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This paper presents an integrating decision support system (IDSS) for food security in the United Kingdom. In ever-larger dynamic systems, such as the food system, it is increasingly difficult for decision makers (DMs) to effectively account for all the variables within the system that may influence the outcomes of interest under enactments of various candidate policies. Each of the influencing variables is likely, themselves, to be dynamic subsystems with expert domains supported by sophisticated probabilistic models. Recent increases in food poverty in the United Kingdom have raised the questions about the main drivers of food insecurity, how this may be changing over time and how evidence can be used in evaluating policy for decision support. In this context, an IDSS is proposed for household food security to allow DMs to compare several candidate policies which may affect the outcome of food insecurity at the household level.

In any crisis, there is a great deal of uncertainty, often geographical uncertainty or, more precisely, spatiotemporal uncertainty. Examples include the spread of contamination from an industrial accident, drifting volcanic ash, and the path of a hurricane. Estimating spatiotemporal probabilities is usually a difficult task, but that is not our primary concern. Rather, we ask how analysts can communicate spatiotemporal uncertainty to those handling the crisis. We comment on the somewhat limited literature on the representation of spatial uncertainty on maps. We note that many cognitive issues arise and that the potential for confusion is high. We note that in the early stages of handling a crisis, the uncertainties involved may be deep, i.e., difficult or impossible to quantify in the time available. In such circumstance, we suggest the idea of presenting multiple scenarios.

Evolutionary models of languages are usually considered to take the form of trees. With the development of so-called tree constraints the plausibility of the tree model assumptions can be assessed by checking whether the moments of observed variables lie within regions consistent with Gaussian latent tree models. In our linguistic application, the data set comprises acoustic samples (audio recordings) from speakers of five Romance languages or dialects. The aim is to assess these functional data for compatibility with a hereditary tree model at the language level. A novel combination of canonical function analysis (CFA) with a separable covariance structure produces a representative basis for the data. The separable-CFA basis is formed of components which emphasize language differences whilst maintaining the integrity of the observational language-groupings. A previously unexploited Gaussian tree constraint is then applied to component-by-component projections of the data to investigate adherence to an evolutionary tree. The results highlight some aspects of Romance language speech that appear compatible with an evolutionary tree model but indicates that it would be inappropriate to model all features as such.

We introduce a subclass of chain event graphs that we call stratified chain event graphs, and present a dynamic programming algorithm for the optimal selection of such chain event graphs that maximizes a decomposable score derived from a complete independent sample. We apply the algorithm to such a dataset, with a view to deducing the causal structure of the variables under the hypothesis that there are no unobserved confounders. We show that the algorithm is suitable for small problems. Similarities with and differences to a dynamic programming algorithm for MAP learning of Bayesian networks are highlighted, as are the relations to causal discovery using Bayesian networks.

UK military commanders have a degree of devolved decision\ud authority delegated from command and control (C2) regulators,\ud and they are trained and expected to act rationally and accountably. Therefore from a Bayesian perspective they should be subjective expected utility maximizers. In fact they largely appear\ud to be so. However when current tactical objectives conflict with\ud broader campaign objective there is a strong risk that fielded\ud commanders will lose rationality and coherence. By systematically analysing the geometry of their expected utilities, arising\ud from a utility function with two attributes, we demonstrate in\ud this paper that even when a remote C2 regulator can predict\ud only the likely broad shape of her agents' marginal utility functions it is still often possible for her to identify robustly those\ud settings where the commander is at risk of making inappropriate\ud decisions.

In this paper we investigate the geometry of a discrete Bayesian\ud network whose graph is a tree all of whose variables are binary and the\ud only observed variables are those labeling its leaves. We provide the full\ud geometric description of these models which is given by a set of polynomial\ud equations together with a set of complementary implied inequalities induced\ud by the positivity of probabilities on hidden variables. The phylogenetic\ud invariants given by the equations can be useful in the construction of simple\ud diagnostic tests. However, in this paper we point out the importance of also\ud incorporating the associated inequalities into any statistical analysis. The\ud full characterization of these inequality constraints derived in this paper\ud helps us determine now why routine statistical methods can break down\ud for this model class.

