The Mondrian process is a stochastic process that produces a recursive partition of space with random axis-aligned cuts. Random forests and Laplace kernel approximations built from the Mondrian process have led to efficient online learning methods and Bayesian optimization. By viewing the Mondrian process as a special case of the stable under iterated tessellation (STIT) process, we utilize tools from stochastic geometry to resolve three fundamental questions concern generalizability of the Mondrian process in machine learning. First, we show that the Mondrian process with general cut directions can be efficiently simulated, but it is unlikely to give rise to better classification or regression algorithms. Second, we characterize all possible kernels that generalizations of the Mondrian process can approximate. This includes, for instance, various forms of the weighted Laplace kernel and the exponential kernel. Third, we give an explicit formula for the density estimator arising from a Mondrian forest. This allows for precise comparisons between the Mondrian forest, the Mondrian kernel and the Laplace kernel in density estimation. Our paper calls for further developments at the novel intersection of stochastic geometry and machine learning.

The aim of probabilistic programming is to automatize every aspect of probabilistic inference in arbitrary probabilistic models (programs) so that the user can focus her attention on modeling, without dealing with ad-hoc inference methods. Gradient based automatic differentiation stochastic variational inference offers an attractive option as the default method for (differentiable) probabilistic programming as it combines high performance with high computational efficiency. However, the performance of any (parametric) variational approach depends on the choice of an appropriate variational family. Here, we introduced a fully automatic method for constructing structured variational families inspired to the closed-form update in conjugate models. These pseudo-conjugate families incorporate the forward pass of the input probabilistic program and can capture complex statistical dependencies. Pseudo-conjugate families have the same space and time complexity of the input probabilistic program and are therefore tractable in a very large class of models. We validate our automatic variational method on a wide range of high dimensional inference problems including deep learning components.

In this work, we develop a novel Bayesian estimation method for the Dirichlet process (DP) mixture of the inverted Dirichlet distributions, which has been shown to be very flexible for modeling vectors with positive elements. The recently proposed extended variational inference (EVI) framework is adopted to derive an analytically tractable solution. The convergency of the proposed algorithm is theoretically guaranteed by introducing single lower bound approximation to the original objective function in the VI framework. In principle, the proposed model can be viewed as an infinite inverted Dirichelt mixture model (InIDMM) that allows the automatic determination of the number of mixture components from data. Therefore, the problem of pre-determining the optimal number of mixing components has been overcome. Moreover, the problems of over-fitting and under-fitting are avoided by the Bayesian estimation approach. Comparing with several recently proposed DP-related methods, the good performance and effectiveness of the proposed method have been demonstrated with both synthesized data and real data evaluations.

Several countries are currently investigating issues of neglect, poor quality care and abuse in the aged care sector. In most cases it is the State who license and monitor aged care providers, which frequently introduces a serious conflict of interest because the State also operate many of the facilities where our most vulnerable peoples are cared for. Where issues are raised with the standard of care being provided, the State are seen by many as a deep-pockets defendant and become the target of high-value lawsuits. This paper draws on cases and circumstances from one jurisdiction based on the English legal tradition, Australia, and proposes a Bayesian solution capable of determining probability for success for citizen plaintiffs who bring negligence claims against a public authority defendant. Use of a Bayesian network trained on case audit data shows that even when the plaintiff case meets all requirements for a successful negligence litigation, success is not often assured. Only in around one-fifth of these cases does the plaintiff succeed against a public authority as defendant.

Item Response Theory is a ubiquitous algorithm used around the world to understand humans based on their responses to questions in fields as diverse as education, medicine and psychology. However, for medium to large datasets, contemporary solutions pose a tradeoff: either have bayesian, interpretable, accurate estimates or have fast computation. We introduce variational inference and deep generative models to Item Response Theory to offer the best of both worlds. The resulting algorithm is (a) orders of magnitude faster when inferring on the classical model, (b) naturally extends to more complicated input than binary correct/incorrect, and more expressive deep bayesian models of responses. Applying this method to five large-scale item response datasets from cognitive science and education, we find improvements in imputing missing data and better log likelihoods. The open-source algorithm is immediately usable.

