Gene expression datasets consist of thousand of genes with relatively small samplesizes (i.e. are large-$p$-small-$n$). Moreover, dependencies of various orders co-exist in the datasets. In the Undirected probabilistic Graphical Model (UGM) framework the Glasso algorithm has been proposed to deal with high dimensional micro-array datasets forcing sparsity. Also, modifications of the default Glasso algorithm are developed to overcome the problem of complex interaction structure. In this work we advocate the use of a simple score-based Hill Climbing algorithm (HC) that learns Gaussian Bayesian Networks (BNs) leaning on Directed Acyclic Graphs (DAGs). We compare HC with Glasso and its modifications in the UGM framework on their capability to reconstruct GRNs from micro-array data belonging to the Escherichia Coli genome. We benefit from the analytical properties of the Joint Probability Density (JPD) function on which both directed and undirected PGMs build to convert DAGs to UGMs. We conclude that dependencies in complex data are learned best by the HC algorithm, presenting them most accurately and efficiently, simultaneously modelling strong local and weaker but significant global connections coexisting in the gene expression dataset. The HC algorithm adapts intrinsically to the complex dependency structure of the dataset, without forcing a specific structure in advance. On the contrary, Glasso and modifications model unnecessary dependencies at the expense of the probabilistic information in the network and of a structural bias in the JPD function that can only be relieved including many parameters.

In former articles, we have been looking at many different concepts related to Bayesian Networks: how to represent them in a compact manner, their reasoning patterns, and I-Map. In this article, we will take a look at D-separation and try to understand it both intuitively and mathematically.

Variational inference (VI) is a technique to approximate difficult to compute posteriors by optimization. In contrast to MCMC, VI scales to many observations. In the case of complex posteriors, however, state-of-the-art VI approaches often yield unsatisfactory posterior approximations. This paper presents Bernstein flow variational inference (BF-VI), a robust and easy-to-use method, flexible enough to approximate complex multivariate posteriors. BF-VI combines ideas from normalizing flows and Bernstein polynomial-based transformation models. In benchmark experiments, we compare BF-VI solutions with exact posteriors, MCMC solutions, and state-of-the-art VI methods including normalizing flow based VI. We show for low-dimensional models that BF-VI accurately approximates the true posterior; in higher-dimensional models, BF-VI outperforms other VI methods. Further, we develop with BF-VI a Bayesian model for the semi-structured Melanoma challenge data, combining a CNN model part for image data with an interpretable model part for tabular data, and demonstrate for the first time how the use of VI in semi-structured models.

We establish new generalisation bounds for multiclass classification by abstracting to a more general setting of discretised error types. Extending the PAC-Bayes theory, we are hence able to provide fine-grained bounds on performance for multiclass classification, as well as applications to other learning problems including discretisation of regression losses. Tractable training objectives are derived from the bounds. The bounds are uniform over all weightings of the discretised error types and thus can be used to bound weightings not foreseen at training, including the full confusion matrix in the multiclass classification case.

In this paper, we study the long-time convergence and uniform strong propagation of chaos for a class of nonlinear Markov chains for Markov chain Monte Carlo (MCMC). Our technique is quite simple, making use of recent contraction estimates for linear Markov kernels and basic techniques from Markov theory and analysis. Moreover, the same proof strategy applies to both the long-time convergence and propagation of chaos. We also show, via some experiments, that these nonlinear MCMC techniques are viable for use in real-world high-dimensional inference such as Bayesian neural networks.

The (probabilistic) graphical model consists of a collection of probability distributions that factorize according to the structure of an underlying graph [52]. The graphical model captures the complex dependencies among random variables and build large-scale multivariate statistical models, which has been used in many research areas such as hierarchical Bayesian models [27], contingency table analysis [20, 53] in categorical data analysis [1, 23, 37], constraint satisfaction [16, 15], language and speech processing [11, 31], image processing [17, 24, 28] and spatial statistics more generally [8]. In our work, we focus on the undirected graphical models, where the probability distribution factorizes according to the function defined on the cliques of the graph. The undirected graphical models have a variety of applications, including statistical physics [32], natural language processing [38], image analysis [54] and spatial statistics [43]. Specifically, we pay attention to the undirected graphical models which can be described as exponential families, a broad class of probability distributions elaborately studied in many statistical literature [4, 21, 13]. The properties of the exponential families provide some connections between the inference methods and the convex analysis [12, 29]. There are many well-known examples that are undirected graphical models viewed as exponential families, such as Ising model [32, 5], Gaussian MRF [46] and latent Dirichlet allocation [11].

We will understand how our assumptions on the data enable us to create meaningful optimization problems. In fact, we will derive commonly used criteria such as cross-entropy in classification and mean square error in regression. Finally, I am trying to answer an interview question that I encountered: What would happen if we use MSE on binary classification? To begin, let's start with a fundamental question: what is the difference between likelihood and probability? The data xxx are connected to the possible models θ\thetaθ by means of a probability P(x,θ)P(x,\theta)P(x,θ) or a probability density function (pdf) p(x,θ)p(x,\theta)p(x,θ).

In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model. The given targets are statistical moments for which the underlying realizations are not available. Under these conditions, the Kullback-Leibler divergence minimum principle is used for estimating the posterior probability measure. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. The MCMC generator and the necessary numerical elements are given to facilitate the implementation of the methodology in a parallel computing framework. In a second part, an application is presented to illustrate the proposed theory and is also, as such, a contribution to the three-dimensional stochastic homogenization of heterogeneous linear elastic media in the case of a non-separation of the microscale and macroscale. For the construction of the posterior probability measure by using the probabilistic learning inference, in addition to the constraints defined by given statistical moments of the random effective elasticity tensor, the second-order moment of the random normalized residue of the stochastic partial differential equation has been added as a constraint. This constraint guarantees that the algorithm seeks to bring the statistical moments closer to their targets while preserving a small residue.

Bayesian Neural Networks (BNNs) offer a mathematically grounded framework to quantify the uncertainty of model predictions but come with a prohibitive computation cost for both training and inference. In this work, we show a novel network architecture search (NAS) that optimizes BNNs for both accuracy and uncertainty while having a reduced inference latency. Different from canonical NAS that optimizes solely for in-distribution likelihood, the proposed scheme searches for the uncertainty performance using both in- and out-of-distribution data. Our method is able to search for the correct placement of Bayesian layer(s) in a network. In our experiments, the searched models show comparable uncertainty quantification ability and accuracy compared to the state-of-the-art (deep ensemble). In addition, the searched models use only a fraction of the runtime compared to many popular BNN baselines, reducing the inference runtime cost by $2.98 \times$ and $2.92 \times$ respectively on the CIFAR10 dataset when compared to MCDropout and deep ensemble.

Simulator-based models are models for which the likelihood is intractable but simulation of synthetic data is possible. They are often used to describe complex real-world phenomena, and as such can often be misspecified in practice. Unfortunately, existing Bayesian approaches for simulators are known to perform poorly in those cases. In this paper, we propose a novel algorithm based on the posterior bootstrap and maximum mean discrepancy estimators. This leads to a highly-parallelisable Bayesian inference algorithm with strong robustness properties. This is demonstrated through an in-depth theoretical study which includes generalisation bounds and proofs of frequentist consistency and robustness of our posterior. The approach is then assessed on a range of examples including a g-and-k distribution and a toggle-switch model.