Daniel Simpson

Penalising model component complexity:A principled, practical approach to constructing priors

In this paper, we introduce a new concept for constructing prior distributions. We exploit the natural nested structure inherent to many model components, which defines the model component to be a flexible extension of a base model. Proper priors are defined to penalise the complexity induced by deviating from the simpler base model and are formulated after the input of a user-defined scaling parameter for that model component, both in the univariate and the multivariate case. These priors are invariant to reparameterisations, have a natural connection to Jeffreys’ priors, are designed to support Occam’s razor and seem to have excellent robustness properties, all which are highly desirable and allow us to use this approach to define default prior distributions. Through examples and theoretical results, we demonstrate the appropriateness of this approach and how it can be applied in various situations.

Origins of stereoselectivity in optically pure phenylethaniminopyridinetris-chelates M(NN′)3n+ (M = Mn, Fe, Co, Ni and Zn)

One-pot reactions of 2-pyridinecarboxaldehyde, chiral phenylethanamines and Fe(II) give single diastereomer fac diimine complexes at thermodynamic equilibrium so that no chiral separations are required (d.r. > 200 : 1). The origins of this stereoselectivity are partly steric and partly a result of the presence of three sets of inter-ligand parallel-offset π-stacking interactions. Mn(II), Co(II), Co(III), Ni(II) and Zn(II) give similar fac structures, alongside the imidazole analogues for Fe(II). While most of the complexes are paramagnetic, the series of molecular structures allows us to assess the influence of the π-stacking present, and there is a strong correlation between this and the M-N bond length. Fe(II) is close to optimal. For the larger Zn(II) ion, very weak π-stacking leads to poorer measured stereoselectivity (NMR) but this is improved with increased solvent polarity. The mechanism of stereoselection is further investigated via DFT calculations, chiroptical spectroscopy and the use of synthetic probes.

On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods

A large number of statistical models are “doubly-intractable”: the likelihood normalising term, which is a function of the model parameters, is intractable, as well as the marginal likelihood (model evidence). This means that standard inference techniques to sample from the posterior, such as Markov chain Monte Carlo (MCMC), cannot be used. Examples include, but are not confined to, massive Gaussian Markov random fields, autologistic models and Exponential random graph models. A number of approximate schemes based on MCMC techniques, Approximate Bayesian computation (ABC) or analytic approximations to the posterior have been suggested, and these are reviewed here. Exact MCMC schemes, which can be applied to a subset of doubly-intractable distributions, have also been developed and are described in this paper. As yet, no general method exists which can be applied to all classes of models with doubly-intractable posteriors. In addition, taking inspiration from the Physics literature, we study an alternative method based on representing the intractable likelihood as an infinite series. Unbiased estimates of the likelihood can then be obtained by finite time stochastic truncation of the series via Russian Roulette sampling, although the estimates are not necessarily positive. Results from the Quantum Chromodynamics literature are exploited to allow the use of possibly negative estimates in a pseudo-marginal MCMC scheme such that expectations with respect to the posterior distribution are preserved. The methodology is reviewed on well-known examples such as the parameters in Ising models, the posterior for Fisher–Bingham distributions on the d-Sphere and a largescale Gaussian Markov Random Field model describing the Ozone Column data. This leads to a critical assessment of the strengths and weaknesses of the methodology with pointers to ongoing research.

Constructing Priors that Penalize the Complexity of Gaussian Random Fields

ABSTRACT Priors are important for achieving proper posteriors with physically meaningful covariance structures for Gaussian random fields (GRFs) since the likelihood typically only provides limited information about the covariance structure under in-fill asymptotics. We extend the recent penalized complexity prior framework and develop a principled joint prior for the range and the marginal variance of one-dimensional, two-dimensional, and three-dimensional Matérn GRFs with fixed smoothness. The prior is weakly informative and penalizes complexity by shrinking the range toward infinity and the marginal variance toward zero. We propose guidelines for selecting the hyperparameters, and a simulation study shows that the new prior provides a principled alternative to reference priors that can leverage prior knowledge to achieve shorter credible intervals while maintaining good coverage. We extend the prior to a nonstationary GRF parameterized through local ranges and marginal standard deviations, and introduce a scheme for selecting the hyperparameters based on the coverage of the parameters when fitting simulated stationary data. The approach is applied to a dataset of annual precipitation in southern Norway and the scheme for selecting the hyperparameters leads to conservative estimates of nonstationarity and improved predictive performance over the stationary model. Supplementary materials for this article are available online.

