Coherent combination of probabilistic outputs for group decision making: an algebraic approach
aa r X i v : . [ s t a t . O T ] J u l Annals of Operations Research manuscript No. (will be inserted by the editor)
Coherent combination of probabilistic outputs forgroup decision making: an algebraic approach
Manuele Leonelli · Eva Riccomagno · Jim Q. Smith
Received: date / Accepted: date
Abstract
Current decision support systems address domains that are het-erogeneous in nature and becoming progressively larger. Such systems oftenrequire the input of expert judgement about a variety of different fields andan intensive computational power to produce the scores necessary to rank theavailable policies. Recently, integrating decision support systems have beenintroduced to enable a formal Bayesian multi-agent decision analysis to bedistributed and consequently efficient. In such systems, where different pan-els of experts oversee disjoint but correlated vectors of variables, each expertgroup needs to deliver only certain summaries of the variables under theirjurisdiction to properly derive an overall score for the available policies. Herewe present an algebraic approach that makes this methodology feasible for awide range of modelling contexts and that enables us to identify the summariesneeded for such a combination of judgements. We are also able to demonstratethat coherence, in a sense we formalize here, is still guaranteed when panelsonly share a partial specification of their model with other panel members. Weillustrate this algebraic approach by applying it to a specific class of Bayesiannetworks and demonstrate how we can use it to derive closed form formulae for
We acknowledge that J.Q. Smith was partly supported by EPSRC grant EP/K039628/1and The Alan Turing Institute under EPSRC grant EP/N510129/1, whilst E. Riccomagnowas supported by the GNAMPA-INdAM 2017 project.Manuele LeonelliSchool of Mathematics and Statistics, University of Glasgow, UKE-mail: [email protected] RiccomagnoDipartimento di Matematica, Universit`a degli Studi di Genova, ItaliaE-mail: [email protected] Q. SmithDepartment of Statistics, University of Warwick, UKE-mail: [email protected] Manuele Leonelli et al. the computations of the joint moments of variables that determine the scoreof different policies.
Keywords
Bayesian networks · Integrating decision support systems · Polynomial algebra · Structural equation models
Although still being refined, probabilistic decision support tools for singleagents are now well developed and used in practice in a variety of domains.One of the most common probabilistic models for multivariate systems areBayesian networks (BNs) (Jensen and Nielsen, 2013; Pearl, 1988) and theirdynamic and object-oriented extensions (Koller and Pfeffer, 1997; Murphy,2002). However, these are not the only frameworks around which probabilisticmodels have been built. Other well-established models comprise, among oth-ers, Bayesian hierarchical spatio-temporal models (Blangiardo and Cameletti,2015), asymmetric probability trees (Smith and Anderson, 2008) and proba-bilistic emulators (Kennedy and O’Hagan, 2001).However, the size and complexity of current applications often require thatsupporting systems consist of component modules which, encoding the judge-ments of panels of domain experts, describe a particular sub-domain of theoverall system. In these contexts decision makers need a tool that can coher-ently paste together the outputs of each of these modules to provide a com-prehensive picture of the whole process. Often in practice, because of bothcomputational and methodological constraints, the modules’ outputs end upbeing collated together in a simple, essentially deterministic way by transfer-ring from one module to another a single vector of means about what mighthappen and hence effectively ignoring any associated uncertainty. However,such a na¨ıve method can be very misleading and guide decision makers tochoose a non-optimal course of action (see e.g. Leonelli and Smith, 2013, 2015).Recently, integrating decision support systems (IDSSs) (Leonelli and Smith,2015; Smith et al, 2015) have been defined to extend coherence requirementstraditionally applied within a Bayesian decision support system for singleagents so that these apply to this new multi-expert setting. IDSSs embeda methodology, similar to a standard Bayesian one, where decisions can beguaranteed to be coherent, i.e. expected utility maximising for some utilityand probability distribution derived from individual but connected suites ofmodels. Before briefly reviewing the theory of IDSSs in Section 2.2, we dis-cuss in Section 2.1 a domain of application where we have found it necessaryto knit together a suite of models. We then introduce in Section 2.3 a real-world example that illustrates our methodology. In Section 2.4 we highlightthe contributions of the paper. oherent combination of probabilistic outputs for group decision making 3
Manuele Leonelli et al. – the decision centre responsible for the implementation of any policy needsto consist of individuals who act collaboratively and strive to behave as asingle coherent unit would. In the food poverty (Barons et al, 2016) andnuclear emergency management (Leonelli and Smith, 2013) applicationsthis condition was broadly met. We suppose the centre consists of m panelsof experts denoted by G , . . . , G m ; – there must be a consensus about the policies d that could be scrutinizedand eventually implemented by the centre. In other words, all individualsin the centre must agree on a set D of decision rules whose efficacy might beexamined by the IDSS. The choice of D is usually resolved using decisionconferencing (French et al, 2009) across panel representatives, users andstakeholders. We refer to this condition as policy consensus ; – there must also be a consensus about the appropriate utility structureunderlying a set of agreed attributes against which the efficacy of anypolicy is evaluated. So all individuals in the centre need to agree on theclass U of utility functions supported by the IDSS. For instance, this con-sensus might be that the centre’s utility function has utility independentattributes (Keeney and Raiffa, 1976). Again the choice of U is often re-solved through decision conferencing. We refer to this condition as utilityconsensus ; – consensus also needs to be found about an overarching description of thedynamics driving the process. We assume that all panellists make their in-ferences in a parametric setting where a random vector Y is parametrisedby a vector θ , where Y defines the variables of the process whilst θ isthe parameter vector on which inference is made. Then such a consensusconsists of an agreement of all involved on the variables Y , where, foreach policy d ∈ D , each utility function u ∈ U is a function of Y togetherwith a set of qualitative statements about the dependence between var-ious functions of Y and θ . This can take a variety of forms dependingon the domain of application. In this paper we mainly focus on depen-dence structures represented by BNs, although our methods apply equallywell to other frameworks (Smith et al, 2015). We refer to this condition as structural consensus . oherent combination of probabilistic outputs for group decision making 5 The union of the policy, utility and structural consensus is called the common-knowledge class (CK-class) and describes the agreement of all in-dividuals on the components of the system and their relationships with eachother. The CK-class defines the qualitative structure of the domain investi-gated and therefore more easily provides a framework for the group’s agree-ment (Smith, 1996). Protocols to guide the construction of the sets D and U ,and the identification of an overarching probabilistic model for the structuralconsensus have been recently defined (Barons et al, 2016).Given this overarching qualitative structure has been agreed by the centreand represented by the CK-class, then an agreement on how to populate thisclass with quantitative statements must be found. To this end, we assume thefollowing condition holds: – the centre must find a consensus about who is expert about what. In aformal sense, this implies that all panellists are prepared to adopt thebeliefs of the designated expert panel in a specific sub-domain of the processas their own.Thus in an IDSS beliefs’ specifications are delegated to the most informedpanel. Each panel then, given a CK-class, individually delivers the necessaryquantities for the computation of expected utilities concerning the variablesunder their jurisdiction. However, as illustrated by influence of cost of oil andweather on food production and accessability to food, there is in general noguarantee that the individual beliefs of the panels can be combined to givea probabilistic coherent overall picture of the process. For the purposes of aformal Bayesian decision analysis an IDSS needs to entertain the followingproperty. Definition 1
