42]. While most of the discrepancy between the model and the experimentaldata at our global best-fit point (living in the FP region) comes from δa SUSY µ ( ∼ BR ( B → X s γ ) contributes only about 0 .
1% to the total χ , these two observables partiallyswitch roles at the COA point. This confirms that BR ( B → X s γ ) in general favours large– 21 – artial χ (fractional contribution to the total χ in %)observable GA global BFP GA COA BFP MN global BFP MN global BFPlocated in FP region with flat priors with log priorsnuisance parameters 0 .
12 (1 . .
35 (3 . .
48 (3 . .
81 (6 . m W .
21 (12 . .
83 (7 . .
48 (10 . .
69 (5 . θ eff .
024 (0 . ∼ − (0 . .
07 (0 . . . δa SUSY µ .
09 (75 . .
86 (25 . .
21 (68 . .
40 (20 . BR ( B → X s γ ) 0 .
010 (0 . .
03 (26 . .
10 (0 . .
83 (32 . M B s .
028 (0 . .
26 (2 . .
09 (0 . .
29 (2 . BR ( B u → τ ν ) ∼ − (10 − %) 0 .
050 (0 . .
91 (14 . .
043 (0 . χ h . . ∼ − (10 − %) 0 .
03 (0 . .
13 (1 . BR ( B s → µ + µ − ) 0 .
016 (0 . .
00 (0 . .
00 (0 . .
00 (0 . m h .
85 (9 . .
96 (34 . .
15 (1 . .
70 (31 . .
00 (0 . .
00 (0 . .
00 (0 . .
00 (0 . all 9.35 (100 %) 11.34 (100 %) 13.51 (100 %) 11.90 (100 %) Table 4:
Contributions to the total χ by different observables employed in the scans (Table 1). Con-tributions are shown for the GA global best-fit point (BFP) located in the focus point (FP) region andthe GA best-fit point in the stau co-annihilation (COA) region. Fractional contributions are also given inpercent. Similar quantities for both MultiNest scans with flat (linear) and logarithmic priors are also listedfor comparison, where the former is in the FP region and the latter is in the COA region. m (the FP region), while δa SUSY µ favours smaller masses (the COA region). A similarfeature is also visible in the two MultiNest best-fit points for flat and log priors, as theyreside in the FP and COA regions, respectively.In the case where our best-fit point is placed in the FP region, the total χ from allobservables except δa SUSY µ and BR ( B → X s γ ) is a remarkably smaller fraction ( ∼ ∼ MultiNest points in the table, with contributions of ∼
31% and ∼ . δa SUSY µ , the data is largely consistent with the global best-fit point being inthe FP region. That is, if one ignores δa SUSY µ , it is much easier to fulfil all the experimentalconstraints on the CMSSM by moving towards larger m . If one wants to satisfy also theextra constraint coming from δa SUSY µ , this might be possible by moving back towards lowermasses (the COA region), but at the price of reducing the total likelihood.It is important to stress that our global best-fit point is in fact part of the FP region,with high m (i.e. ∼ δa SUSY µ , the FP is still favoured over the COA region in our analysis, in clear con-tradiction with some recent claims [31, 37] that the latter is favoured by existing data.These analyses were performed in a frequentist framework, and based on MCMC scans.However, the large discrepancy with our findings probably comes more from differences inthe likelihood functions themselves than the scanning algorithms, i.e. in the calculationsof physical observables and their contributions to the likelihood. A direct comparison withthese works would only be possible if we were to also work with exactly the same routinesfor the calculation of the likelihood as in Refs. 31 and 37, changing only the scanningalgorithm (as we have here in comparing with Ref. 14). The difference we see in this case– 22 –ostly reflects the discrepancy between the results of Ref. 14 and Refs. 31 and 37.Nontheless, it is important to note that some differences could be due to the scanningtechnique. We have shown in this paper that at least for the specific physics setup imple-mented in SuperBayeS , GAs find better-fit points than nested sampling, which in turn isknown to find essentially the same points as MCMCs. It is therefore quite reasonable toexpect that GAs could find many points missed by MCMCs. There are even some otherFP points found by GAs with masses of about 2800 GeV and located in the 1 σ region(see Fig. 1a), supporting the conclusion that although low masses are favoured over highmasses in the previous MultiNest and MCMC scans using
SuperBayeS , the opposite holdsin our GA scans. This means that there exist many high-likelihood points in the FP regionentirely missed by
MultiNest and MCMC scans performed in the
SuperBayeS analyses. Itseems that those algorithms do not sample this region of the parameter space very well,at least when
SuperBayeS routines are used for physics and likelihood calculations. We seeno reason why a similar situation could not also occur when different codes are used toevaluate the likelihood function.Since we have not used exactly the same physics and likelihood setup, nor the samenumerical routines for calculating different quantities as employed in Refs. 31 and 37 (andwe cannot do that in a consistent way as the code employed in those studies is not publiclyavailable), we cannot make a definitive statement as to the overall impact of the scan-ning algorithm in the discrepancy we see with their results. One should however be verycautious in general when attempting to draw strong conclusions about e.g. the FP beingexcluded by existing data. The complex structure of the CMSSM parameter space makesthe corresponding likelihood surface very sensitive to small changes in the codes and exper-imental data used to construct the full likelihood, which in turn can introduce a significantdependence upon the scanning algorithm.
