In this paper, based on a fuzzy entropy feature selection framework, different methods have been implemented and compared to improve the key components of the framework. Those methods include the combinations of three ideal vector calculations, three maximal similarity classifiers and three fuzzy entropy functions. Different feature removal orders based on the fuzzy entropy values were also compared. The proposed method was evaluated on three publicly available biomedical datasets. From the experiments, we concluded the optimized combination of the ideal vector, similarity classifier and fuzzy entropy function for feature selection. The optimized framework was also compared with other six classical filter-based feature selection methods. The proposed method was ranked as one of the top performers together with the Correlation and ReliefF methods. More importantly, the proposed method achieved the most stable performance for all three datasets when the features being gradually removed. This indicates a better feature ranking performance than the other compared methods.

In this paper, we propose a novel weighted combination feature selection method using bootstrap and fuzzy sets. The proposed method mainly consists of three processes, including fuzzy sets generation using bootstrap, weighted combination of fuzzy sets and feature ranking based on defuzzification. We implemented the proposed method by combining four state-of-the-art feature selection methods and evaluated the performance based on three publicly available biomedical datasets using five-fold cross validation. Based on the feature selection results, our proposed method produced comparable (if not better) classification accuracies to the best of the individual feature selection methods for all evaluated datasets. More importantly, we also applied standard deviation and Pearson's correlation to measure the stability of the methods. Remarkably, our combination method achieved significantly higher stability than the four individual methods when variations and size reductions were introduced to the datasets.

Consider a policymaker who wants to decide which intervention to perform in order to change a currently undesirable situation. The policymaker has at her disposal a team of experts, each with their own understanding of the causal dependencies between different factors contributing to the outcome. The policymaker has varying degrees of confidence in the experts' opinions. She wants to combine their opinions in order to decide on the most effective intervention. We formally define the notion of an effective intervention, and then consider how experts' causal judgments can be combined in order to determine the most effective intervention. We define a notion of two causal models being \emph{compatible}, and show how compatible causal models can be merged. We then use it as the basis for combining experts' causal judgments. We also provide a definition of decomposition for causal models to cater for cases when models are incompatible. We illustrate our approach on a number of real-life examples.

We introduce a theoretical model of information acquisition under resource limitations in a noisy environment. An agent must guess the truth value of a given Boolean formula $\varphi$ after performing a bounded number of noisy tests of the truth values of variables in the formula. We observe that, in general, the problem of finding an optimal testing strategy for $\phi$ is hard, but we suggest a useful heuristic. The techniques we use also give insight into two apparently unrelated, but well-studied problems: (1) \emph{rational inattention}, that is, when it is rational to ignore pertinent information (the optimal strategy may involve hardly ever testing variables that are clearly relevant to $\phi$), and (2) what makes a formula hard to learn/remember.

Learning a Gaussian Mixture Model (GMM) is hard when the number of parameters is too large given the amount of available data. As a remedy, we propose restricting the GMM to a Gaussian Markov Random Field Mixture Model (GMRF-MM), as well as a new method for estimating the latter's sparse precision (i.e., inverse covariance) matrices. When the sparsity pattern of each matrix is known, we propose an efficient optimization method for the Maximum Likelihood Estimate (MLE) of that matrix. When it is unknown, we utilize the popular Graphical LASSO (GLASSO) to estimate that pattern. However, we show that even for a single Gaussian, when GLASSO is tuned to successfully estimate the sparsity pattern, it does so at the price of a substantial bias of the values of the nonzero entries of the matrix, and we show that this problem only worsens in a mixture setting. To overcome this, we discard the non-zero values estimated by GLASSO, keep only its pattern estimate and use it within the proposed MLE method. This yields an effective two-step procedure that removes the bias. We show that our "debiasing" approach outperforms GLASSO in both the single-GMRF and the GMRF-MM cases. We also show that when learning priors for image patches, our method outperforms GLASSO even if we merely use an educated guess about the sparsity pattern, and that our GMRF-MM outperforms the baseline GMM on real and synthetic high-dimensional datasets. Our code is available at \url{https://github.com/shahaffind/GMRF-MM}.