An IDSS is said to be adequate for a CK-class if it can unam-bigously calculate the expected utility score of any decision d ∈ D and anyutility function u ∈ U from the beliefs of the panels G , . . . , G m .It is vital for an IDSS to be adequate since otherwise it could not produce aranking of the available polices (Keeney and Raiffa, 1976) and would thereforenot be of any help to the decision centre for implementing and justifying anypolicy choice.2.3 A real-world exampleAfter a series of decision conferences with local authorities from Warwick-shire County Council, stakeholders and potential decision makers, Barons et al(2016) identified three areas that are impacted by increasing household foodinsecurity: health ( Y ), educational attainment ( Y ) and social cohesion ( Y ).Of course the cost ( Y ) associated to the enactment of any policy is deemedrelevant in this domain. Measurable indices were developed for each of theseareas - for instance, educational attainment is assessed by the percentage ofpupils not failing a combination of UK school examinations. Further details Manuele Leonelli et al. ?>=<89:; ?>=<89:; (cid:15) (cid:15) ?>=<89:; O O @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) / / ?>=<89:; Fig. 1
BN representing the relationship between the four attributes in the food insecurityexample, where the vertex i is associated to the random variable Y i , i = 1 , . . . , about the form of the attributes are beyond the scope of this paper and we referto Barons et al (2016) for a discussion of these (see also Leonelli and Smith,2017).Notice that of course in a reliable description of the food system any de-cision support system needs to account for the probabilistic dependence overa much larger vector of variables. But for the illustrative purposes of thisexample, we assume the decision centre agrees that these four random vari-ables provide an overall, sufficient description of the household food system.The structural consensus of an IDSS with these four random variables thenconsists of a conditional independence structure that we suppose here to be de-picted by the BN in Fig. 1. This states that given different levels of the healthattribute, the associated costs are independent of both educational attainmentand social cohesion.Even in such a simple example we can highlight the heterogeneity of thefood system and the consequent need of an IDSS. So for instance beliefs aboutthe health attribute are delivered by doctors and public health experts; ed-ucational attainment is under the jurisdiction of school representatives andteachers; social unrest is overseen by sociologists, whilst judgements aboutcosts are given by politicians and policymakers.We consider a decision space D comprising of three possible policies: ei-ther an increase ( d ), a decrease ( d ) or not a change ( d ) of the number ofpupils eligible for free school meals in Warwickshire. The UK government hasalready implemented this type of policy to give pupils a healthy start in life,since evidence seems to point towards an improvement of development andsocial skills of young children that eat a healthy meal together at lunchtime(Kitchen et al, 2013). We suppose henceforth that the decision centre, consist-ing of local authorities and stakeholders, agrees to consider only these threepolicies.Lastly, the utility consensus might correspond to an agreement of a spe-cific utility factorization over these four attributes. For instance, letting y =( y , y , y , y ), where y i is an instantiation of Y i , the centre might find anagreement that the utility function factorizes additively. Specifically, u ( d, y ) = k ( d ) u ( y , d ) + k ( d ) u ( y , d ) + k ( d ) u ( y , d ) + k ( d ) u ( y , d ) , (1)where k i ( d ) ∈ (0 ,
1) and d ∈ D . Given this agreed factorization, the specificform of the functions u i ( y i ) is then elicited by the appropriate expert panel. oherent combination of probabilistic outputs for group decision making 7 In Section 7 below we illustrate the methodology we introduce in this paperusing this simple IDSS for household food security.2.4 ContributionsRecent advances in the integration of distributed expert judgements in com-plex systems have been reported in Leonelli and Smith (2015) and Smith et al(2015). It has been demonstrated there that, perhaps surprisingly, it is com-mon to be able to define a coherent system by only specifying qualitativerelationships between its random variables and quantifying a few of their as-sociated summaries. Although some work has addressed the difficulties asso-ciated to the combination of expert judgments in multivariate systems (e.g.Faria and Smith, 1997; Farr et al, 2014), none of these formally took into ac-count the heterogeneity of the domain to be modelled.In Smith et al (2015) we focused on the inferential full-distributional dif-ficulties associated to this integration. However, a formal Bayesian decisionanalysis is based on the maximization of an expected utility (EU) functionand this often only depends on some simple summaries of key output vari-ables, for example a few low order moments. By requesting from the relevantpanels only the value of these expectations, the implementation of an IDSScan become orders of magnitude more manageable. Panels then just need tocommunicate a few summaries of their analysis: a trivial and fast task to per-form within most inferential systems. In these cases real time decision supportis thus feasible even when the system is huge.We demonstrate below that the EUs of such an IDSS are usually polyno-mials whose indeterminates are functions of the panels’ delivered summaries.This polynomial structure enables us to identify new separation conditions, of-ten implicit in standard conditional independence over the parameters of cer-tain graphical models (Freeman and Smith, 2011; Spiegelhalter and Lauritzen,1990) and milder than those of Smith et al (2015), sufficient to guarantee thatan IDSS is adequate. An adequate IDSS is then capable of supporting deci-sion centres by providing a sound and coherent ranking of the available policiestogether with their associated EU scores.Under the conditions derived above, we develop new propagation algo-rithms for BNs, here called algebraic substitutions , for the distributed compu-tations of an IDSS EU scores. These generalize the theory of the computationof moments of decomposable functions (Cowell et al, 1999; Nilsson, 2001) tomultilinear ones. Algebraic substitutions mirror the recursions of Lauritzen(1992) for the computation of the first two moments of chain graph models.Here, focusing only on specific BN models, we are able to explicitly computeany joint moment and provide an intuitive graphical interpretation of the as-sociated propagation rules.Importantly, the recognition of the polynomial nature of EUs also enablesus to analyze efficiently as well as exactly even large problems using softwarefor symbolic manipulations (or computer algebra software), e.g. Mathematica
