In a routinely occurring estimation problem, the experimenter can often consider the interested parameters themselves as random variables with unknown prior distribution. Without knowledge of the exact prior distribution the Bayes estimator cannot be obtained. However, as long as independent repetitions of the experiment occur, the empirical Bayes approach can then be applied. A general strategy underlying the empirical Bayes estimator consists of finding the Bayes estimator in a form which can be estimated sequentially by using the past data. Such use of the data circumvents knowledge of the prior distribution. Three different types of sampling distributions of cell counts of 2x2 contingency tables were considered. In the Independent Poisson Case, an empirical Bayes estimator for the cross-product ratio is presented. If the squared error loss function is used, this empirical Bayes estimator, ᾶ, has asymptotic risk only ε > 0 larger than the true Bayes risk. For the Product Binomial and Multinomial situations, several empirical Bayes estimators for α are proposed. Basically, these 'empirical' Bayes estimators by-pass the prior distribution by estimating the marginal probabilities P(X₁₁,X₂₁,X₁₂,X₂₂) and P(X₁₁+1,X₂₁-1,X₁₂-1,X₂₂+1), where (X₁₁,X₂₁,X₁₂,X₂₂) is the set of current cell counts. Furthermore, because of the assumption of varying sample size(s), they will have asymptotic risk only ε > 0 away from the true Bayes risk if both the number of past experiences and the sample size(s) are sufficiently large. Results of Monte Carlo simulation of empirical Bayes estimators are presented for the carefully selected prior distributions. Mean squared errors for these estimators and classical estimators were compared. The improvement of empirical Bayes over classical estimators was found to be dependent upon the prior means, the prior variances, the prior distribution of the parameters considered as random variables, and sample size(s). These conclusions are summarized, and tables are provided. The empirical Bayes estimators of α start to show significant improvement over classical estimators for as few as only ten past experiences. In many instances, the improvement is something on the order of 15% with only ten past experiences and sample size(s) larger than twenty. However, for the cases where the prior variances are very large, the empirical Bayes estimator indicates neither better nor worse over the classical. Greater improvement is shown for more past experiences until around thirty when the improvement appears stabilized. Finally, the other existing estimators for a which also take into account past experiences are discussed and compared to the corresponding empirical Bayes estimator. They were proposed respectively by Birch, Goodman, Mantel and Haenszel, Woolf, etc. The simulation study for comparisons indicate that empirical Bayes estimators outmatch them even with small prior variance. A test for deciding when empirical Bayes estimators of α should be used is also suggested and discussed. / Ph. D.

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Bayesian probabilistic modeling provides a powerful framework for building flexible models to incorporate latent structures through likelihood model and prior. When we specify a model, we make certain assumptions about the underlying data-generating process with respect to these latent structures. For example, the latent Dirichlet allocation (LDA) model assumes that when generating a document, we first select a latent topic and then select a word that often appears in the selected topic. We can uncover the latent structures conditioned on the observed data via posterior inference. In this dissertation, we apply the tools of probabilistic latent variable models and try to understand complex real-world data about music semantics and user behavior. We first look into the problem of automatic music tagging -- inferring the semantic tags (e.g., "jazz'', "piano'', "happy'', etc.) from the audio features. We treat music tagging as a matrix completion problem and apply the Poisson matrix factorization model jointly on the vector-quantized audio features and a "bag-of-tags'' representation. This approach exploits the shared latent structure between semantic tags and acoustic codewords. We present experimental results on the Million Song Dataset for both annotation and retrieval tasks, illustrating the steady improvement in performance as more data is used. We then move to the intersection between music semantics and user behavior: music recommendation. The leading performance in music recommendation is achieved by collaborative filtering methods which exploit the similarity patterns in user's listening history. We address the fundamental cold-start problem of collaborative filtering: it cannot recommend new songs that no one has listened to. We train a neural network on semantic tagging information as a content model and use it as a prior in a collaborative filtering model. The proposed system is evaluated on the Million Song Dataset and shows comparably better result than the collaborative filtering approaches, in addition to the favorable performance in the cold-start case. Finally, we focus on general recommender systems. We examine two different types of data: implicit and explicit feedback, and introduce the notion of user exposure (whether or not a user is exposed to an item) as part of the data-generating process, which is latent for implicit data and observed for explicit data. For implicit data, we propose a probabilistic matrix factorization model and infer the user exposure from data. In the language of causal analysis (Imbens and Rubin, 2015), user exposure has close connection to the assignment mechanism. We leverage this connection more directly for explicit data and develop a causal inference approach to recommender systems. We demonstrate that causal inference for recommender systems leads to improved generalization to new data. Exact posterior inference is generally intractable for latent variables models. Throughout this thesis, we will design specific inference procedure to tractably analyze the large-scale data encountered under each scenario.