We showed in the previous section that compared to the state-of-the-art Bayesian algorithm
MultiNest , GAs are a very powerful tool for finding high-likelihood points in the CMSSMparameter space. It is therefore interesting to examine how strongly these results impactpredictions for future experimental tests at e.g. the LHC. We have calculated the 1Dprofile likelihoods corresponding to the gluino mass m ˜ g (as a popular representative of thesparticles) and the lightest Higgs mass m h , both of which will be searched for at the LHC.The resultant plots are given in Fig. 4. These plots are generated in the same way as thosein Fig. 3, indicating the differences between the two scanning strategies. Here, we onceagain see that the GA has found much better fits in the mass ranges covered by MultiNest .Looking first at the gluino mass prediction (left-hand plot of Fig. 4), we confirm earlierfindings [14, 37] that the LHC will probe all likely CMSSM gluino masses if its reach extendsbeyond ∼ ∼
900 GeV). These values are well within the reach of even theearly operations of the LHC. The detailed CMSSM mass spectra computed at both of thesepoints are presented in Table 5. It can be seen from these spectra that our global best-fit– 23 – krami, Scott, Edsjö, Conrad & Bergström (2010) )GeV(m g~ m ax P L / P L Akrami, Scott, Edsjö, Conrad & Bergström (2010) )GeV(m h m ax P L / P L Figure 4:
As in Fig. 3, but for the gluino mass m ˜ g and the lightest Higgs mass m h .GA global BFP GA COA BFP GA global BFP GA COA BFP(GeV) located in FP region (GeV) located in FP region m ˜ e L m ˜ d R m ˜ e R m ˜ s L m ˜ µ L m ˜ s R m ˜ µ R m ˜ b m ˜ τ m ˜ b m ˜ τ m ˜ χ m ˜ ν e m ˜ χ m ˜ ν µ m ˜ χ m ˜ ν τ m ˜ χ m ˜ u L m ˜ χ ± m ˜ u R m ˜ χ ± m ˜ c L m h m ˜ c R m H m ˜ t m A m ˜ t m H ± m ˜ d L m ˜ g Table 5:
Mass spectra of the GA global best-fit point (BFP) located in the focus point (FP) region andthe GA best-fit point in the stau co-annihilation (COA) region. point favours rather high masses for the sfermions, while the COA best-fit point favourslow masses.Looking at the right-hand plot of Fig. 4, corresponding to the likelihood of differentvalues of m h , we notice that although a large number of good points have Higgs masseshigher than the SM limit from the Large Electron-Positron Collider (LEP; i.e. m h ≥ . m h = 115 .
55 GeV), there are also– 24 –any other important ones which violate this limit, including the best-fit COA point (with m h = 111 .
11 GeV). These points with low-mass Higgs bosons have been allowed by thesmoothed likelihood function that we employed for the LEP limit (cf. Sec. 2.1). Insteadof this treatment of the Higgs sector, one could use a more sophisticated method, such asimplemented by
HiggsBounds [90]. This would apply the collider bounds on the Higgs massin a SUSY-appropriate manner, and give more accurate likelihoods at low masses aroundthe 114 . m h as a percentage of the total χ isconsiderably larger in the COA case than the FP. This becomes clear when we compare theircorresponding values for m h (i.e. 115 .