Manuele Leonelli et al. (Wolfram Research, Inc., 2017). Assuming the panels are able to deliver a vec-tor of required summaries from the complex probabilistic model they plan touse, the software is then capable of combining them using algebraic substi-tutions to compute the associated EU scores almost instantaneously and inreal-time to evaluate the candidate policies available to a decision centre. Thisis a critical property of any decision support system and in Section 7 we givean illustration of how this can be achieved with computer algebra software.
We start by giving a polynomial description of the EUs of an IDSS. Consider arandom vector Y = ( Y i ) i ∈ [ m ] , [ m ] = { , . . . , m } , where a subvector Y i of Y isunder the jurisdiction of a panel of experts G i , i ∈ [ m ]. Let y ∈ Y and y i ∈ Y i be instantiations of Y and Y i , respectively. Assume each panel of expertsdelivers beliefs about θ i , the parameter of the density f i over Y i | ( θ i , d ),where d ∈ D is one of the available policies in the decision space D . Suppose θ i takes values in Θ i and let θ = ( θ i ) i ∈ [ m ] take values in Θ . Let f , π i and π denote densities over Y | ( θ , d ), θ i | d and θ | d , respectively.The IDSS processes the panels’ judgements in order to calculate variousstatistics of an attribute vector , usually some function of Y . For simplicity andwith no loss of generality we assume in this paper that attributes coincide with Y . For the purpose of a formal Bayesian analysis the IDSS computes the setof EU scores { ¯ u ( d ) : d ∈ D} as a function of both the utility function u ( y , d )and the probability statements of the individual panels. The IDSS would thenrecommend to follow the policy d ∗ with the highest EU score, ¯ u ( d ∗ ), wherethe EU is computed as ¯ u ( d ) = Z Θ ¯ u ( d | θ ) π ( θ | d )d θ , and ¯ u ( d | θ ) = Z Y u ( y , d ) f ( y | θ , d )d y , is the conditional expected utility (CEU) .By approaching the theory of IDSSs from an algebraic viewpoint, we areable to identify the necessary panels’ summaries and the required assumptionsfor adequacy. In order to do this we first need to define the EU polynomials. Definition 2
The CEU ¯ u ( d | θ ) of an IDSS is called algebraic in the panels if, for each d ∈ D and for each panel G i in charge of Y i with parameter θ i , i ∈ [ m ], there exist functions λ i ( θ i , d ) of θ i and d such that ¯ u ( d | θ ) is asquare-free polynomial q d of the λ i ¯ u ( d | θ ) = q d ( λ ( θ , d ) , · · · , λ m ( θ m , d )) . oherent combination of probabilistic outputs for group decision making 9 Each λ i is a vector of length s i , where s i is the number of summaries eachpanel is required to deliver. Let λ i ( θ i , d ) = ( λ ji ( θ i , d )) j ∈ [ s i ] , [ s i ] = [ s i ] ∪ { } and b ∈ B = × i ∈ [ m ] [ s i ] . For a given b = ( b i ) i ∈ [ m ] and j ∈ [ s i ], define b j,i = 0if j = b i , b j,i = 1 if j = b i and b ,i = 1, for i ∈ [ m ]. It follows that b j,i is notzero if and only if either j = 0 or j equals the i -th entry of b . Let λ i ( θ i , d ) = 1,for every θ i ∈ Θ i , d ∈ D and i ∈ [ m ]. Example 1
Let m = 2, s = s = 1, i.e. there are two panels each deliveringone summary only. Then B = { (0 , , (0 , , (1 , , (1 , } . For b = (0 , ∈ B we have that b , = 1, b , = 0, b , = 1 and b , = 1. Definition 3
The CEU ¯ u ( d | θ ) of an IDSS is called algebraic if, for each d ∈ D , q d is a square-free polynomial of the λ ji , i ∈ [ m ], j ∈ [ s i ] , such that q d ( λ ( θ , d ) , . . . , λ m ( θ m , d )) = X b ∈ B k b ( d ) λ b ( θ , d ) , (2)with k b ( d ) ∈ R and λ b ( θ , d ) = Y i ∈ [ m ] Y j ∈ [ s i ] λ ji ( θ i , d ) b j,i . Thus, λ b is a monomial having at most one term not unity delivered by eachpanel and k b ( d ) is a weight. Example 2
Let the CK-class specify that Y = ( Y i ) i ∈ [ m ] , where each variable Y i is binary and overseen by panel G i . Assume that for all decisions d ∈D , θ i = P ( Y i | θ i , d ), θ = ( θ i ) i ∈ [ m ] , and that the CK-class includes the beliefthat ⊥⊥ i ∈ [ m ] Y i | θ , d , where ⊥⊥ denotes conditional independence (Dawid, 1979).Suppose the utility consensus consists of a utility factorization of the form u ( y ) = u ( y , . . . , y m ) = X i ∈ [ m ] k i ( d ) y i + X i ∈ [ m ] X i A more complex example is given by two dependent continuousrandom variables Y and Y such that E ( Y ) = θ , V ( Y ) = ψ , E ( Y | Y = y ) = θ + θ y and V ( Y | Y = y ) = V ( Y ) = ψ . Here E stands for ex-pectation and V for variance. Assume the utility consensus includes an addi-tive utility factorization u ( y , y , d ) = k ( d ) u ( y , d )+ k ( d ) u ( y , d ), where eachmarginal utility function is in the family of quadratic utility functions (Wakker,2008), i.e. u ( y i , d ) = a i ( d ) y i − b i ( d ) y i , where a i ( d ) ∈ R and b i ( d ) ∈ R > , d ∈ D and i ∈ [2]. Using standard properties of conditional moments (see e.g.Brillinger, 1969), by leaving implicit the dependence on d the CEU can bewritten as¯ u ( d | θ ) = k ( a θ − b θ − b ψ )+ k ( a θ + a θ θ − b θ − b θ θ − b θ ψ − b ψ − b θ θ θ ) . In this second example the CEU is again algebraic and λ ( θ , d ) = (1 , a θ , b θ , b ψ , θ , θ , ψ ) , λ ( θ , d ) = (1 , a θ , a θ , b θ , b θ , b ψ , b θ θ ) . To achieve adequacy we need the following property. Definition 4 Let µ ji ( d ) = E (cid:0) λ ji ( θ i , d ) b j,i (cid:1) , for a given b ∈ B . We call anIDSS score separable if, in the notation above, all panellists agree that, for alldecisions d ∈ D and all indices b ∈ B such that k b ( d ) = 0, E ( λ b ( θ , d )) = Y i ∈ [ m ] Y j ∈ [ s i ] µ ji ( d ) . A score separable IDSS can then determine the EU score of any policy d ∈ D from the summaries µ ij ( d ) individually delivered by the panels, i ∈ [ m ], j ∈ [ s i ]. This implies adequacy as formalized in Lemma 1 below. For every d ∈ D , let µ i ( d ) = ( µ ji ( d )) j ∈ [ s i ] . Lemma 1 Suppose G i delivers its vectors of expectations µ i ( d ) , i ∈ [ m ] , d ∈D . For an algebraic CEU, if the IDSS is score separable then it is adequate. The proof of this result follows from the definition of algebraic CEU inequation (2) and the definition of score separability. Example 4 (Example 2 continued) From equation (3) we can deduce that thescore separability condition corresponds to the factorization of the expecta-tions E ( θ i θ j ), i, j ∈ [ m ], i = j , into E ( θ i ) E ( θ j ). Example 5 (Example 3 continued) Score separability corresponds to the con-ditions E ( θ θ ) = E ( θ ) E ( θ ) E ( θ θ ) = E ( θ ) E ( θ ) E ( θ ψ ) = E ( θ ) E ( ψ ) E ( θ θ θ ) = E ( θ θ ) E ( θ ) . oherent combination of probabilistic outputs for group decision making 11 Lemma 1 shows that adequacy is guaranteed whenever score separability holdsfor algebraic CEUs. This implies that the expectation of certain functions ofthe panels’ parameters separate appropriately. We first introduce conditionsthat ensure this type of separability and then in Section 5 identify classes ofmodels that give rise to algebraic CEUs. Definition 5 Let q d ( λ ( θ , d ) , . . . , λ m ( θ m , d )) be the algebraic CEU of anIDSS. An IDSS is called quasi independent if E ( q d ( λ ( θ , d ) , . . . , λ m ( θ m , d ))) = q d ( E ( λ ( θ , d )) , . . . , E ( λ m ( θ m , d ))) . This condition requires the expectation of the product of certain functions ofthe parameters overseen by different panels to be equal to the product of theindividual expectations.Often the λ ji , i ∈ [ m ], j ∈ [ s i ], are monomial functions of the panels’parameters. This was the case in Examples 2 and 3 above. It is thereforehelpful to