In contemporary military endeavours, Command and Control (C2) arrangements generally aim to\ud ensure an appropriate regulation of command-decision autonomy such that decision-takers are able to\ud act in a way that is consistent with the overall set of commanders' intents and according to the nature\ud of the unfolding situation. This can be a challenge, especially in situations with increasing degrees of\ud uncertainty, ambiguity and complexity, also where individual commanders are faced with con\ud icting\ud objectives. Increasingly it seems that command decisions are being taken under conditions of internal\ud command contention; for example, when the likely successful outcome of a tactical mission can often be\ud at odds with the overall strategic and political aims of the campaign. The work in the paper builds on our\ud previous research in decision-taking under uncertainty and con\ud icting objectives, where we analysed the\ud responses of military commanders in decision experiments. We demonstrated how multi-attribute utility\ud theory could be used to represent and understand the effects of uncertainty and con\ud icting objectives on a\ud particular commander's choices. In this paper, we further develop and generalise the theory to show that\ud the geometrical forms of expected utilities, which arise from the assumption of commander rationality,\ud are qualitatively stable in a wide range of scenarios. This opens out into further analysis linking to\ud Catastrophe Theory as it relates to C2 regulatory frameworks for devolving command decision freedoms.\ud We demonstrate how an appreciation of this geometry can aid understanding of the relationship between\ud socially complex operational environments and the prevailing C2, which can also inform selection and\ud training of personnel, to address issues of devolving command decision-rights, as appropriate for the\ud endeavour as a whole. The theory presented in the paper, therefore, provides a means to explore and\ud gain insight into different approaches to regulation of C2 decision-taking aimed ultimately at achieving\ud C2 agility, or at least at a conceptual language to allow its formal representation. C2 regulatory agents\ud are discussed in terms of detailed functions for moderating command decision-taking, as appropriate for\ud the degrees of uncertainty and goal contention being faced. The work also begins to address implications\ud of any lack of experience and any differences in personality-type of the individual commanders with\ud respect to risk-taking, open-minded-ness and creativity.

Over the last 50 years there has been an enormous explosion in developing full distributional analogues of the Kalman filter. In this paper we\ud explore some of the ways analogues of the orginal second order processes\ud discovered by Kalman have their analogues in Bayesian state space models. Many of these analogues need to be calculated using numerical methods like MCMC so they retain, or even enhance the descriptive power of\ud the Kalman Filter, but at a cost of transparency. However, if the analogues are drawn properly, elegant recurrence relationships - like those of\ud the Kalman Filter - can still be developed that apply at least for one step\ud ahead forecast distribution. In this paper we explore the variety of ways\ud such models have been built, in particular with respect to graphical time\ud series models.

The main feature of the paper is to show that Algebraic Statistics\ud is a natural framework to address issues of causality and to help discern a total\ud cause. Indeed identifiability of an effect of a cause in discrete models is almost\ud algebraic rather than graphical in nature. It is useful to think of it as such and\ud it leads to the definition of a large class of discrete models which comprises\ud popular ones.

In this paper we investigate the geometry of undirected discrete\ud graphical models of trees when all the variables in the system are binary, where\ud leaves represent the observable variables and where the inner nodes are unobserved.\ud We obtain a full geometric description of these models which is given\ud by polynomial equations and inequalities. We also give exact formulas for their\ud parameters in terms of the marginal probability over the observed variables.\ud Our analysis is based on combinatorial results generalizing the notion of cumulants\ud and introduce a novel use of Mobius functions on partially ordered sets.\ud The geometric structure we obtain links to the notion of a tree metric considered\ud in phylogenetic analysis and to some interesting determinantal formulas\ud involving hyperdeterminants of 2 x 2 x 2 tables as defined in [19].