55 GeV and 111 .
11 GeV, respectively) with the LEPlimit. The lower Higgs mass in the COA region is a reflection of the correlation between m and m h in the CMSSM, confirming once more that moving to low m (i.e. approaching theCOA region) causes models to become less compatible with all experimental data except δa SUSY µ . As a natural continuation of our discussion of the consequences of our results for presentand upcoming experiments, we turn now to dark matter, beginning with direct detection(DD) experiments. One interesting quantity for these experiments is the spin-independentscattering cross-section σ SIp of the neutralino and a proton. This cross-section is often plot-ted against the neutralino mass m ˜ χ when comparing limits from different direct detectionexperiments. Predictions are given in this plane from both the MultiNest and GA scansin Fig. 5, drawn similarly to Figs. 1 and 2. σ SIp is shown in units of pb (i.e. 10 − cm ).Contours shown in the upper (lower) panels are generated according to the GA ( MultiNest )best-fit point, and the points with highest likelihoods are marked as before. Although noconstraints from direct detection measurements have been used in forming the model likeli-hood in this paper (mainly in order to work with the same set of quantities and constraintsas employed in Ref. 14), we have also included the current best DD limits for comparison.These are limits at the 90% confidence level from CDMS-II [91] and XENON10 [92].Looking at Fig. 5, we first notice that all the conclusions we made earlier are recon-firmed here: there are many important points in the parameter space that have appearedby the use of GAs, having a strong impact on the statistical conclusions. For example, in-stead of a rather spread and sparse 1 σ confidence region produced by MultiNest (Fig. 5d),GAs (Fig. 5a) reveal a more compact region, sharply peaked around the best-fit points.It is interesting to see that in the latter case, most of the 1 σ FP region around the dot-ted circle, including the point itself, is already excluded by CDMS-II and XENON10 un-der standard halo assumptions. The global best-fit point has quite a large cross-section( σ SIp ∼ × − pb) compared to the MultiNest global best-fit point ( σ SIp ∼ . × − pb),making it much more easily probed by direct detection (Table 6). On the contrary, thebest-fit COA point has a much lower σ SIp ( ∼ . × − pb), and is still well below theseexperimental limits. With future experiments planned to reach cross-sections as low as10 − pb, this point will eventually be tested as well. Even if we do not consider thehighest-likelihood point found by the GA, and just compare the two lower panels in Fig. 5,– 25 – a) XENON10CDMS-II
Akrami, Scott, Edsjö, Conrad & Bergström (2010)
GA points + GA levels )] pb ([ l og S I p10 (cid:2) )GeV(m ~ (cid:2) (b) XENON10CDMS-II
Akrami, Scott, Edsjö, Conrad & Bergström (2010)
MN points + GA levels )] pb ([ l og S I p10 (cid:2) )GeV(m ~ (cid:2) (c) XENON10CDMS-II
Akrami, Scott, Edsjö, Conrad & Bergström (2010)
GA points + MN levels )] pb ([ l og S I p10 (cid:2) )GeV(m ~ (cid:2) (d) XENON10CDMS-II
Akrami, Scott, Edsjö, Conrad & Bergström (2010)
MN points + MN levels )] pb ([ l og S I p10 (cid:2) )GeV(m ~ (cid:2) Figure 5:
As in Fig. 1 and Fig. 2, but representing the best-fit points and high-likelihood regions forthe spin-independent scattering cross-section of the neutralino and a proton σ SIp versus the neutralino mass m ˜ χ . Panels (a) and (d) show the statistically-consistent results of the GA and MultiNest scans, respectively.Panels (b) and (c) are given for comparative purposes only. The latest experimental limits from CDMS-II [91] and XENON10 [92] are also shown, plotted as red and black curves respectively. These curves areexclusion limits at the 90% confidence level under the assumption of a standard local halo configuration.
MultiNest has obviously only explored a small fraction of its 1 σ high-likelihood regionin the parameter space. It has also missed most of its 1 σ and 2 σ points in the region σ SIp > − pb. This is a particularly interesting area, being within the reach of currentdark matter DD experiments.In Table 6, we also give the calculated values for the spin-dependent scattering cross-sections of the neutralino with a proton ( σ SDp ) and a neutron ( σ SDn ), for both the FP andCOA best-fit points.Considering implications for indirect detection (ID) of dark matter, one is often in-– 26 – irect detectionGA global BFP GA BFP in COA region σ SIp . × − pb 2 . × − pb σ SDp . × − pb 4 . × − pb σ SDn . × − pb 3 . × − pbindirect detectionGA global BFP GA BFP in COA region (cid:104) σv (cid:105) . × − cm s − . × − cm s − Table 6:
