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Potential violent criminals will often need to go through a sequence of preparatory steps before they can execute their plans. During this escalation process police have the opportunity to evaluate the threat posed by such people through what they know, observe and learn from intelligence reports about their activities. In this paper we customise a three-level Bayesian hierarchical model to describe this process. This is able to propagate both routine and unexpected evidence in real time. We discuss how to set up such a model so that it calibrates to domain expert judgments. The model illustrations include a hypothetical example based on a potential vehicle based terrorist attack., Comment: 57 pages, 20 figures, 8 tables

In this paper a new class of graphical Bayesian discrete semi-Markov models is developed to describe the various pathways that might lead someone into perpetrating various kinds of crime involving assault and violence. Our discrete probability models are crafted to embody various theory and empirical studies by psychologists and sociologists explaining and describing this development. This probability model is then used to formally structure a new decision support system to help authorities evaluate public risks guided by not only archived but real time data. We argue here that such systems will be able to provide provisional quanti ed evaluations of the impacts of various policy and policing decisions into the short and medium term. The construction of such probability models is illustrated throughout by examples. We end the paper with a more detailed description of a model built to support authorities to frustrate populations of criminals who have been radicalised into violent extremism.

In a number of applications, we argue that standard Bayes factor model comparison and selection may be inappropriate for decision making under specific, utility-based, criteria. It has been suggested that the use of scoring rules in this context allows greater exibility: scores can be customised to a client's utility and model selection can proceed on the basis of the highest scoring model. We argue here that the approach of comparing the cumulative scores of competing models is not ideal because it tends to ignore a model's ability to 'catch up' through parameter learning. An alternative approach of selecting a model on its maximum posterior score based on a plug in or posterior expected value is problematic in that it uses the data twice in estimation and evaluation. We therefore introduce a new Bayesian posterior score information criterion (BPSIC), which is a generalisation of the Bayesian predictive information criterion proposed by Ando (2007). This allows the analyst both to tailor an appropriate scoring function to the needs of the ultimate decision maker and to correct appropriately for bias in using the data on a posterior basis to revise parameter estimates. We show that this criterion can provide a convenient method of initial model comparison when the number of models under consideration is large or when computational burdens are high. We illustrate the new methods with simulated examples and real data from the UK electricity imbalance market.

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

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

We introduce a new modelling representation, the Decision Event Graph (DEG), for asymmetric multistage decision problems. The DEG explicitly encodes conditional independences and has additional significant advantages over other representations of asymmetric decision problems. The colouring of edges makes it possible to identify conditional independences on decision trees, and these coloured trees serve as a basis for the construction of the DEG. We provide an efficient backward-induction algorithm for finding optimal decision rules on DEGs, and work through an example showing the efficacy of these graphs. Simplifications of the topology of a DEG admit analogues to the sufficiency principle and barren node deletion steps used with influence diagrams.

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

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

Over the last 50 years there has been an enormous explosion in developing full distributional analogues of the Kalman filter. In this paper we explore some of the ways analogues of the orginal second order processes 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 the Kalman Filter, but at a cost of transparency. However, if the analogues are drawn properly, elegant recurrence relationships - like those of the Kalman Filter - can still be developed that apply at least for one step ahead forecast distribution. In this paper we explore the variety of ways such models have been built, in particular with respect to graphical time series models.

As the Chain Event Graph (CEG) has a topology which represents sets of conditional independence statements, it becomes especially useful when problems lie naturally in a discrete asymmetric non-product space domain, or when much context-specific information is present. In this paper we show that it can also be a powerful representational tool for a wide variety of causal hypotheses in such domains. Furthermore, we demonstrate that, as with Causal Bayesian Networks (CBNs), the identifiability of the effects of causal manipulations when observations of the system are incomplete can be verified simply by reference to the topology of the CEG. We close the paper with a proof of a Back Door Theorem for CEGs, analogous to Pearl's Back Door Theorem for CBNs.

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

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

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

The class of chain event graph models is a generalisation of the class of discrete Bayesian networks, retaining most of the structural advantages of the Bayesian network for model interrogation, propagation and learning, while more naturally encoding asymmetric state spaces and the order in which events happen. In this paper we demonstrate how with complete sampling, conjugate closed form model selection based on product Dirichlet priors is possible, and prove that suitable homogeneity assumptions characterise the product Dirichlet prior on this class of models. We demonstrate 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 C2's overall objectives. This can be a challenge, especially in scenarios where individual commanders face conflicting objectives: the success of a mission verses the compromise of a campaign. Building on our experiences in observing the reactions of military in simulated conflict, in this paper we demonstrate how multiattribute utility theory can be used to model the effects of such conflicting objectives on a particular commander's actions. We show that the geometrical forms of expected utilities that arise from the assumption of commander rationality are qualitatively stable in a wide range of scenarios and therefore open to analysis. We proceed to demonstrate how an appreciation of this geometry can aid understanding of the relationship between complex operational environments and the C2 arrangements and also inform selection and training of personnel to address such conflicts appropriately.

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

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

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

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

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

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

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

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

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

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

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

This item is contained in a letter book created by Hill Ferguson documenting The University of Alabama history between 1906 and 1911. Ferguson presented the bound book to University President Richard D. Foster. The title on the spine is The Greater University of Alabama, 1906-1911, Hill Ferguson.

This item is contained in a letter book created by Hill Ferguson documenting The University of Alabama history between 1906 and 1911. Ferguson presented the bound book to University President Richard D. Foster. The title on the spine is The Greater University of Alabama, 1906-1911, Hill Ferguson.