Thesis (MSc)--Stellenbosch University, 2004. / ENGLISH ABSTRACT: The Bayesian approach to decision-making is considered in this thesis for reliability/survival models pertaining to a Weibull class of distributions. A generalised right censored sampling scheme has been assumed and implemented. The Jeffreys' prior for the inverse mean lifetime and the survival function of the exponential model were derived. The consequent posterior distributions of these two parameters were obtained using this non-informative prior. In addition to the Jeffreys' prior, the natural conjugate prior was considered as a prior for the parameter of the exponential model and the consequent posterior distribution was derived. In many reliability problems, overestimating a certain parameter of interest is more detrimental than underestimating it and hence, the LINEX loss function was used to estimate the parameters and their consequent risk measures. Moreover, the same analogous derivations have been carried out relative to the commonly-used symmetrical squared error loss function. The risk function, the posterior risk and the integrated risk of the estimators were obtained and are regarded in this thesis as the risk measures. The performance of the estimators have been compared relative to these risk measures. For the Jeffreys' prior under the squared error loss function, the comparison resulted in crossing-over risk functions and hence, none of these estimators are completely admissible. However, relative to the LINEX loss function, it was found that a correct Bayesian estimator outperforms an incorrectly chosen alternative. On the other hand for the conjugate prior, crossing-over of the risk functions of the estimators were evident as a result. In comparing the performance of the Bayesian estimators, whenever closed-form expressions of the risk measures do not exist, numerical techniques such as Monte Carlo procedures were used. In similar fashion were the posterior risks and integrated risks used in the performance compansons. The Weibull pdf, with its scale and shape parameter, was also considered as a reliability model. The Jeffreys' prior and the consequent posterior distribution of the scale parameter of the Weibull model have also been derived when the shape parameter is known. In this case, the estimation process of the scale parameter is analogous to the exponential model. For the case when both parameters of the Weibull model are unknown, the Jeffreys' and the reference priors have been derived and the computational difficulty of the posterior analysis has been outlined. The Jeffreys' prior for the survival function of the Weibull model has also been derived, when the shape parameter is known. In all cases, two forms of the scalar estimation error have been t:. used to compare as much risk measures as possible. The performance of the estimators were compared for acceptability in a decision-making framework. This can be seen as a type of procedure that addresses robustness of an estimator relative to a chosen loss function. / AFRIKAANSE OPSOMMING: Die Bayes-benadering tot besluitneming is in hierdie tesis beskou vir betroubaarheids- / oorlewingsmodelle wat behoort tot 'n Weibull klas van verdelings. 'n Veralgemene regs gesensoreerde steekproefnemingsplan is aanvaar en geïmplementeer. Die Jeffreyse prior vir die inverse van die gemiddelde leeftyd en die oorlewingsfunksie is afgelei vir die eksponensiële model. Die gevolglike aposteriori-verdeling van hierdie twee parameters is afgelei, indien hierdie nie-inligtingge-wende apriori gebruik word. Addisioneel tot die Jeffreyse prior, is die natuurlike toegevoegde prior beskou vir die parameter van die eksponensiële model en ooreenstemmende aposteriori-verdeling is afgelei. In baie betroubaarheidsprobleme het die oorberaming van 'n parameter meer ernstige nagevolge as die onderberaming daarvan en omgekeerd en gevolglik is die LINEX verliesfunksie gebruik om die parameters te beraam tesame met ooreenstemmende risiko maatstawwe. Soortgelyke afleidings is gedoen vir hierdie algemene simmetriese kwadratiese verliesfunksie. Die risiko funksie, die aposteriori-risiko en die integreerde risiko van die beramers is verkry en word in hierdie tesis beskou as die risiko maatstawwe. Die gedrag van die beramers is vergelyk relatief tot hierdie risiko maatstawwe. Die vergelyking vir die Jeffreyse prior onder kwadratiese verliesfunksie het op oorkruisbare risiko funksies uitgevloei en gevolglik is geeneen van hierdie beramers volkome toelaatbaar nie. Relatief tot die LINEX verliesfunksie is egter gevind dat die korrekte Bayes-beramer beter vaar as die alternatiewe beramer. Aan die ander kant is gevind dat oorkruisbare risiko funksies van die beramers verkry word vir die toegevoegde apriori-verdeling. Met hierdie gedragsvergelykings van die beramers word numeriese tegnieke toegepas, soos die Monte Carlo prosedures, indien die maatstawwe nie in geslote vorm gevind kan word nie. Op soortgelyke wyse is die aposteriori-risiko en die integreerde risiko's gebruik in die gedragsvergelykings. Die Weibull waarskynlikheidsverdeling, met skaal- en vormingsparameter, is ook beskou as 'n betroubaarheidsmodel. Die Jeffreyse prior en die gevolglike aposteriori-verdeling van die skaalparameter van die Weibull model is afgelei, indien die vormingsparameter bekend is. In hierdie geval is die beramingsproses van die skaalparameter analoog aan die afleidings van die eksponensiële model. Indien beide parameters van die Weibull modelonbekend is, is die Jeffreyse prior en die verwysingsprior afgelei en is daarop gewys wat die berekeningskomplikasies is van 'n aposteriori-analise. Die Jeffreyse prior vir die oorlewingsfunksie van die Weibull model is ook afgelei, indien die vormingsparameter bekend is. In al die gevalle is twee vorms van die skalaar beramingsfoute gebruik in die vergelykings, sodat soveel as moontlik risiko maatstawwe vergelyk kan word. Die gedrag van die beramers is vergelyk vir aanvaarbaarheid binne die besluitnemingsraamwerk. Hierdie kan gesien word as 'n prosedure om die robuustheid van 'n beramer relatief tot 'n gekose verliesfunksie aan te spreek.