Dark matter direct and indirect detection observables. These are the spin-independent andspin-dependent scattering cross-sections of the neutralino with nucleons in pb (10 − cm ), and the velocity-averaged neutralino self-annihilation cross-section. These are calculated at the global best-fit point (BFP)located in the focus point (FP) region, and at the best-fit point in the stau co-annihilation (COA) region. terested in a plot very similar to the one presented for the DD case, but now for thevelocity-averaged neutralino self-annihilation cross-section (cid:104) σv (cid:105) (instead of the scatteringcross-section in the previous case), again versus the neutralino mass m ˜ χ . Such plots areshown in Fig. 6 for all different cases in the same style as in Fig. 5. Their general charac-teristics resemble very much those we enumerated for the DD case.We first notice the strong correlation between the DD and ID plots, such as the genericsimilarities between the corresponding high-likelihood regions and the best-fit points. Oneinteresting feature visible in these plots is the existence of a new, considerably large, high-likelihood region spanned by the approximate ranges of 350 GeV < m ˜ χ <
500 GeV and − . < log ( (cid:104) σv (cid:105) ) < − . MultiNest . To our knowl-edge, this region has not been introduced so far by any other Bayesian or frequentist anal-ysis. Further investigations show that these points are located in the stau co-annihilationregion, but with very high m (up to ∼ Fermi gamma-ray space telescope to cover part of the high-likelihood FP region,including the global best-fit point with (cid:104) σv (cid:105) ∼ . × − cm s − (Table 6). It is in factexpected from pre-launch estimates [94] that the instrument will cover some fraction ofthe parameter space below (cid:104) σv (cid:105) ∼ − cm s − , depending on the neutralino mass. Adetailed MultiNest global fit of the CMSSM parameter space using 9 months of
Fermi datahas already been performed [45], using the dwarf spheroidal galaxy Segue 1 as a target;constraints are already quite close to becoming interesting. A similar analysis can also bedone using GAs. The other best-fit (COA) point, however, has (cid:104) σv (cid:105) ∼ . × − cm s − ,which is well below what is realistically detectable by Fermi .– 27 – a) Akrami, Scott, Edsjö, Conrad & Bergström (2010)
GA points + GA levels )GeV(m ~ (cid:2) )] sc m ( v [ l og -
13 10 (cid:2) (b)
Akrami, Scott, Edsjö, Conrad & Bergström (2010)
MN points + GA levels )GeV(m ~ (cid:2) )] sc m ( v [ l og -
13 10 (cid:2) (c)
Akrami, Scott, Edsjö, Conrad & Bergström (2010)
GA points + MN levels )GeV(m ~ (cid:2) )] sc m ( v [ l og -
13 10 (cid:2) (d)
Akrami, Scott, Edsjö, Conrad & Bergström (2010)
MN points + MN levels )GeV(m ~ (cid:2) )] sc m ( v [ l og -
13 10 (cid:2)
Figure 6:
As in Fig. 1, Fig. 2 and Fig. 5 but representing the best-fit points and high-likelihood regions forthe velocity-averaged neutralino self-annihilation cross-section (cid:104) σv (cid:105) versus the neutralino mass m ˜ χ . Again,panels (a) and (d) show the statistically-consistent GA and MultiNest results, respectively, and panels (b)and (c) are given for comparative purposes only.
Finally, we have shown in Fig. 7 the 1D profile likelihood for the neutralino mass m ˜ χ .The plot is produced in the same way as Figs. 3 and 4, and compares the results of bothGA and MultiNest scans. It is again important to notice the higher-likelihood points foundby the GA almost everywhere in the interesting mass range. We also observe that both theFP and COA best-fit points have quite similar and low neutralino masses ( ∼
150 GeV),similar to what was seen for m ˜ g in Fig. 4. Throughout the previous sections, we compared our GA results mostly with those of the– 28 – krami, Scott, Edsjö, Conrad & Bergström (2010) )GeV(m ~ (cid:2) m ax P L / P L Figure 7:
As in Fig. 3 and Fig. 4, but for the neutralino mass m ˜ χ . linear-prior MultiNest scan of the CMSSM, especially when we discussed the resultant 1Dand 2D profile likelihoods. We mentioned several times that
MultiNest implemented withlog priors gives a better value of 11 .