A Bayesian network is a graphical model that encodes probabilistic relationships among variables of interest. When used in conjunction with statistical techniques, the graphical model has several advantages for data analysis. One, because the model encodes dependencies among all variables, it readily handles situations where some data entries are missing. Two, a Bayesian network can be used to learn causal relationships, and hence can be used to gain understanding about a problem domain and to predict the consequences of intervention. Three, because the model has both a causal and probabilistic semantics, it is an ideal representation for combining prior knowledge (which often comes in causal form) and data. Four, Bayesian statistical methods in conjunction with Bayesian networks offer an efficient and principled approach for avoiding the overfitting of data. In this paper, we discuss methods for constructing Bayesian networks from prior knowledge and summarize Bayesian statistical methods for using data to improve these models. With regard to the latter task, we describe methods for learning both the parameters and structure of a Bayesian network, including techniques for learning with incomplete data. In addition, we relate Bayesian-network methods for learning to techniques for supervised and unsupervised learning. We illustrate the graphical-modeling approach using a real-world case study.

For instance, penalties are used to improve generalization, prune neurons or reduce the rank of tensors of weights. Therefore, usual penalties are mostly empirical and user-defined, and integrated to the loss as follows: L( w) null( w) r (w), with w the vector of all parameters in the network, null( w) the error term and r (w) the penalty term. From a Bayesian point of view, optimizing such a loss L is equivalent to finding the Maximum A Posteriori (MAP) of the parameters w given the training data and a prior α exp( r). Indeed, assuming that the loss null is a log-likelihood loss, namely, null(w) ln p w( D) with dataset D, then minimizing L is equivalent to minimizing L MAP(w) ln p w(D) ln(α (w)). Thus, within the MAP framework, we can interpret the penalty term r as the influence of a prior α [14]. However, the MAP approximates the Bayesian posterior very roughly, by taking its maximum. Variational Inference (VI) provides a variational posterior distribution rather than a single value, hopefully representing the Bayesian posterior much better. VI looks for the best posterior approximation within a family β u(w) of approximate posteriors over w, parameterized Inria, Team TAU, Gif-sur-Yvette, France † Facebook, France 1 arXiv:2002.00178v1

Nested sampling is an efficient algorithm for the calculation of the Bayesian evidence and posterior parameter probability distributions. It is based on the step-by-step exploration of the parameter space by Monte Carlo sampling with a series of values sets called live points that evolve towards the region of interest, i.e. where the likelihood function is maximal. In presence of several local likelihood maxima, the algorithm converges with difficulty. Some systematic errors can also be introduced by unexplored parameter volume regions. In order to avoid this, different methods are proposed in the literature for an efficient search of new live points, even in presence of local maxima. Here we present a new solution based on the mean shift cluster recognition method implemented in a random walk search algorithm. The clustering recognition is integrated within the Bayesian analysis program NestedFit. It is tested with the analysis of some difficult cases. Compared to the analysis results without cluster recognition, the computation time is considerably reduced. At the same time, the entire parameter space is efficiently explored, which translates into a smaller uncertainty of the extracted value of the Bayesian evidence.

We consider distributions arising from a mixture of causal models, where each model is represented by a directed acyclic graph (DAG). We provide a graphical representation of such mixture distributions and prove that this representation encodes the conditional independence relations of the mixture distribution. We then consider the problem of structure learning based on samples from such distributions. Since the mixing variable is latent, we consider causal structure discovery algorithms such as FCI that can deal with latent variables. We show that such algorithms recover a "union" of the component DAGs and can identify variables whose conditional distribution across the component DAGs vary. We demonstrate our results on synthetic and real data showing that the inferred graph identifies nodes that vary between the different mixture components. As an immediate application, we demonstrate how retrieval of this causal information can be used to cluster samples according to each mixture component.

Approximate Bayesian Computation is widely used in systems biology for inferring parameters in stochastic gene regulatory network models. Its performance hinges critically on the ability to summarize high-dimensional system responses such as time series into a few informative, low-dimensional summary statistics. The quality of those statistics critically affect the accuracy of the inference. Existing methods to select the best subset out of a pool of candidate statistics do not scale well with large pools. Since it is imperative for good performance this becomes a serious bottleneck when doing inference on complex and high-dimensional problems. This paper proposes a convolutional neural network architecture for automatically learning informative summary statistics of temporal responses. We show that the proposed network can effectively circumvent the statistics selection problem as a preprocessing step to ABC for a challenging inference problem learning parameters in a high-dimensional stochastic genetic oscillator. We also study the impact of experimental design on network performance by comparing different data richness and different data acquisition strategies.