Multi-species functional response models are required to model the predation of generalist preda- tors, which consume more than one prey species. In chapter 2, a new model for the multi-species functional response is presented. This model can describe generalist predators that exhibit func- tional responses of Holling type II to some of their prey and of type III to other prey. In chapter 3, I review some of the theoretical distinctions between Bayesian and frequentist statistics and show how Bayesian statistics are particularly well-suited for the fitting of functional response models because uncertainty can be represented comprehensively. In chapters 4 and 5, the multi- species functional response model is fitted to field data on two generalist predators: the hen harrier Circus cyaneus and the harp seal Phoca groenlandica. I am not aware of any previous Bayesian model of the multi-species functional response that has been fitted to field data. The hen harrier's functional response fitted in chapter 4 is strongly sigmoidal to the densities of red grouse Lagopus lagopus scoticus, but no type III shape was detected in the response to the two main prey species, field vole Microtus agrestis and meadow pipit Anthus pratensis. The impact of using Bayesian or frequentist models on the resulting functional response is discussed. In chapter 5, no functional response could be fitted to the data on harp seal predation. Possible reasons are discussed, including poor data quality or a lack of relevance of the available data for informing a behavioural functional response model. I conclude with a comparison of the role that functional responses play in behavioural, population and community ecology and emphasise the need for further research into unifying these different approaches to understanding predation with particular reference to predator movement. In an appendix, I evaluate the possibility of using a functional response for inferring the abun- dances of prey species from performance indicators of generalist predators feeding on these prey. I argue that this approach may be futile in general, because a generalist predator's energy intake does not depend on the density of any single of its prey, so that the possibly unknown densities of all prey need to be taken into account.