A new tree-based graphical model — the dynamic staged tree — is used to model\ud discrete-valued discrete-time multivariate processes which are hypothesised to exhibit certain symmetries\ud concerning how situations might unfold. We define and implement a one-step-ahead\ud prediction algorithm using multi-process modelling and the power steady model. This is robust\ud to short-term variations in the data yet sensitive to underlying system changes. We demonstrate\ud that the whole analysis can be performed in a conjugate way so that the vast model space can be\ud traversed quickly and results communicated transparently. We also demonstrate how to analyse\ud causal hypotheses on this model class. Our techniques are illustrated using a simple educational\ud example.

Because of the huge number of partitions of even\ud a moderately sized dataset, even when Bayes factors\ud have a closed form, a comprehensive search\ud for the highest scoring (MAP) partition is usually\ud impossible. Therefore, both deterministic or random\ud search algorithms traverse only a small proportion\ud of partitions of a large dataset. The main\ud contribution of this paper is to encode the formal\ud Bayes factor search on partitions as a weighted\ud MAX-SAT problem and use well-known solvers\ud for that problem to search for partitions. We\ud demonstrate how, with the appropriate priors\ud over the partition space, this method can be used\ud to fully search the space of partitions in smaller\ud problems and how it can be used to enhance the\ud performance of more familiar algorithms in large\ud problems. We illustrate our method on clustering\ud of time-course microarray experiments.

The class of chain event graph models is a generalisation of the class of discrete Bayesian networks,\ud retaining most of the structural advantages of the Bayesian network for model interrogation,\ud propagation and learning, while more naturally encoding asymmetric state spaces and the order in\ud which events happen. In this paper we demonstrate how with complete sampling, conjugate closed\ud form model selection based on product Dirichlet priors is possible, and prove that suitable homogeneity\ud assumptions characterise the product Dirichlet prior on this class of models. We demonstrate\ud our techniques using two educational examples.

Command and Control (C2) arrangements aim to ensure their autonomous but rational commanders act in a way that is consistent with\ud C2's overall objectives. This can be a challenge, especially in scenarios\ud where individual commanders face conflicting objectives: the success of a\ud mission verses the compromise of a campaign. Building on our experiences\ud in observing the reactions of military in simulated conflict, in this paper\ud we demonstrate how multiattribute utility theory can be used to model\ud the effects of such conflicting objectives on a particular commander's actions. We show that the geometrical forms of expected utilities that arise\ud from the assumption of commander rationality are qualitatively stable in\ud a wide range of scenarios and therefore open to analysis. We proceed to\ud demonstrate how an appreciation of this geometry can aid understanding\ud of the relationship between complex operational environments and the\ud C2 arrangements and also inform selection and training of personnel to\ud address such conflicts appropriately.\ud

Conjugacy assumptions are often used in Bayesian selection over a partition because\ud they allow the otherwise unfeasibly large model space to be searched very quickly. The\ud implications of such models can be analysed algebraically. In this paper we use the explicit\ud forms of the associated Bayes factors to demonstrate that such methods can be unstable under\ud common settings of the associated hyperparameters. We then prove that the regions of\ud instability can be removed by setting the hyperparameters in an unconventional way. Under\ud this family of assignments we prove that model selection is determined by an implicit separation\ud measure: a function of the hyperparameters and the sufficient statistics of clusters in\ud a given partition. We show that this family of separation measures has desirable properties.\ud The proposed methodology is illustrated through the selection of clusters of longitudinal gene\ud expression profiles.

Under local DeRobertis (LDR) separation measures, the posterior distances between two densities is the same as between the prior densities. Like Kullback -\ud Leibler separation they also are additive under factorization. These two properties\ud allow us the prove that the precise specification of the prior will not be critical with\ud respect to the variation distance on the posteriors under the following conditions.\ud The genuine and approximating prior need to be similarly rough, the approximating\ud prior has concentrated on a small ball on the margin of interest, not on the boundary of the probability space, and the approximating prior has similar or fatter tails\ud to the genuine prior. Robustness then follows for all likelihoods, even ones that\ud are misspecified. Furthermore, the variation distances can be bounded explicitly by\ud a easy to calculate function the prior LDR and simple summary statistics of the\ud functioning posterior. In this paper we apply these results to study the robustness\ud of prior specification to learning Bayesian Networks.