90 for the best-fit χ , compared to the best fit whenimplemented with linear priors (13.51). It is true that one can achieve better fits in certainregions of the parameter space by utilising e.g. a logarithmic prior in the search algorithm(see e.g. Ref. 14 and references therein for a discussion of the effects of priors on best-fitpoints and high-likelihood regions, as well as Bayesian posterior means and high-probabilityregions). However, there are good reasons not to use the log-prior MultiNest results for themain performance comparisons in this paper.Firstly, in order to make any comparison of the two algorithms reasonable, one shouldput both on the same footing. On the one hand, the likelihood function, as defined interms of the original model parameters, is the only statistical measure employed in anyfrequentist study. Our genetic analysis is no exception. Our GA scans the parameter spaceaccording to the likelihood, as a function of the original model parameters. On the otherhand,
MultiNest , similarly to every other sampling technique optimised for Bayesian scans,performs the scan based on the posterior PDF (i.e. likelihood times prior) rather than thelikelihood alone. Consequently, a very natural way of comparing the two is to make thelatter sampling algorithm also proceed according to the likelihood function only. This canbe achieved by choosing a flat prior in this case.Imposing any other nontrivial prior (or equivalently, changing the scanning metric),although entirely justified in the Bayesian framework, is a very ad hoc approach in afrequentist framework. In the Bayesian case, this simply means that the algorithm samplesthe regions containing larger prior volumes better, producing more sample points in theseregions. This is exactly what one requires for a Bayesian scan in which the density of– 29 –amples reflects the posterior density at different points in the parameter space. In theprofile likelihood analysis however, we are interested in having reasonable maps of thelikelihood function in terms of the given model parameters. Imposing any prior in this casemeans nothing but giving different scanning weights to different parts of the parameterspace, i.e. forcing the algorithm to scan some regions with higher resolutions than theothers; this can make the algorithm miss important points in some regions.In the frequentist language, the effect of imposing a non-flat prior is the same asreparameterising the model. This for example means that, in the case of the log-priorscan, the likelihood function is redefined in terms of the logarithmically-scaled parametersrather than the original model parameters. Results of a profile likelihood analysis shouldin principle be independent of the specific parameterisation of the model; it should notmatter if one works with e.g. one set of coordinates or another. This statement is howevercorrect only if one has perfect knowledge of the likelihood function. No numerical scanningalgorithm provides this perfect knowledge, as its resolution is always limited. This meansthat different parameterisations of the model do give different results until the limit of‘perfect sampling’ has been reached. Any specific choice should then be justified ‘a priori’.One can for example argue that a specific scaling is theoretically better justified comparedto others (e.g. that a log prior is geometrically preferred to a flat one). In principle this isa Bayesian statement, as it places an implicit measure on the parameter space, but it doeshave a practical impact upon frequentist profile likelihood scans. If one wanted to explorethe effects of such reparameterisations, it would be entirely possible to do this by way of aGA, implemented in terms of genomes encoding the rescaled parameters. We suspect thatby using a logarithmically-encoded genome, we would for example find the funnel regionproperly and probably even some better-fitting points than our current best-fit. Sinceour primary intention in the current work has been to look at the CMSSM model as itis, we have therefore adhered to the likelihood function defined in terms of the originalparameters (and thus employed a linearly-encoded genome). We leave the investigation oflogarithmically-encoded genomes for future work, where we intend to compare results withthose of log-prior
MultiNest scans.
The results presented in this work are based on 3 million sample points in total, corre-sponding to the same number of likelihood evaluations. The samples have been generatedthrough 10 separate runs with different initial populations and 3000 generations each. Theresultant samples were then combined to obtain the final set of points. Compared to a typ-ical number of likelihood evaluations required in a
MultiNest scan (around 500 , MultiNest , has globally converged;there are indeed several reasons making us believe that it probably has not.– 30 – m / A tan β m t m b ( m b ) MS α s ( m Z ) MS /α em ( m Z ) MS χ min (GeV) (GeV) (GeV) (GeV) (GeV)Run 1 1900 . . . . . .
20 0 . . Run 2 133 . . . . . .
20 0 . . Run 3 198 . . . . . .
20 0 . . Run 4 2817 . . . . . .
20 0 . . Run 5 2693 . . . . . .
20 0 . . Run 6 2737 . . . . . .
20 0 . . Run 7 2775 . . . . . .
20 0 . . Run 8 3102 . . . . . .
20 0 . . Run 9 3158 . . . . . .
21 0 . . Run 10 3159 . . . . . .
20 0 . . Table 7:
Parameter and χ values at the best-fit points found by Genetic Algorithms in each of 10 runs.The final inference is based on all points found in all runs. One way of seeing this is to look at the results for each of the 10 runs separately. Toillustrate this, we have given in Fig. 8 the corresponding two-dimensional profile likelihoodsin the m - m / plane. The iso-likelihood contours in each panel show the statistically-consistent 1 σ and 2 σ confidence regions, based on the best-fit point found in that specificscan. Panels are sorted according to the best-fit χ values. The values of all model andnuisance parameters at the best-fit points for all 10 runs are also given in Table 7, togetherwith the best-fit χ values.Our global best-fit point ( χ = 9 .
35) is found only in one run. The other 9 runs havenot found best-fit points better than χ = 11 .
34. However, all these other best-fit values( χ = 11 .
34, 11 .
45, 11 .
55, 11 .
61, 11 .
63, 11 .
65, 11 .
76, 11 .
78, and 11 .