An influence diagram is a graphical representation of sequential decision-making under uncertainty, defining a structured decision problem by conditional probability functions and additive utility functions over discrete state and action variables. The task of finding the maximum expected utility of influence diagrams is closely related to the cost-optimal probabilistic planning, stochastic programmings, or model-based reinforcement learning. In this position paper, we address the heuristic search for solving influence diagram, where we generate admissible heuristic functions from graph decomposition schemes. Then, we demonstrate how such heuristics can guide an AND/OR branch and bound search. Finally, we briefly discuss the future directions for improving the quality of heuristic functions and search strategies.

This thesis describes work on two applications of probabilistic programming: the learning of probabilistic program code given specifications, in particular program code of one-dimensional samplers; and the facilitation of sequential Monte Carlo inference with help of data-driven proposals. The latter is presented with experimental results on a linear Gaussian model and a non-parametric dependent Dirichlet process mixture of objects model for object recognition and tracking. In Chapter 1 we provide a brief introduction to probabilistic programming. In Chapter 2 we present an approach to automatic discovery of samplers in the form of probabilistic programs. We formulate a Bayesian approach to this problem by specifying a grammar-based prior over probabilistic program code. We use an approximate Bayesian computation method to learn the programs, whose executions generate samples that statistically match observed data or analytical characteristics of distributions of interest. In our experiments we leverage different probabilistic programming systems to perform Markov chain Monte Carlo sampling over the space of programs. Experimental results have demonstrated that, using the proposed methodology, we can learn approximate and even some exact samplers. Finally, we show that our results are competitive with regard to genetic programming methods. In Chapter 3, we describe a way to facilitate sequential Monte Carlo inference in probabilistic programming using data-driven proposals. In particular, we develop a distance-based proposal for the non-parametric dependent Dirichlet process mixture of objects model. We implement this approach in the probabilistic programming system Anglican, and show that for that model data-driven proposals provide significant performance improvements. We also explore the possibility of using neural networks to improve data-driven proposals.

Markov decision processes are of major interest in the planning community as well as in the model checking community. But in spite of the similarity in the considered formal models, the development of new techniques and methods happened largely independently in both communities. This work is intended as a beginning to unite the two research branches. We consider goal-reachability analysis as a common basis between both communities. The core of this paper is the translation from Jani, an overarching input language for quantitative model checkers, into the probabilistic planning domain definition language (PPDDL), and vice versa from PPDDL into Jani. These translations allow the creation of an overarching benchmark collection, including existing case studies from the model checking community, as well as benchmarks from the international probabilistic planning competitions (IPPC). We use this benchmark set as a basis for an extensive empirical comparison of various approaches from the model checking community, variants of value iteration, and MDP heuristic search algorithms developed by the AI planning community. On a per benchmark domain basis, techniques from one community can achieve state-ofthe-art performance in benchmarks of the other community. Across all benchmark domains of one community, the performance comparison is however in favor of the solvers and algorithms of that particular community. Reasons are the design of the benchmarks, as well as tool-related limitations. Our translation methods and benchmark collection foster crossfertilization between both communities, pointing out specific opportunities for widening the scope of solvers to different kinds of models, as well as for exchanging and adopting algorithms across communities.

We propose a framework called HyperVAE for encoding distributions of distributions. When a target distribution is modeled by a VAE, its neural network parameters \theta is drawn from a distribution p(\theta) which is modeled by a hyper-level VAE. We propose a variational inference using Gaussian mixture models to implicitly encode the parameters \theta into a low dimensional Gaussian distribution. Given a target distribution, we predict the posterior distribution of the latent code, then use a matrix-network decoder to generate a posterior distribution q(\theta). HyperVAE can encode the parameters \theta in full in contrast to common hyper-networks practices, which generate only the scale and bias vectors as target-network parameters. Thus HyperVAE preserves much more information about the model for each task in the latent space. We discuss HyperVAE using the minimum description length (MDL) principle and show that it helps HyperVAE to generalize. We evaluate HyperVAE in density estimation tasks, outlier detection and discovery of novel design classes, demonstrating its efficacy.