This paper illustrates the theory and applications of a methodology\ud for non-stationary time series modeling which combines sequential\ud parametric Bayesian estimation with non-parametric change-point\ud testing. A novel Kullback-Leibler divergence between posterior distributions\ud arising from different sets of data is proposed as a nonparametric\ud test statistic. A closed form expression of this test statistic\ud is derived for exponential family models whereas Markov chain Monte\ud Carlo simulation is used in general to approximate its value and that\ud of its critical region. The effects of detecting a change-point using\ud our method are assessed analytically for the one-step ahead predictive\ud distribution of a linear dynamic Gaussian time series model. Conditions\ud under which our approach reduces to fully parametric state-space\ud modeling are illustrated.\ud The method is applied to estimating the functional dynamics of a\ud wide range of neural data, including multi-channel electroencephalogram\ud recordings, the learning performance in longitudinal behavioural\ud experiments and in-vivo multiple spike trains. The estimated dynamics\ud are related to the presentation of visual stimuli, to the generation\ud of motor responses and to variations of the functional connections between\ud neurons across different experiments.

Here a new class of local separation measures over prior densities is\ud studied and their usefulness for examining prior to posterior robustness\ud under a sequence of observed likelihoods, possibly erroneous, illustrated.\ud It is shown that provided an approximation to a prior distribution satisfies certain mild smoothness and tail conditions then prior to posterior\ud inference for large samples is robust, irrespective of whether the priors\ud are grossly misspecified with respect to variation distance and irrespective of the form or the validity of the observed likelihood. Furthermore\ud it is usually possible to specify error bounds explicitly in terms of statistics associated with the posterior associated with the approximating prior\ud and asumed prior error bounds. These results apply in a general multivariate setting and are especially easy to interpret when prior densities\ud are approximated using standard families or multivariate prior densities\ud factorise.

Under a new family of separations the distance between two posterior densities is the same as the distance between their prior densities\ud whatever the observed likelihood when that likelihood is strictly positive. Local versions of such separations form the basis of a weak\ud topology having close links to the Euclidean metric on the natural\ud parameters of two exponential family densities. Using these local separation measures it is shown that when the tails of the approximating\ud density have appropriate properties, the variation distance between an\ud approximating posterior density to a genuine density can be bounded\ud explicitly. These bounds apply irrespective of whether the prior densities are grossly misspecified with respect to variation distance and\ud irrespective of the form or the validity of the observed likelihood.

The relationship between algebraic geometry and the inferential framework\ud of the Bayesian Networks with hidden variables has now been fruitfully explored\ud and exploited by a number of authors. More recently the algebraic formulation of Causal\ud Bayesian Networks has also been investigated in this context. After reviewing these\ud newer relationships, we proceed to demonstrate that many of the ideas embodied in the\ud concept of a "causal model" can be more generally expressed directly in terms of a partial\ud order and a family of polynomial maps. The more conventional graphical constructions,\ud when available, remain a powerful tool.

Discrete Bayesian Networks (BN’s) have been very successful as\ud a framework both for inference and for expressing certain causal hypotheses.\ud In this paper we present a class of graphical models called\ud the chain event graph (CEG) models, that generalises the class of\ud discrete BN models. It provides a flexible and expressive framework\ud for representing and analysing the implications of causal hypotheses,\ud expressed in terms of the effects of a manipulation of the generating\ud underlying system.We prove that, as for a BN, identifiability analyses\ud of causal effects can be performed through examining the topology\ud of the CEG graph, leading to theorems analogous to the back-door\ud theorem for the BN.

Bayesian networks (BNs) are useful for coding conditional independence statements between a given set of measurement variables.\ud On the other hand, event trees (ETs) are convenient for representing asymmetric structure and how situations unfold. In this paper\ud we report the development of a new graphical framework for discrete\ud probability models called the Chain Event Graph (CEG). The class of\ud CEG models contains finite BNs as a special case. Unlike the BN, the\ud CEG is equally appropriate for representing conditional independencies in asymmetric systems and does not need dependent variables to\ud be specified in advance. As with the BN, it also provides a framework\ud for learning relevant conditional probabilities and propagation. Furthermore, being a function of an ET, the CEG is a more \ud exible way of\ud representing various causal hypotheses than the BN. This new framework is illustrated throughout by a biological regulatory network: the\ud tryptophan metabolic pathway in the bacterium E. coli.

Background: An increasing number of microarray experiments produce time series of expression levels for many\ud genes. Some recent clustering algorithms respect the time ordering of the data and are, importantly, extremely\ud fast. The focus of this paper is the development of such an algorithm on a microarray data set consisting of\ud 22,810 genes of the plant Arabidopsis thaliana measured at 13 time points over two days. Circadian rhythms\ud control the timing of various physiological and metabolic processes and are regulated by genes acting in\ud feedback loops. The aim is to cluster and classify the expression profiles in order to identify genes potentially\ud involved in, and regulated by, the circadian clock.\ud Results: A greedy search over time series of expression levels (where series are compared pairwise, the two most\ud similar put in the same cluster and so forth) will get a fast result but will only explore a very limited number of\ud the possible partitions of the profiles. We propose an improved, deterministic method based on a multi-step\ud application of a conjugate Bayesian clustering algorithm. It allows the entire space to be searched more fully and\ud intelligently. The values of the summary statistics are used to not only score clusters of genes, but also to guide\ud the search of the vast partition space. By following this procedure, we are able to cluster genes that are known\ud to be rhythmically expressed with genes of previously unknown function; thus suggesting potentially interesting\ud targets for future experiments.

Discrete Bayesian Networks (BNs) have been very successful as\ud a framework both for inference and for expressing certain causal hypotheses. In this paper we present a class of graphical models called\ud the chain event graph (CEG) models, that generalises the class of discrete BN models. This class is suited for representing conditional independence and sample space structures of asymmetric models. It retains many useful properties of discrete BNs, in particular admitting\ud conjugate estimation. It provides a flexible and expressive framework\ud for representing and analysing the implications of causal hypotheses,\ud expressed in terms of the effects of a manipulation of the generating\ud underlying system.We prove that, as for a BN, identifiability analyses\ud of causal effects can be performed through examining the topology\ud of the CEG graph, leading to theorems analogous to the Backdoor\ud theorem for the BN.

Particle Filters are now regularly used to obtain the filter distributions associated with state space financial time series. Most commonly used nowadays is the auxiliary particle filter method in conjunction with a first order Taylor expansion of the log-likelihood. We argue in this paper that for series such as stock returns, which exhibit fairly frequent and extreme outliers, filters based on this first order approximation can easily break down. However, an auxiliary particle filter based on the much more rarely used second order approximation appears to perform well in these circumstances. To detach the issue of algorithm design from problems related to model misspecification and parameter estimation, we demonstrate the lack of robustness of the first order approximation and the feasibility of a specific second order approximation using simulated data.

Bayesian networks (BNs) are useful for coding conditional independence statements, especially in discrete symmetric models. On\ud the other hand, event trees (ETs) are convenient for representing asymmetric structure\ud and how situations unfold. In this paper\ud we report the development of a new graphical framework called the chain event graph\ud (CEG). For symmetric models, all conditional independencies in a BN can be expressed through the topology of a CEG. However, unlike the BN, the CEG is equally appropriate for representing conditional independencies in asymmetric systems and does\ud not need dependent variables to be specified\ud in advance. As with the BN, it also provides\ud a framework for learning relevant conditional\ud probabilities. Furthermore, being a function\ud of an ET, the CEG is a more \ud exible way of\ud representing various causal hypotheses than\ud the BN. This new framework is illustrated\ud throughout by a biological regulatory network: the tryptophan metabolic pathway in\ud the bacterium E. coli.