86) are also significantlybetter than the one found by
MultiNest with linear priors ( χ = 13 . MultiNest withlogarithmic priors ( χ = 11 . σ region does not include the global best-fit regionof run 1, is very likely to extend and eventually uncover that region if the run continues.The other difficulty is that if the current version of the algorithm finds the global best-fitpoint (or some high-likelihood points in its vicinity) relatively quickly, the chance that itwill then scan other points with lower likelihoods within the interesting confidence regionsis dramatically reduced. This indicates a drawback of the algorithm in correctly mappingconfidence intervals around the global best-fit point. This problem is not unexpected, asthe primary purpose of GAs is to find global maxima or minima for a specific function asquickly as possible; if they succeed in this, there is no reason for them to start mappingthe other less important surrounding points. This issue obviously becomes more serious Other two- and also one-dimensional profile likelihood plots for the parameters and observables exhibitsimilar features, so add little to the discussion. – 31 – TeV)(m
1 2 3 4
Akrami,Scott,Edsjö,Conrad&Bergström(2010)
Run1, min (cid:2) = 9.35 T e V )( m Akrami,Scott,Edsjö,Conrad&Bergström(2010) TeV)(m
1 2 3 4 T e V )( m Run2, min (cid:2) = 11.34 Akrami,Scott,Edsjö,Conrad&Bergström(2010) TeV)(m
1 2 3 4 T e V )( m Run3, min (cid:2) = 11.45 Akrami,Scott,Edsjö,Conrad&Bergström(2010) TeV)(m
1 2 3 4 T e V )( m Run4, min (cid:2) = 11.55 Akrami,Scott,Edsjö,Conrad&Bergström(2010) TeV)(m
1 2 3 4 T e V )( m Run5, min (cid:2) = 11.61 Akrami,Scott,Edsjö,Conrad&Bergström(2010) TeV)(m
1 2 3 4 T e V )( m Run6, min (cid:2) = 11.63 Akrami,Scott,Edsjö,Conrad&Bergström(2010) TeV)(m
1 2 3 4 T e V )( m Run7, min (cid:2) = 11.65 Akrami,Scott,Edsjö,Conrad&Bergström(2010) TeV)(m
1 2 3 4 T e V )( m Run8, min (cid:2) = 11.76 Akrami,Scott,Edsjö,Conrad&Bergström(2010) TeV)(m
1 2 3 4 T e V )( m Run9, min (cid:2) = 11.78 Akrami,Scott,Edsjö,Conrad&Bergström(2010) TeV)(m
1 2 3 4 T e V )( m Run10, min (cid:2) = 11.86 Figure 8:
Individual two-dimensional profile likelihoods in the m - m / plane for each of the 10 runsemployed in our analysis. Each panel shows the statistically-consistent results of each scan based on thebest-fit point found in that scan. As in previous figures, the inner and outer contours represent 68 .
3% (1 σ )and 95 .
4% (2 σ ) confidence regions, respectively. The dotted circles show the best-fit points of each run.The sample points have been divided into 75 ×
75 bins in all plots. Panels are sorted according to the valuesfor the best-fit χ . The final two-dimensional profile likelihood in Fig. 1a is obtained by combining all 10scans and drawing iso-likelihood contours relative to the global best fit, i.e. χ = 9 . when spike-like best-fit regions exist, a characteristic which appears to be the case for theCMSSM. For example, the existence of relatively large high-likelihood regions in panels– 32 –,5,6,7,8 and 10 is probably due to the fact that there are many points with likelihoodvalues very close to the highest one, distributed in a large area; this is not the case for runs1,2, or 3. This means that if the GAs continue to run in those cases and it so happens thatthe global best-fit point of run 1 is found at some stage, their mapping of the interestingregions will be much better than the present case in run 1. This is again an indication thatthere is a trade-off between how quickly we want the algorithm to find the actual globalbest-fit point and how accurately it is supposed to map the confidence regions; employingmore advanced operators and strategies in the algorithms might improve the situation.As far as our current implementation of GAs is concerned, all the above imply thatthe algorithms have not converged properly in every individual run. Firstly, they havenot been able to find the actual global best-fit point in all (or even necessarily any) ofthe scans, and secondly, in the run with the highest-likelihood best-fit point (run 1), themapping of the confidence regions around that point is rather unsatisfactory. On theother hand, most of the unwanted features discussed here are alleviated by combining theresults of all 10 runs. This demonstrates the important role that parallelisation couldplay in improving the efficiency of GAs. Although our full set of sample points seems toprovide better results than MultiNest , we are still not sure that this parallel version ofthe algorithm has converged either. This is again because the number of separate runsemployed in our analysis is chosen rather arbitrarily. The arbitrariness in both the numberof runs and the termination criteria in each run means that no result-driven convergencecondition exists in our GAs. One could possibly utilise more sophisticated criteria in thetermination condition, giving rise to better estimates of the convergence in each run, but toour knowledge no problem-independent such alternatives exist; we leave the investigationof such possibilities for future work.As discussed in Sec. 3.1, even though such a convergence condition does exist for the
MultiNest scans, making the algorithm terminate after less total likelihood evaluations thanour GAs, it still misses many points that are important in the frequentist framework. Theconvergence criterion for