introduce the following independence condition specific for monomialfunctions. Let < lex denote a lexicographic order (Cox et al, 2007). Definition 6 Let θ = ( θ i ) i ∈ [ m ] ∈ R m be a parameter vector and c = ( c i ) i ∈ [ n ] ∈ Z n ≥ . We say that θ entertains moment independence of order c if for any a = ( a i ) i ∈ [ n ] < lex c , a ∈ Z n ≥ , and letting θ a = θ a · · · θ a n n , it holds E ( θ a ) = Y i ∈ [ n ] E ( θ a i i ) . Example 6 In Example 5 we say that score separability holds for the EU ofExample 3 if e.g. E ( θ θ ) = E ( θ ) E ( θ ) and E ( θ θ ) = E ( θ ) E ( θ ). Thefirst requirement correspond to a moment independence of degree (1 , , θ and θ . Assume a CEU is equal to θ θ and that a momentindependence of order (2 , 2) holds. Then E (cid:0) θ θ (cid:1) = E (cid:0) θ (cid:1) E (cid:0) θ (cid:1) = E ( θ ) E ( θ ) + E ( θ ) V ( θ ) + E ( θ ) V ( θ ) + V ( θ ) V ( θ ) . (4) The same expression is obtained when using sequentially the tower rule of ex-pectations and the law of total variance under the assumption of independenceof the two parameters above (Brillinger, 1969). Therefore, the expression ob-tained under moment independence is reasonable and coincides with the oneimplied by the independence of θ and θ . However the condition we need forequation (4) to hold does not require θ and θ to be independent. Given the above definitions of new independence concepts tailored to IDSSs,we can now study situations where these can be shown to be adequate andtherefore provide a coherent, operational support tool to decision centres. Proposition 1 Let q d ( λ ( θ , d ) , . . . , λ m ( θ m , d )) be an algebraic CEU of aquasi independent IDSS. The IDSS is adequate if panel G i delivers the vectorsof expectations µ i ( d ) , for all i ∈ [ m ] and d ∈ D . This result follows by noting that quasi independence implies score sepa-rability since¯ u ( d ) = q d ( E ( λ ( θ , d )) , . . . , E ( λ m ( θ m , d ))) = X b ∈ B k b ( d ) Y i ∈ [ m ] Y j ∈ [ s i ] µ ji ( d ) . Assuming the CEU is a polynomial in the panels’ parameters, under aspecific moment independence assumption we have a more operational result. Corollary 1 Let q d ( λ ( θ , d ) , . . . , λ m ( θ m , d )) be an algebraic CEU of an IDSS, θ i = ( θ ji ) j ∈ [ s i ] and λ ji ( θ i , d ) = θ a ji i , with a ji ∈ Z s i ≥ , i ∈ [ m ] , j ∈ [ s i ] . Let a ∗ i = ( a ∗ ji ) j ∈ [ s i ] , where a ∗ ji is the greatest element in { a ji : j ∈ [ s i ] } , i ∈ [ m ] ,and let a ∗ = ( a ∗ i ) i ∈ [ m ] . Let θ = ( θ i ) i ∈ [ m ] and assume the CK-class includes amoment independence assumption of order a ∗ . The IDSS is then adequate ifpanel G i delivers the vectors of expectations µ i ( d ) , for all i ∈ [ m ] and d ∈ D . The proof of this result is given in Appendix A.1. Proposition 1 and Corol-lary 1 formalize the independence conditions required for an IDSS to be ade-quate, under the assumption of an algebraic CEU. In practice it is often thecase that a CEU is algebraic (e.g. Madsen and Jensen, 2005, and in Examples2 and 3). However, there are particular families of utility factorizations andstatistical models that ensure the associated CEU is algebraic. We define theseclasses below and prove that their associated CEU is algebraic. Definition 7 Let Y i be the vector overseen by panel G i , i ∈ [ m ]. A utilityfunction over Y , . . . , Y m is called panel separable if it factorizes as u ( y , . . . , y m , d ) = X I ∈P ([ m ]) k I ( d ) Y i ∈ I u i ( y i , d ) , where P is the power set without the empty set and k I ( d ) is a criterion weight. oherent combination of probabilistic outputs for group decision making 13 Definition 8 Under the conditions of Definition 7, a utility function over Y , . . . , Y m is called additive panel separable if it factorizes as u ( y , . . . , y m , d ) = X i ∈ [ m ] k i ( d ) u i ( y i , d ) . Under the assumption of an (additive) panel separable utility, each panelcan model its preferences over the variables under its jurisdiction using amarginal utility function of its choice. A large class of utilities, often usedin practice, are polynomial (M¨uller and Machina, 1987). For simplicity, wehere consider only the case when marginal utility functions have univariatearguments. Definition 9 A polynomial utility function over y i of degree n i is defined as u ( y i , d ) = X j ∈ [ n i ] ρ ij ( d ) y ji , where the coefficients ρ ij ( d ) ∈ R and the domain of y i need to entertain someconstraints. An explicit derivation of the required constraints can be found in Keeney and Raiffa(1976) and M¨uller and Machina (1987).The probabilistic model class we consider here is a specific structural equa-tion model (SEM) (Bowen and Guo, 2011; Wall and Amemiya, 2000), whereeach variable is defined through a polynomial function. Henceforth we callthese a polynomial SEM . SEMs were first introduced as a modelling approachin the social sciences (Westland, 2015) and are nowadays widely used especiallyin the causal literature (Pearl, 2000). Definition 10 Let Y = ( Y i ) i ∈ [ m ] be a random vector. A polynomial SEM isdefined by Y i = X a i ∈ A i θ i a i Y a i [ i − + ε i , i ∈ [ m ] , where A i ⊂ Z i − ≥ , ε i is a random error with mean zero and variance ψ i , θ i a i isa parameter, i ∈ [ m ], a i ∈ A i , and Y [ i − = ( Y j ) j ∈ [ i − , with [0] = ∅ .An alternative formulation of a polynomial SEM in terms of distributions is Y i | ( θ i , Y [ i − ) ∼ X a i ∈ A i θ i a i Y a i [ i − , ψ i ! , where θ i = ( θ i a i ) a i ∈ A i and i ∈ [ m ]. These models are suitable candidates fora CK-class since their definition is qualitative in nature and requires only thespecification of the relationships between the random variables together witha few selected moments.For polynomial SEMs and panel separable utilities the following holds. For simplicity, we assume the intercept to be equal to zero since utilities are unique upto positive affine transformations.4 Manuele Leonelli et al. Theorem 1 Assume panel G i is responsible for Y i , i ∈ [ m ] and that the CK-class of an IDSS includes a panel separable utility and a polynomial SEM.Assume also that each panel agreed to model its marginal utility with a poly-nomial utility function. Then, under quasi independence, the IDSS is scoreseparable. The proof of this theorem is given in Appendix A.2. Theorem 1 togetherwith Lemma 1 shows that IDSSs, whose CK-class includes polynomial SEMsand panel separable utilities, can uniquely compute EU scores from the in-dividual judgements of the panels. By construction, the quasi independencecondition of Theorem 1 actually corresponds to a moment independence. Theorder of such independence depends on the polynomial form of both the SEMand the utility function. In Section 6 we identify the order of the moment inde-pendence condition required for adequacy in a subclass of polynomial SEMs. The subclass of polynomial SEMs we study next consists of BN models whereeach vertex is defined by a linear regression over its parents. For this modelclass we are able to deduce the exact moment independence required for ade-quacy. Definition 11 A BN over a directed acyclic graph (DAG) G with vertex set V ( G ) = { i : i ∈ [ m ] } and edge set E ( G ) is a linear SEM if each variable Y i isdefined as Y i = θ i + X j ∈ Π i θ ji Y j + ε i , (5)where Π i is the parent set of i in G , ε i is a random error with mean zero andvariance ψ i and θ i , θ ji ∈ R .Although such models are often multivariate Gaussian, in general this doesnot need to be the case.As Sullivant et al (2010), we consider regression parameters as indetermi-nates in a polynomial function. We associate these to edges and vertices of theunderlying DAG. For i ∈ [ m ], let θ ′ i = θ i + ε i be the indeterminate associatedto the vertex i , whilst θ ij is associated to the edge ( i, j ) ∈ E ( G ). Define P i to be the set of rooted paths in G ending