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / Social Sciences / Doctoral / Doctor of Philosophy

When justice goes wrong, grave consequences entail. They are damaging to the standing of the legal system and people’s lives. Humans are not good at assessing uncertainties. Parties to a legal proceeding adduce evidence to support or reject hypotheses. Errors happen when the tribunal fails to consider properly all the evidence in the context of inherent probabilities or improbabilities. This research work does not advocate trials by mathematics or statistics. This work extends the understanding of the application of Bayesian Networks in the law domain. The original contribution to knowledge is the analysis of evidence by Bayesian Network in the context of specific legal requirements of Hong Kong. The research questions are: 1. What are the legal requirements for the analysis of evidence in a criminal trial in Hong Kong? 2. How can a Bayesian Network be constructed for the purpose of such analysis? 3. Is such a Bayesian Network effective for the analysis? In answering the questions, this research work examined the feasibility of generic models created for digital crime scene investigations and concluded that each case must be, for the purpose of analysis of evidence in the trial, represented by a different Bayesian Network. This research work examined the trial processes, the tasks of tribunal of facts of criminal trials and some appellate decisions in Hong Kong. The work also created models of reasoning processes for the juries in Hong Kong. The work then compared the properties of Bayesian Networks with the processes of evaluation of evidence during trials. This research work also considered the reluctance of courts in the United Kingdom to allow experts to express their opinions on the bases of Bayesian calculations; even though trials are practically evaluations of uncertainties and assignments of degrees of beliefs. This research work then constructed a schedule of levels of proof and proposed a schematic method, to be used with the schedule, to construct Bayesian Networks for any types of trial in Hong Kong. The method requires an analyst to go through a mass of evidence systematically, and analyse their relationships amongst the ultimate probandum, the penultimate probanda, the intermediate propositions, the facts in issue and the facts relevant to the issue. This work then demonstrated the applications by two criminal cases in Hong Kong. The analyses show that the construction of Bayesian Network by the schematic method enables an analyst to take precaution to reach an assessment rationally and to approximate as far as capable his or her belief to the facts in issue. / published_or_final_version / Computer Science / Doctoral / Doctor of Philosophy

The accident rate at an intersection is one attribute used to evaluate the hazard associated with the intersection. Two techniques traditionally used to make such evaluations are the rate-quality technique and a technique based on the confidence interval of classical statistics. Both of these techniques label intersections as hazardous if their accident rate is greater than some critical accident rate determined by the technique. An alternative technique is one based on a Bayesian analysis of available accident number and traffic volume data. In contrast to the two classic techniques, the Bayesian technique identifies an intersection as hazardous based on a probabilistic assessment of accident rates. The goal of this thesis is to test and compare the ability of the three techniques to accurately identify traffic intersections known to be hazardous. Test data is generated from an empirical distribution of accident rates. The techniques are then applied to the generated data and compared based on the simulation results.

It is well-known that the observation of a variable in a Bayesian network can affect the effective connectivity of the network, which in turn affects the efficiency of inference. Unfortunately, the observed variables may not be known until runtime, which limits the amount of compile-time optimization that can be done in this regard. This thesis considers how to improve inference when users know the likelihood of a variable being observed. It demonstrates how these probabilities of observation can be exploited to improve existing heuristics for choosing elimination orderings for inference. Empirical tests over a set of benchmark networks using the Variable Elimination algorithm show reductions of up to 50% and 70% in multiplications and summations, as well as runtime reductions of up to 55%. Similarly, tests using the Elimination Tree algorithm show reductions by as much as 64%, 55%, and 50% in recursive calls, total cache size, and runtime, respectively. / xi, 88 leaves : ill. ; 29 cm