MultiNest is defined in terms of the Bayesian evidence; even ifa run is properly converged in terms of the evidence, this convergence makes sense onlyin the context of the Bayesian posterior PDF, not a frequentist profile likelihood analysis.That is, many isolated spikes might have been missed, and consequently the likelihoodmight not have been mapped effectively. Even tuning the convergence parameters (suchas the tolerance) may not help. This is because if the
MultiNest algorithm does not find ahigh-likelihood point on its first approach to a region, it is given no chance to go back andfind it at a later stage. This means that even if the convergence parameter for
MultiNest istuned in such a way that the algorithm runs for the same number of likelihood evaluationsas a GA (i.e. 3 million here), this does not help in finding better-fit points. One should thusbe very careful in introducing convergence criteria to any scanning techniques (includingGAs and
MultiNest ) dealing with a complex model such as the CMSSM; the criteria shouldbe carefully defined depending on which statistical measure is employed.We also point out that the isolated likelihood spikes missed by Bayesian algorithmswill only fail to affect the posterior mass if a limited number of them exist. If there are asignificant number of spikes, they could add up to a large portion of the posterior mass, and– 33 –ffect even the Bayesian inference. Our GA results indicate that many such missed pointsactually exist, hinting that
MultiNest might not have even mapped the entire posteriorPDF correctly. Given the frequentist framework of this paper, this must unfortunatelyremain mere speculation, as our results do not allow any good estimate to be made of theposterior mass contained in the extra points.We conclude this section by emphasising the role of parallelisation in terms of requiredcomputational power. Not only does the parallelisation enhance the scanning efficiency ofGAs by reducing the probability of premature convergence and trapping in local maxima,it also increases the speed significantly. This can be done in different ways. Firstly, GAswork with a population of points instead of a single individual, providing extensive op-portunity for treating different individuals in parallel. As an immediate consequence, therequired time in a typical simple GA run with full generational replacement (the schemewe have used) can in principle decrease by a factor of n p , the number of individuals in eachpopulation (100 in our case). The other way to parallelise GAs is to have separate popula-tions evolve in parallel. By employing more advanced genetic operators and strategies suchas ‘immigration’, individuals in different populations can even interact with each other.Although we have not employed such advanced parallel versions of the algorithm in ouranalysis, the samples have been generated in a parallel manner, i.e. through 10 separateruns with 3000 generations each.
4. Summary and conclusions
Constraining the parameter space of the MSSM using existing data is under no circum-stances an easy or straightforward task. Even in the case of the CMSSM, a highly simplifiedand economical version of the model, the present data are not sufficient to constrain theparameters in a way completely independent of computational and statistical techniques.There have been several efforts to study properties and predictions of different versionsof the MSSM. Many recent activities in this field have used scanning methods optimisedfor calculating the Bayesian evidence and posterior PDF. Those analyses have been highlysuccessful in revealing the complex structure of SUSY models, demonstrating that somepatience will be required before we can place any strong constraints on their parameters.The same Bayesian scanning methods have also been employed for frequentist analyses ofthe problem, particularly in the framework of the profile likelihood. These methods arenot optimised for such frequentist analyses, so care should be taken in applying them tosuch tasks.We have employed a completely new scanning algorithm in this paper, based on GeneticAlgorithms (GAs). We have shown GAs to be a powerful tool for frequentist approachesto the problem of scanning the CMSSM parameter space. We compared the outcomesof GA scans directly with those of the state-of-the-art Bayesian algorithm
MultiNest , inthe framework of the CMSSM. For this comparison, we mostly considered
MultiNest scanswith flat priors, but kept in mind that e.g. logarithmic priors give rise to higher-likelihoodpoints at low masses in the CMSSM parameter space; we justified this choice of priors.– 34 –ur results are very promising and quite surprising. We found many new high-likelihood CMSSM points, which have a strong impact on the final statistical conclusionsof the study. These not only influence considerably the inferred high-likelihood regionsand confidence levels on the parameter values, but also indicate that the applicability ofthe conventional Bayesian scanning techniques is highly questionable in a frequentist con-text. Although our initial motivation in using GAs was to gain a correct estimate of thelikelihood at the global best-fit point, which is crucial in a profile likelihood analysis, wealso realised that they can find many new and interesting points in almost all the relevantregions of parameter space. These points strongly affect the inferred confidence regionsaround the best-fit point. Even though we cannot be confident of exactly how completelyour algorithm is really mapping these high-likelihood