Jahn–Teller effects on π-stacking and stereoselectivity in the phenylethaniminopyridine tris-chelates Cu(NN′)32+

Optically pure phenylethaniminopyridine (S(C)-L) tris-chelates of Fe(II) and other first row transition metal systems have previously been shown to give exclusively the fac structures in the solid state. Here it is shown by powder X-ray diffraction that the complex [CuL(3)][ClO(4)](2) crystallises exclusively as the mer isomer, although--for a given absolute configuration of the ligand--of the same helicity (Δ/Λ) as that displayed by the other metal complexes. The similar ligand R(C)-L(F), which contains a peripheral (19)F spin label, gave [CuL(F)(3)][ClO(4)](2) which also adopts exclusively the mer structure in the crystal, but is shown by NMR spectroscopy to have a fac:mer ratio of 1:6 in solution at low temperature. Molecular mechanics calculations for a number of isomers and conformers are consistent with the presence of such a mixture of isomers in solution for both complexes. The origin of the difference in behaviour between Fe(II) and Cu(II) is the presence of a Jahn-Teller distortion (and the generally longer M-N bonds) in the Cu(II) complexes. This disturbs intra-ligand π-stacking, leading to the poor fac/mer stereoselectivity while leaving enantioselectivity Δ/Λ apparently unaffected.

Computationally efficient spatial modeling of annual maximum 24 hour precipitation. An application to data from Iceland

We propose a computationally efficient statistical method to obtain distributional properties of annual maximum 24 hour precipitation on a 1 km by 1 km regular grid over Iceland. A latent Gaussian model is built which takes into account observations, spatial variations and outputs from a local meteorological model. A covariate based on the meteorological model is constructed at each observational site and each grid point in order to assimilate available scientific knowledge about precipitation into the statistical model. The model is applied to two data sets on extreme precipitation, one uncorrected data set and one data set that is corrected for phase and wind. The observations are assumed to follow the generalized extreme value distribution. At the latent level, we implement SPDE spatial models for both the location and scale parameters of the likelihood. An efficient MCMC sampler which exploits the model structure is constructed, which yields fast continuous spatial predictions for spatially varying model parameters and quantiles.

Using Stacking to Average Bayesian Predictive Distributions

The widely recommended procedure of Bayesian model averaging is flawed in the M-open setting in which the true data-generating process is not one of the candidate models being fit. We take the idea of stacking from the point estimation literature and generalize to the combination of predictive distributions, extending the utility function to any proper scoring rule, using Pareto smoothed importance sampling to efficiently compute the required leave-one-out posterior distributions and regularization to get more stability. We compare stacking of predictive distributions to several alternatives: stacking of means, Bayesian model averaging (BMA), pseudo-BMA using AIC-type weighting, and a variant of pseudo-BMA that is stabilized using the Bayesian bootstrap. Based on simulations and real-data applications, we recommend stacking of predictive distributions, with BB-pseudo-BMA as an approximate alternative when computation cost is an issue.

The Prior Can Often Only Be Understood in the Context of the Likelihood

A key sticking point of Bayesian analysis is the choice of prior distribution, and there is a vast literature on potential defaults including uniform priors, Jeffreys’ priors, reference priors, maximum entropy priors, and weakly informative priors. These methods, however, often manifest a key conceptual tension in prior modeling: a model encoding true prior information should be chosen without reference to the model of the measurement process, but almost all common prior modeling techniques are implicitly motivated by a reference likelihood. In this paper we resolve this apparent paradox by placing the choice of prior into the context of the entire Bayesian analysis, from inference to prediction to model evaluation.

Bayesian adaptive smoothing splines using stochastic differential equations

The smoothing spline is one of the most popular curve-fitting methods, partly because of empirical evidence supporting its effectiveness and partly because of its elegant mathematical formulation. However, there are two obstacles that restrict the use of smoothing spline in practical statistical work. Firstly, it becomes computationally prohibitive for large data sets because the number of basis functions roughly equals the sample size. Secondly, its global smoothing parameter can only provide constant amount of smoothing, which often results in poor performances when estimating inhomogeneous functions. In this work, we introduce a class of adaptive smoothing spline models that is derived by solving certain stochastic differential equations with finite element methods. The solution extends the smoothing parameter to a continuous data-driven function, which is able to capture the change of the smoothness of underlying process. The new model is Markovian, which makes Bayesian computation fast. A simulation study and real data example are presented to demonstrate the effectiveness of our method.

Computationally efficient spatial modeling of annual maximum 24‐h precipitation on a fine grid

We propose a computationally efficient statistical method to obtain distributional properties of annual maximum 24 hour precipitation on a 1 km by 1 km regular grid over Iceland. A latent Gaussian model is built which takes into account observations, spatial variations and outputs from a local meteorological model. A covariate based on the meteorological model is constructed at each observational site and each grid point in order to assimilate available scientific knowledge about precipitation into the statistical model. The model is applied to two data sets on extreme precipitation, one uncorrected data set and one data set that is corrected for phase and wind. The observations are assumed to follow the generalized extreme value distribution. At the latent level, we implement SPDE spatial models for both the location and scale parameters of the likelihood. An efficient MCMC sampler which exploits the model structure is constructed, which yields fast continuous spatial predictions for spatially varying model parameters and quantiles.