in Y i . A rooted path of length n + 1from i to j n is a sequence comprising of a vertex in V ( G ) and n distinct edgesin E ( G ) is such that ( i , ( i , j ) , . . . , ( i k , j k ) , ( i k +1 , j k +1 ) , . . . , ( i n , j n )) , where j k = i k +1 , k ∈ [ n − i k , j k ∈ [ m ]. For every element P ∈ P i we define θ P as θ P = Y i ∈ P θ ′ i Y ( i,j ) ∈ P θ ij , and, as Sullivant et al (2010), we call θ P the path monomial . We think of θ ′ i as a parameter although this consists of the sum of a parameter θ andan error ε i . Note however that from a Bayesian viewpoint these are both random variables.oherent combination of probabilistic outputs for group decision making 15 Example 7 Consider the DAG in Figure 1 associated to the food security ex-ample. For instance, the set P is equal to { (3) , (2 , (2 , , (1 , (1 , , (1 , (1 , , (2 , } , and θ ′ , θ ′ θ , θ ′ θ and θ ′ θ θ are the corresponding path monomials.We call algebraic substitution the process of substituting the linear regres-sion expression of a random variable of the DAG, as in equation (5), into theone of the child variable. An example illustrates this process. Example 8 For the DAG in Figure 1, a linear SEM is defined as Y = θ + θ Y + ε , Y = θ + θ Y + θ Y + ε ,Y = θ + θ Y + ε , Y = θ + ε . (6)An algebraic substitution of the variables in the definition of Y entails Y = θ + θ ( θ + ε ) + θ ( θ + θ Y + ε ) + ε = θ ′ + θ θ ′ + θ θ ′ + θ θ Y . The additional algebraic substitution of Y gives Y = θ ′ + θ θ ′ + θ θ ′ + θ θ θ ′ . (7)It is of special interest that after this substitution Y is now uniquely definedin equation (7) in terms of path monomials. Proposition 2 formalizes that thisoccurs for any variable of a DAG defined as a linear SEM and links algebraicsubstitutions to conditional expectation operators. Proposition 2 For a linear SEM over a DAG G , through algebraic substitu-tions each variable Y i , i ∈ [ m ] , can be written as Y i = X P ∈ P i θ P , (8) and letting θ i = ( θ ′ i , θ ji ) j ∈ Π i and θ = ( θ i ) i ∈ [ m ] , i ∈ [ m ] , we then have that E ( Y i | θ , d ) = X P ∈ P i θ P . (9)The proof of this result is given in Appendix A.3. Lemma 2 Consider a linear SEM over a DAG G . Assume that u ( y ) can bewritten as u i ( y ) = X i ∈ [ m ] k i ( d ) u i ( y i ) . and that u i is a polynomial utility function of degree n i . Then the CEU isalgebraic and can be written as ¯ u ( d | θ ) = X i ∈ [ m ] k i ( d ) X j ∈ [ n i ] ρ ij ( d ) X | a i | = j (cid:18) j a i (cid:19) θ a i P i , (10) where a i = ( a ij ) j ∈ [ P i ] ∈ Z P i ≥ , θ P i = Q P ∈ P i θ P , (cid:0) j a i (cid:1) is a multinomial coeffi-cient, P i is the number of elements in P i and | a i | = P j ∈ P i a ij . This result follows by noting, from Equation (9), that the CEU equals E (¯ u ( d | θ )) = X i ∈ [ m ] k i ( d ) X j ∈ [ n i ] ρ ij ( d ) (cid:16) X P ∈ P i θ P (cid:17) j , and then applying the Multinomial Theorem (Cox et al, 2007).Equation (10) is an instance of the computation of the moments of a de-composable function as studied in Cowell et al (1999) and Nilsson (2001). InLemma 2 we explicitly deduce the required monomials and their degree andin Section 6.2 we generalise these results to multilinear functions.Lemma 2 has an appealing intuitive graphical interpretation which is par-ticularly useful for the computation of the EU’s monomials. The j -th noncentral moment of any Y i can be written as the sum of the monomials θ P i with degree j . By the properties of multinomial coefficients, this sum can bethought of as the sum over the set of unordered j -tuples of rooted paths end-ing in Y i . Let P ji be the set of unordered j -tuples from P i . For a P ∈ P ji , themultinomial coefficient in equation (10) counts the distinct permutations ofthe elements of P , denoted as n P i . We then have that, X | a i | = j (cid:18) j a i (cid:19) θ a i P i = X P ∈ P ji n P i Y p ∈ P θ p . (11)Equation (11) shows an intuitive graphical interpretation of equation (10), asillustrated in the following example. oherent combination of probabilistic outputs for group decision making 17 Example 9 For the vertex 4 in the DAG of Figure 1 the set P is equal to { (4) , (1 , (1 , } . From the left hand side of equation (11), Y can be writtenas θ ′ + θ ′ θ + 2 θ ′ θ θ ′ . (12)This polynomial can be also deduced by simply looking at the DAG. Note that P = n(cid:0) (4) , (4) (cid:1) , (cid:0) (1 , (1 , , (1 , (1 , (cid:1) , (cid:0) (4) , (1 , (1 , (cid:1)o . The first and second monomial in equation (12) correspond to the first andsecond element of P respectively, whilst the last elements of this set, havingtwo distinct permutations, is associated to the third monomial in equation(12).From Lemma 2 we can deduce the independences needed for adequacy inIDSSs whose CK class includes a BN defined as a linear SEM. Note that θ P i ,defined as Q P ∈ P i θ P , might include multiple times the same parameter, θ say,if θ is associated to a vertex/edge appearing in different paths ending in Y i .We let θ G i be the simplified version of θ P i where each parameter appears onlyonce and θ c i G i is the simplified version of θ a i P i where each element of c i equalsthe sum of the a ij associated to the same parameter. Let r i be the number ofdistinct parameters in θ P i . Theorem 2 Suppose the CK-class of an IDSS includes a linear SEM over aDAG G , where panel G i oversees Y i , i ∈ [ m ] , and an additive panel separableutility function. Suppose panel G i agreed to use a polynomial utility functionof degree n i , i ∈ [ m ] . If θ G i entertains moment independence of order c i forevery c i ∈ Z r i ≥ such that | c i | = n i and i ∈ [ m ] , then the IDSS is adequate. The proof of this result is given in Appendix A.4. Theorem 2 gives the spe-cific moment independences necessary for the IDSS’s adequacy. By requestingthe collective to agree on these independences, the IDSS can then quickly pro-duce a unique EU score for each policy. Panels are informed on the summariesthey need to deliver to the IDSS since these are the only quantities of whichthe EU is a function. An illustration of the usefulness of this result in practiceis given in Section 7.6.2 Multilinear factorizations.By approaching the combination of outputs in BN models from an algebraicviewpoint, we are able to generalize in a straightforward manner the resultsin Section 6.1 about additive/decomposable factorizations so that they applyto multilinear functions. Let P i = m i , i.e. there are m i rooted paths endingin Y i . Let l i = ( l ij ) j ∈ [ m i ] ∈ Z m i ≥ be the vector listing the lengths of such pathsand l = ( l i ) i ∈ [ m ] . For a vector a = ( a i ) i ∈ [ m ] ∈ Z m , we write l ≃ a if both | a | = | l | and, for all i ∈ [ m ], | l i | = a i . Lemma 3 For a linear SEM over a DAG G , suppose the utility function u ( y , d ) can be written u ( y , d ) = X I ∈P ([ m ]) k I ( d ) Y i ∈ I u i ( y i , d ) . Now suppose u i is a polynomial utility function of degree n i , n = ( n i ) i ∈ [ m ] , i ∈ [ m ] and is a vector of dimension m with only zero entries. The CEU isthen algebraic and can be written as ¯ u ( d | θ ) = X < lex a ≤ lex n c a ( d ) X l ≃ a (cid:18) | a | l (cid:19) θ l P , (13) where c a ( d ) = k J ( d ) Q j ∈ J ρ ja j ( d ) , J = { j ∈ [ m ] : a j = 0 } , and θ P = Q i ∈ [ m ] θ P i . The proof of Lemma 3 is given in Appendix A.5 Lemma 3 makes a signif-icant generalization to the theory of the computation of moments in decom-posable/additive functions of Cowell et al (1999) and Nilsson (2001) extendingthese well-known formulae so that they apply in the much wider context ofmultilinear functions of BNs defined as linear SEMs. It is interesting to notethat Lemma 3 is connected to the propagation algorithms first developed inLauritzen (1992) to compute the first two moments of certain chain graphs.Here, focusing on a specific class of continuous