regions, it has certainly covered largeparts of them better than any previous algorithm.We think that by improving the different ingredients of GAs, such as the crossover andmutation schemes, this ability might even be enhanced further. We largely employed thestandard, simplest versions of the genetic operators in our analysis, as well as very typicalgenetic parameters. These turned out to work sufficiently well for our purposes. Althoughwe believe that tuning the algorithm might produce even more interesting results, it isgood news that satisfactory results can be produced even with a very generic version. Thislikely means that one can apply the method to more complicated SUSY models withoutextensive fine-tuning.One interesting outcome of our scan is that the global best-fit point is found to belocated in the focus point region, with a likelihood significantly larger than the best-fitpoint in the stau co-annihilation region (which in turn actually still has a higher likelihoodthan the global best-fit value obtained with
MultiNest , even with logarithmic priors). Thefocus point region is favoured in our analysis over the co-annihilation region, in contrastto findings from some recent MCMC studies [31, 37], where the opposite was stronglyclaimed. We also found a rather large part of the stau co-annihilation region, consistentwith all experimental data, located at high m . This part of the co-annihilation regionseems to have been missed in other recent scans. All these results show that, at least inour particular setup, high masses, corresponding either to the FP or the COA regions,are by no means disfavoured by current data (except perhaps direct detection of darkmatter). The discrepancy between this finding and those of some other authors that theFP is disfavoured might originate in the different scanning algorithms employed, or in thedifferent physics and likelihood calculations performed in each analysis. We have howevershown, by comparing our results with others produced using exactly the same setup exceptfor the scanning algorithm, that one should not be at all confident that all the relevantpoints for a frequentist analysis can be found by scanning techniques optimised for Bayesianstatistics, such as nested sampling and MCMCs.We have also calculated some of the quantities most interesting in searches for SUSY atthe LHC, and in direct and indirect searches for dark matter. We showed that GAs foundmuch better points compared to MultiNest almost everywhere in the interesting mass rangesof the lightest Higgs boson, gluino and neutralino. We confirmed previous conclusions thatthe LHC is in principle able to investigate a large fraction of the high-likelihood points– 35 –n the CMSSM parameter space if it explores sparticle masses up to around 3 TeV. Asfar as the Higgs mass is concerned, there are many points with rather low masses that,although sitting just below the low mass limit given by LEP, are globally very well fitto the experimental data. In the context of dark matter searches, we noticed that theglobal best-fit point and much of the surrounding 1 σ confidence level region at high cross-sections are actually already mostly ruled out by direct detection limits, if one assumesthe standard halo model to be accurate. We also argued that some of these points maybe tested by upcoming indirect detection experiments, in particular the Fermi gamma-rayspace telescope. Finally, we realised that the high-likelihood stau co-annihilation region atlarge m introduces a new allowed region in the combination of the neutralino mass andself-annihilation cross-section, which (to our knowledge) has not been observed previously.We also compared our algorithm with MultiNest in terms of speed and convergence, andargued that GAs are no worse than
MultiNest in this respect. GAs have a large potential forparallelisation, reducing considerably the time required for a typical run. This property,as well as the fact that the computational effort scales linearly (i.e. as kN for an N -dimensional parameter space), also makes GAs an excellent method for the frequentistexploration of higher-dimensional SUSY parameter spaces.Finally, perhaps the bottom line of the present work is that we once again see thateven the CMSSM, despite its simplicity, possesses a highly complex and poorly-understoodstructure, with many small, fine-tuned regions. This makes investigation of the modelparameter space very difficult and still very challenging for modern statistical scanningtechniques. Although the method proposed in this paper seems to outperform the usualBayesian techniques in a frequentist analysis, it is important to remember that it may byno means be the final word in this direction. Dependence of the results on the chosenstatistical framework, measure and method calls for caution in drawing strong conclusionsbased on such scans. The situation will of course improve significantly with additionalconstraints provided by forthcoming data. Acknowledgments
The authors are grateful to the Swedish Research Council (VR) for financial support. YAwas also supported by the Helge Axelsson Johnson foundation. JC is a Royal SwedishAcademy of Sciences Research Fellow supported by a grant from the Knut and Alice Wal-lenberg Foundation. We thank Roberto Trotta for helpful discussions. We are also thankfulto the authors of Ref. 31 for useful comments on a previous version of the manuscript.
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