DAG models, we are able toexplicitly compute, through algebraic substitution, not only the first two mo-ments, but also any other higher order moment of the distribution associatedwith the graph.Using again the properties of multinomial coefficients, we can relate equa-tion (13) to the topology of the graph and its rooted paths. For an a ∈ Z m ≥ ,let P a = × a i =0 P a i i , where × denotes the Cartesian product. This set consistsof the unordered | a | -tuples of rooted paths, where in each tuple there are a i paths ending at Y i . For each element P ∈ P a , let n P = P a i =0 n P i . Then wehave that, following the same reasoning outlined for additive factorizations, X l ≃ a (cid:18) | a | l (cid:19) θ l P = X P ∈ P a n P Y p ∈ P θ p . Here n P counts the total number of permutations in the sets P i , i ∈ [ m ].This representation of non-central moments in terms of paths extends thecomputation of the second central moment of Sullivant et al (2010) via thetrek rule to generic non-central moments. Example 10 Consider the expectation E ( Y Y ) for the variables in the DAGof Figure 1. This expectation, being the associated monomial of degree 4, canbe computed by looking at all distinct tuples of rooted paths of dimension four,where two paths end in Y and two in Y . These are summarized in Table 1. The oherent combination of probabilistic outputs for group decision making 19((2) , (2) , (4) , (4))((1 , (1 , , (2) , (4) , (4))((1 , (1 , , (1 , (1 , , (4) , (4))((2) , (2) , (1 , (1 , , (4))((1 , (1 , , (2) , (1 , (1 , , (4))((1 , (1 , , (1 , (1 , , (1 , (1 , , (4))((2) , (2) , (1 , (1 , , (1 , (1 , , (1 , , (2) , (1 , (1 , , (1 , (1 , , (1 , , (1 , (1 , , (1 , (1 , , (1 , (1 , Table 1 Tuples of dimension 4 with two paths ending in Y and two more ending in Y inthe graph in Figure 1. associated conditional expectation can be written as the following polynomial,where the i -th monomial corresponds to the tuple in the i -th row of Table 1:¯ u ( d | θ ) = θ ′ θ ′ + 2 θ θ ′ θ ′ + θ θ ′ + 2 θ ′ θ θ ′ + 4 θ θ ′ θ θ + 2 θ θ θ ′ + θ ′ θ + 2 θ θ ′ θ + θ θ . Note for example that θ θ ′ θ ′ has coefficient 2 since the paths ( Y ) and( Y , ( Y , Y )) can be permuted, whilst θ θ ′ θ θ has coefficient 4 since bothpairs of paths ( Y ) and ( Y , ( Y , Y )) and ( Y ) and ( Y , ( Y , Y )) can be per-muted.Just as in the additive case, we are now able to deduce the independencesrequired for score separability of an IDSS whose structural consensus includesa BN. We let θ b G be the simplified version of θ a P where parameters only appearonce and the exponent are appropriately summed. Theorem 3 Suppose that the CK-class of an IDSS includes a linear SEMover a DAG G , where panel G i oversees Y i , i ∈ [ m ] , and a panel separableutility. Suppose panel G i agreed to use a polynomial utility function of degree n i , i ∈ [ m ] . If, for every b ≃ n , where n = ( n i ) i ∈ [ m ] ∈ Z m ≥ , θ G entertainsmoment independence of order b , then the IDSS is score separable. The proof of this result is given in Appendix A.6. Theorem 3 ensures ade-quacy for a large class of IDSSs based on flexible multilinear utility factoriza-tions and commonly used BNs defined as linear SEMs. To illustrate the application of our results in a real-world example, we considerthe food security network discussed in Section 2.3 and reported in Figure 1.Suppose the variables are each under the jurisdiction of a different panel ofexperts. Jointly they reach a consensus to: k ρ θ , k ρ θ k ρ ψ k ρ θ k ρ θ θ k ρ θ k ρ ψ k ρ θ θ k ρ ψ θ k ρ θ θ θ k ρ θ k ρ θ θ k ρ θ θ θ k ρ θ θ k ρ θ k ρ ψ k ρ θ θ k ρ θ ψ k ρ θ θ k ρ θ θ θ θ k ρ ψ θ k ρ ψ θ θ k ρ θ θ θ k ρ θ θ θ k ρ θ θ θ k ρ ψ θ θ θ k ρ θ θ θ θ k ρ θ θ θ θ k ρ θ θ θ θ k ρ θ k ρ θ θ k ρ θ k ρ ψ k ρ θ θ k ρ ψ θ k ρ θ θ θ Table 2 Monomials of the CEU for the utility class U . – investigate the effectiveness of an increase ( d ), decrease ( d ) or not achange ( d ) of the number of pupils eligible for free school meals, with D = { d , d , d } (decision consensus); – model the conditional dependences between the four random variablesdeemed relevant with the BN reported in Figure 1 (structural consensus); – consider two utility classes of utility factorizations - the first with prefer-entially independent attributes (class U ) as in equation (1), the secondenjoying a multilinear utility factorization (class U ) defined as u ( d, y ) = X I ∈P ([4]) k I ( d ) Y i ∈ I u i ( y i , d ) . These agreements give the CK-class for this application. Next suppose thateach panel decides to model the variable under its jurisdiction via a linear SEMas specified in equation (6) and to model its marginal utility with a polynomialutility function of degree two, i.e. u i ( y i , d ) = ρ i ( d ) y i + ρ i ( d ) y i , i ∈ [4].The two questions that we next address are the following: what indepen-dences do the panels need to be prepared to make for the IDSS to be adequate?What summaries do they have to deliver? The answer depends on the class ofutility functions chosen. We thus first focus on the simpler class U of prefer-entially independent attributes.Through the process of algebraic substitutions, as formalized in Lemma 2,the CEU for the utility class U can be computed as the sum of the monomi-als reported in Table 2 where we left the dependence on the decision d ∈ D implicit. Given this list of monomials, it is then straightforward the identifythe independences required by the IDSS to be able to compute the EU ofany available policy as a function of beliefs individually delivered by panels.Specifically, the moment independences summarized in Table 3 need to hold.Assuming these, then each panel can deliver independently the required be-liefs to derive appropriate EU scores uniquely. So, for instance, if the panelsdelivered the beliefs reported in Appendix B.1, then the IDSS would recom-mend that the number of pupils eligible for free school meals is increased sincethe EU of this policy equals 1 . 87, whilst for d and d this is 0 . 51 and 0 . U of multilinear utilities. Whilst for preferentially independent attributes the oherent combination of probabilistic outputs for group decision making 21 E ( θ θ ) = E ( θ ) E ( θ ) E ( θ θ ) = E ( θ ) E ( θ ) E ( ψ θ ) = E ( ψ ) E ( θ ) E ( θ θ θ ) = E ( θ ) E ( θ θ ) E ( θ θ ) = E ( θ ) E ( θ ) E ( θ θ θ ) = E ( θ ) E ( θ ) E ( θ ) E ( θ θ ) = E ( θ ) E ( θ ) E ( θ θ ) = E ( θ ) E ( θ ) E ( θ ψ ) = E ( θ ) E ( ψ ) E ( θ θ ) = E ( θ ) E ( θ ) E ( θ θ θ θ ) = E ( θ ) E ( θ ) E ( θ θ ) E ( θ ψ ) = E ( θ ) E ( ψ ) E ( ψ θ θ ) = E ( ψ ) E ( θ ) E ( θ ) E ( θ θ θ ) = E ( θ ) E ( θ ) E ( θ ) E ( θ θ θ ) = E ( θ ) E ( θ θ ) E ( θ θ θ ) = E ( θ ) E ( θ θ ) E ( ψ θ θ θ ) = E ( ψ ) E ( θ ) E ( θ θ ) E ( θ θ θ θ ) = E ( θ ) E ( θ ) E ( θ θ ) E ( θ θ θ θ ) = E ( θ ) E ( θ θ ) E ( θ ) E ( θ θ θ θ ) = E ( θ ) E ( θ θ ) E ( θ ) E ( θ θ ) = E ( θ ) E ( θ ) E ( θ θ ) = E ( θ ) E ( θ ) E ( ψ θ ) = E ( ψ ) E ( θ ) E ( θ θ θ ) = E ( θ ) E ( θ θ ) Table 3 Moment independences required by the IDSS for adequacy. CEU has 36 monomials (in Table 2), in this case the CEU can be shown tohave 3869 monomials. In this case the computer algebra software Mathematicainstantaneously gives us the CEU polynomial using the simple code reportedin Appendix C. The output CEU can then be used to identify the requiredmoment independences and panels’ beliefs. For instance, the parameter θ has degree up to 8 in the polynomial CEU when U is used, whilst for U its maximum degree was 2. Using this more general class of utilities and thespecifications given in Appendix B.2, the IDSS would again recommend thatthe number of pupils eligible for free school meals is increased since the EU ofthis policy equals 0 . 97, whilst for d and d this is 0 . 16 and 0 . 37 respectively. The framework of IDSSs is capable of supporting decision making in situationswhere judgements come from different panels of experts having jurisdictionover different aspects of the system. In this paper we have relaxed many ofthe assumptions guaranteeing coherence in this type of systems (Smith et al,2015) by exploiting the polynomial structure of certain statistical models andutility functions and illustrated their usefulness in a practical application.In the particular context where the structural consensus includes a BNmodel, the process of algebraic substitution has proven fundamental in iden-tifying the required summaries and independence relations. We have encour-aging results, to be reported in future work, towards a generalization of such re-cursions in dynamic models, as the multiregression dynamic model (Queen and Smith,1993), where expressions for the moments’ forecasts can be deduced in closedform. Furthermore, when each vertex of the BN is no longer a random variablebut a random vector (for example when a variable is measured at different ge-ographic location), the theory of tensors (McCullagh, 1987) can be employedto concisely report the associated EU expressions. We plan to develop such amethodology in future work. References Barons MJ, Smith JQ, Leonelli M (2015) Decision focused inference on net-worked probabilistic systems: with applications to food security. In: Pro-ceedings of the Joint Statistical Meeting, pp 3220–3233Barons MJ, Wright SK, Smith JQ (2016) Eliciting probabilistic judgementsfor integrating decision support systems. Tech. rep., CRISM 16-05Blangiardo M, Cameletti M (2015) Spatial and spatio-temporal Bayesian mod-els with R-INLA. John Wiley & Sons, ChichesterBowen NK, Guo S (2011) Structural equation modeling. Oxford UniversityPress, OxfordBrillinger DR (1969) The calculation of cumulants via conditioning. Annals ofthe Institute of Statistical Mathematics 21:215–218Cowell RG, Dawid AP, Lauritzen SL, Spiegelhalter DJ (1999) Probabilisticnetworks and expert systems. Springer-Verlag, New YorkCox DA, Little J, O’Shea D (2007) Ideals, varieties and algorithms. Springer,New YorkDawid AP (1979) Conditional independence in statistical theory. Journal ofthe Royal Statistical Society Series B 41:1–31Dowler E, Lambie-Mumford H (2015) How can households eat in austerity?challenges for social policy. Social Policy and Society 14:417–428Drewnowski A, Specter SE (2004) Poverty and obesity: the role of energydensity and energy costs. The American Journal of Clinical Nutrition 79:6–16Efendigil T, ¨On¨ut S, Kahraman C (2009) A decision support system for de-mand forecasting with artificial neural networks and neuro-fuzzy models: Acomparative analysis. Expert Systems with Applications 36:6697–6707Faria AE, Smith JQ (1997) Conditionally externally Bayesian pooling opera-tors in chain graphs. Annals of Statistics 25:1740–1761Farr AC, Simpson D, Ruggeri F, Mengersen K (2014) Combining opinions foruse in Bayesian networks: a measurement error approach. Tech. rep., QUTEprints - 79211Freeman G, Smith JQ (2011) Bayesian MAP model selection of chain eventgraphs. Journal of Multivariate Analysis 102:1152–1165French S, Maule J, Papamichail KN (2009) Decision behaviour, analysis andsupport. Cambridge University Press, CambridgeHernandez T, Bennison D (2000) The art and science of retail location deci-sions. International Journal of Retail & Distribution Management 28:357–367Jensen FV, Nielsen TD (2013) Probabilistic decision graphs for optimizationunder uncertainty. Annals of Operations Research 204:223–248Johnson S, Mengersen K (2012) Integrated Bayesian network framework formodeling complex ecological issues. Integrated Environmental Assessmentand Management 8(3):480–490Keeney RL, Raiffa H (1976) Decisions with multiple objectives: preferencesand value trade-offs. Cambridge University Press, Cambridge oherent combination of probabilistic outputs for group decision making 23 Kennedy MC, O’Hagan A (2001) Bayesian calibration of computer models.Journal of the Royal Statistical Society Series B 63(3):425–464Kitchen S, Tanner E, Brown V, Colin P, Crawford C, Deardon L, Greaves E,Purdon S (2013) Evaluation of the free school meals pilot: impact report.Tech. rep., Department for Education, DFERR227Koller D, Pfeffer A (1997) Object-oriented Bayesian networks. In: Proceedingsof the 13th Conference on Uncertainty in Artificial Intelligence, pp 302–313Lauritzen SL (1992) Propagation of probabilities, means and variances inmixed graphical association models. Journal of the American Statistical As-sociation 87:1098–1108Leonelli M, Smith JQ (2013) Using graphical models and multi-attribute util-ity theory for probabilistic uncertainty handling in large systems, with appli-cation to the nuclear emergency management. In: Proceedings of ICDEW,pp 181–192Leonelli M, Smith JQ (2015) Bayesian decision support for complex systemswith many distributed experts. Annals of Operation Research 235:517–542Leonelli M, Smith JQ (2017) Directed expected utility networks. DecisionAnalysis 17(2):108–125Loopstra R, Reeves A, Taylor-Robinson D, Barr B, McKee M, Stuckler D(2015) Austerity, sanctions, and the rise of food banks in the uk. BMJ350:h1775Madsen AL, Jensen FV (2005) Solving linear-quadratic conditional Gaus-sian influence diagrams. International Journal of Approximate Reasoning38:263–282Mahoney S, Laskey K (1996) Network engineering for complex belief networks.In: Proceedings of the 12th International Conference on Uncertainty in Ar-tificial Intelligence, Morgan Kaufmann Publishers Inc., pp 389–396McCullagh P (1987) Tensor methods in statistics. Chapman and Hall, LondonM¨uller SM, Machina MJ (1987) Moment preferences and polynomial utility.Economics Letters 23:349–353Murphy KP (2002) Dynamic Bayesian networks: representation, inference andlearning. PhD thesis, University of California, BerkeleyNilsson D (2001) The computation of moments of decomposable functionsin probabilistic expert systems. In: Proceedings of the Third InternationalSymposium on Adaptive Systems, pp 116–121Pearl J (1988) Probabilistic inference in intelligent systems. Morgan Kauf-mann, San MateoPearl J (2000) Causality: models, reasoning and inference. Cambridge Univer-sity Press, CambridgeQueen CM, Smith JQ (1993) Multiregression dynamic models. Journal of theRoyal Statistical Society Series B 55(4):849–870Smith JQ (1996) Plausible Bayesian games. In: Bayesian Statistics 5, pp 387–406Smith JQ, Anderson PE (2008) Conditional independence and chain eventgraphs. Artificial Intelligence 172(1):42–68 Smith JQ, Barons MJ, Leonelli M (2015) Coherent frameworks for statis-tical inference serving integrating decision support systems. Tech. rep.,arXiv:1507.07394Spiegelhalter DJ, Lauritzen SL (1990) Sequential updating of conditional prob-abilities on directed graphical structures. Networks 20:579–605Sullivant S, Talaska K, Draisma J (2010) Trek separation for Gaussian graph-ical models. Annals of Statistics 38:1665–1685Wakker PP (2008) Explaining the characteristics of the power (CRRA) utilityfamily. Health Economics 17:1329–1344Wall MM, Amemiya Y (2000) Estimation for polynomial structural equationmodels. Journal of the American Statistical Association 95(451):929–940Westland JC (2015) Structural equation modeling: from paths to networks.Springer, New YorkWolfram Research, Inc (2017) Mathematica, Version 11.1. Champaign A Proofs A.1 Proof of Corollary 1 Adequacy is guaranteed if the EU function can be written in terms of µ ji ( d ) and k b ( d ), i ∈ [ m ], j ∈ [ s i ] and d ∈ D . Note that¯ u ( d ) = E ( q d ( λ ( θ , d ) , . . . , λ m ( θ m , d )))= X b ∈ B k b ( d ) E (cid:16) Y i ∈ [ m ] Y j ∈ [ s i ] λ ji ( θ i , d ) b j,i (cid:17) = X b ∈ B k b ( d ) E (cid:16) Y i ∈ [ m ] Y j ∈ [ s i ] θ a ji i (cid:17) . The argument of this expectation is a monomial of multi-degree lower or equal to a ∗ . Momentindependence then implies that ¯ u ( d ) = P b ∈ B k b ( d ) Q i ∈ [ m ] Q j ∈ [ s i ] µ ji ( d ) , and the resultfollows. A.2 Proof of Theorem 1 Fix a policy d ∈ D and suppress this dependence. Under the assumptions of the theorem,the utility function can be written as u ( y ) = X I ∈P ([ m ]) k I X i ∈ I X j ∈ [ n i ] ρ ij y ji . (14)Note also that we can rewrite (14) as u ( y ) = ˆ u ( y [ m − ) + ˆ u ( y m ) , where ˆ u ( y [ m − ) = X I ∈P ([ m − k I Y i ∈ I (cid:16) X j ∈ [ n i ] ρ ij y ji (cid:17) , ˆ u ( y m ) = X I ∈P m ([ m ]) k I Y i ∈ I (cid:16) X j ∈ [ n i ] ρ ij y ji (cid:17) , (15)oherent combination of probabilistic outputs for group decision making 25and P m ([ m ]) = P ([ m ]) ∩{ m } . Calling θ the overall parameter vector of the IDSS, the CEUfunction can be written applying sequentially the tower rule of expectation as E ( u ( Y ) | θ ) = E Y | θ (cid:16) · · · E Y m − | Y [ m − , θ (cid:0) ˆ u ( y [ m − ) + E Y m | Y [ m − , θ (ˆ u ( y m )) (cid:1)(cid:17) . (16)From equation (15), the definition of a polynomial SEM and observing that the power of apolynomial is still a polynomial function in the same arguments, it follows that E Y m | Y [ m − , θ (ˆ u ( y m )) = p m ( Y [ m − , θ ) , where p m is a generic polynomial function. Thus ˆ u ( Y [ m − ) + E Y m | Y [ m − , θ (ˆ u ( y m )) is alsoa polynomial function in the same arguments. Following the same reasoning, we have that E Y m − | Y [ m − , θ (cid:16) ˆ u ( y [ m − ) + E Y m | Y [ m − , θ (ˆ u ( y m )) (cid:17) = p m − ( Y [ m − , θ ) , where p m − is a generic polynomial function. Therefore the same procedure can be appliedto all the expectations in (16). So E ( u ( Y ) | θ ) = p ( θ ) , where p is a generic polyno-mial function. This defines by construction an algebraic CEU, where the functions λ ij aremonomials. Quasi independence and Lemma 1 then guarantee score separability holds. A.3 Proof of Proposition 2 We prove equation (8) via induction over the indices of the variables. Let Y be a root of G . Thus Y = θ ′ , where θ ′ is the monomial associated to the only rooted path endingin Y , namely ( Y ). Assume the result is true for Y n − and consider Y n . By the inductivehypothesis we have that, if i < j whenever i ∈ Π j , Y n = θ ′ n + X i ∈ Π n θ in Y i = θ ′ n + X i ∈ Π n θ in X P ∈ P i θ P . (17)Note that every rooted path ending in Y n is either ( Y n ) or consists of a rooted path ending in Y i , i ∈ Π n , together with the edge ( Y i , Y n ). From this observation the result then follows byrearranging the terms in equation (17). Equation (9) can be proven via the same inductiveprocess noting that E ( Y | θ , d ) = θ ′ and E ( Y n | θ , d ) = θ ′ n + P i ∈ Π n θ in E ( Y i | θ , d ). A.4 Proof of Theorem 2 Under the assumptions of the theorem, the CEU function can be written as in equation(10). From the linearity of the expectation operator we have that E (¯ u ( d | θ )) = X i ∈ [ m ] ,j ∈ [ n i ] k i ( d ) ρ ij ( d ) X | a i | = j (cid:16) j a i (cid:17) E (cid:16) θ a i P i (cid:17) = X i ∈ [ m ] ,j ∈ [ n i ] k i ( d ) ρ ij ( d ) X | c i | = j (cid:16) j c i (cid:17) E (cid:16) θ c i G i (cid:17) . Applying moment independence and letting V i and E i be the sets of distinct vertices andedges, respectively, for all the elements P ∈ P i , we have that E (¯ u ( d | θ )) = X i ∈ [ m ] ,j ∈ [ n i ] , | c i | = j k i ( d ) ρ ij ( d ) (cid:16) j c i (cid:17) Y l ∈ V i E (cid:16) θ ′ c il l θ c iChl lCh l (cid:17) Y ( j,k ) ∈ E i \ ( l,Ch l ) E (cid:16) θ c ik jk (cid:17) , where c ik is the element of c i associated to θ jk and Ch l is the index of a children of thevertex l . The thesis then follows since each of these expectations is delivered by an individualpanel.6 Manuele Leonelli et al. A.5 Proof of Lemma 3 To prove this result we first show that under the assumptions of the lemma the utilityfunction can be written as u ( y , d ) = X < lex a ≤ lex n c a ( d ) y a , (18)and then prove that Y a = X l ≃ a (cid:16) | a | l (cid:17) θ l P . (19)The lemma then follows by substituting into equation (18) for y a given in equation (19).Fix a policy d ∈ D and suppress this dependence. We prove equation (18) via inductionover the number of vertices of the DAG. If the DAG has only one vertex then u ( y ) = k X i ∈ n ρ i y i . This can be seen as an instance of equation (18). Assume the result holds for a networkwith n − u ( y ) = X I ∈P ([ n − k I Y i ∈ I u i ( y i ) + X I ∈P n ([ n ]) k I Y i ∈ I \{ n } u i ( y i ) u n ( y n ) + k n u n ( y n ) . (20)The first term on the rhs of (20) is by inductive hypothesis equal to the sum of all the possiblemonomial of degree a = ( a , . . . , a n − , 0) where 0 ≤ a i ≤ n i , i ∈ [ n ]. The other terms onlyinclude monomials such that the exponent of y n is not zero. Letting n n − = ( n i ) i ∈ [ n − , y [ n − = Q i ∈ [ n − y i and u ′ = P I ∈P n ([ n ]) k I Q i ∈ I \{ n } u i ( y i ) u n ( y n ) + k n u n ( y n ), we nowhave that u ′ = X < lex a ≤ lex n n − c a y a [ n − (cid:18) X i ∈ [ n n ] ρ ni y in (cid:19) + k n u n ( y n )= X < lex a ≤ lex n n − i ∈ [ n n ] c a ρ ni y a [ n − y in + k n u n ( y n ) = X ′ < lex a ≤ lex n n a n =0 c a y a [ n ] . (21)Therefore, equation (18) follows from equations (20) and (21). To prove equation (19) notethat the monomial Y a can be written as Y α = Y i ∈ [ m ] Y a i i = Y i ∈ [ m ] X | l i | = a i (cid:16) a i l i (cid:17) θ l i P i = X l ≃ a θ l P Y i ∈ [ m ] (cid:16) a i l i (cid:17) . Equation (19) then follows by noting that Y i ∈ [ m ] (cid:16) a i l i (cid:17) = Q i ∈ [ m ] a i ! Q i ∈ [ m ] Q j ∈ [ n i ] l ij ! = (cid:16) | a | l (cid:17) . A.6 Proof of Theorem 3 Under the conditions of the theorem, the CEU function can be written as in (13). Thelinearity of the expectation operator than implies that E (¯ u ( d | θ )) = X < lex a ≤ lex nl ≃ a c a (cid:16) | a | l (cid:17) E (cid:16) θ l P (cid:17) = X < lex b ≤ lex nl ≃ b c b (cid:16) | b | l (cid:17) E (cid:16) θ l G (cid:17) . oherent combination of probabilistic outputs for group decision making 27Applying moment independence and letting V tot and E tot be the sets of distinct verticesand edges, respectively, for all the elements P ∈ P = ∪ i ∈ [ m ] P i , we then have that for any l ≃ b E (cid:16) θ l G (cid:17) = Y t ∈ V tot E (cid:16) θ ′ l it t θ l iCht tCh t (cid:17) Y ( j,k ) ∈ E tot \ ( t,Ch t ) E (cid:16) θ l ik jk (cid:17) . Score separability then follows since each of these expectations is delivered by an individualpanel. B Numerical specifications for the food security example B.1 Utility class U – Probabilistic panel specifications that depend on the decision taken: E ( θ ) E ( ψ ) E ( θ ) E ( ψ ) E ( θ ) E ( ψ ) E ( θ ) d d -2 4 -5 5 -6 20 2 d -0.5 3 10 4 3 15 7 – Probabilistic panel specifications independent of the decision taken: E ( θ ) = 5, E ( θ ) = 17, E ( θ ) = 10, E ( θ ) = 10, E ( ψ ) = 20, V ( θ ) = 1, V ( θ ) = 1, V ( θ ) = 1, V ( θ ) = 1, V ( θ ) = 1, V ( θ ) = 1, V ( θ ) = 3, V ( θ ) = 2, V ( θ ) = 2, – Criterion weights and terms in the utility functions k = 0 . k = 0 . k = 0 . k = 0 . ρ = − ρ = 1, ρ = 2, ρ = 10, ρ = 8, ρ = 0 . ρ = 3, ρ = − B.2 Utility class U In the multilinear case higher moments are required. Here we assume that these can becomputed from the first two moments in Appendix B.1 using the recursions of normaldistributions. The only specifications that change for this second class are the criterionweights given in the following table. k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . k = 0 . C Code for the multilinear factorization y4 := t04 + t14*y1 + e4;y3 := t03 + t13*y1 + t23*y2 + e3;y2 := t02 + t12*y1 + e2;y1 := t01 + e1;u4 := c4*y4 ∧ ∧ ∧ Notice that these values are then normalized to give utility functions between 0 and 1.8 Manuele Leonelli et al. u1 := c1*y1 ∧ { e4 -> 0, e4 ∧ } ];eu3 := ReplaceAll[Collect[eu4, e3], { e3 -> 0, e3 ∧ } ];eu2 := ReplaceAll[Collect[eu3, e2], { e2 -> 0, e2 ∧ } ];eu1 := ReplaceAll[Collect[eu2, e1], { e1 -> 0, e1 ∧ }}