This work investigates the intersection property of conditional independence. It states that for random variables $A,B,C$ and $X$ we have that $X$ independent of $A$ given $B,C$ and $X$ independent of $B$ given $A,C$ implies $X$ independent of $(A,B)$ given $C$. Under the assumption that the joint distribution has a continuous density, we provide necessary and sufficient conditions under which the intersection property holds. The result has direct applications to causal inference: it leads to strictly weaker conditions under which the graphical structure becomes identifiable from the joint distribution of an additive noise model.

Contaminated mixture models are developed for model-based clustering of data with asymmetric clusters as well as spurious points, outliers, and/or noise. Specifically, we introduce a contaminated mixture of contaminated shifted asymmetric Laplace distributions and a contaminated mixture of contaminated skew-normal distributions. In each case, mixture components have a parameter controlling the proportion of bad points (i.e., spurious points, outliers, and/or noise) and one specifying the degree of contamination. A very important feature of our approaches is that these parameters do not have to be specified a priori. Expectation-conditional maximization algorithms are outlined for parameter estimation and the number of components is selected using the Bayesian information criterion. The performance of our approaches is illustrated on artificial and real data.

Sample size determination for a data set is an important statistical process for analyzing the data to an optimum level of accuracy and using minimum computational work. The applications of this process are credible in every domain which deals with large data sets and high computational work. This study uses Bayesian analysis for determination of minimum sample size of vibration signals to be considered for fault diagnosis of a bearing using pre-defined parameters such as the inverse standard probability and the acceptable margin of error. Thus an analytical formula for sample size determination is introduced. The fault diagnosis of the bearing is done using a machine learning approach using an entropy-based J48 algorithm. The following method will help researchers involved in fault diagnosis to determine minimum sample size of data for analysis for a good statistical stability and precision.

Nuclear magnetic resonance (NMR) spectroscopy exploits the magnetic properties of atomic nuclei to discover the structure, reaction state and chemical environment of molecules. We propose a probabilistic generative model and inference procedures for NMR spectroscopy. Specifically, we use a weighted sum of trigonometric functions undergoing exponential decay to model free induction decay (FID) signals. We discuss the challenges in estimating the components of this general model -- amplitudes, phase shifts, frequencies, decay rates, and noise variances -- and offer practical solutions. We compare with conventional Fourier transform spectroscopy for estimating the relative concentrations of chemicals in a mixture, using synthetic and experimentally acquired FID signals. We find the proposed model is particularly robust to low signal to noise ratios (SNR), and overlapping peaks in the Fourier transform of the FID, enabling accurate predictions (e.g., 1% sensitivity at low SNR) which are not possible with conventional spectroscopy (5% sensitivity).

Approximate Bayesian computation (ABC) methods are used to approximate posterior distributions using simulation rather than likelihood calculations. We introduce Gaussian process (GP) accelerated ABC, which we show can significantly reduce the number of simulations required. As computational resource is usually the main determinant of accuracy in ABC, GP-accelerated methods can thus enable more accurate inference in some models. GP models of the unknown log-likelihood function are used to exploit continuity and smoothness, reducing the required computation. We use a sequence of models that increase in accuracy, using intermediate models to rule out regions of the parameter space as implausible. The methods will not be suitable for all problems, but when they can be used, can result in significant computational savings. For the Ricker model, we are able to achieve accurate approximations to the posterior distribution using a factor of 100 fewer simulator evaluations than comparable Monte Carlo approaches, and for a population genetics model we are able to approximate the exact posterior for the first time.

We describe an adaptation of the simulated annealing algorithm to nonparametric clustering and related probabilistic models. This new algorithm learns nonparametric latent structure over a growing and constantly churning subsample of training data, where the portion of data subsampled can be interpreted as the inverse temperature beta(t) in an annealing schedule. Gibbs sampling at high temperature (i.e., with a very small subsample) can more quickly explore sketches of the final latent state by (a) making longer jumps around latent space (as in block Gibbs) and (b) lowering energy barriers (as in simulated annealing). We prove subsample annealing speeds up mixing time N^2 -> N in a simple clustering model and exp(N) -> N in another class of models, where N is data size. Empirically subsample-annealing outperforms naive Gibbs sampling in accuracy-per-wallclock time, and can scale to larger datasets and deeper hierarchical models. We demonstrate improved inference on million-row subsamples of US Census data and network log data and a 307-row hospital rating dataset, using a Pitman-Yor generalization of the Cross Categorization model.

Determinantal point processes (DPPs) are well-suited for modeling repulsion and have proven useful in many applications where diversity is desired. While DPPs have many appealing properties, such as efficient sampling, learning the parameters of a DPP is still considered a difficult problem due to the non-convex nature of the likelihood function. In this paper, we propose using Bayesian methods to learn the DPP kernel parameters. These methods are applicable in large-scale and continuous DPP settings even when the exact form of the eigendecomposition is unknown. We demonstrate the utility of our DPP learning methods in studying the progression of diabetic neuropathy based on spatial distribution of nerve fibers, and in studying human perception of diversity in images.

Hierarchical statistical models are widely employed in information science and data engineering. The models consist of two types of variables: observable variables that represent the given data and latent variables for the unobservable labels. An asymptotic analysis of the models plays an important role in evaluating the learning process; the result of the analysis is applied not only to theoretical but also to practical situations, such as optimal model selection and active learning. There are many studies of generalization errors, which measure the prediction accuracy of the observable variables. However, the accuracy of estimating the latent variables has not yet been elucidated. For a quantitative evaluation of this, the present paper formulates distribution-based functions for the errors in the estimation of the latent variables. The asymptotic behavior is analyzed for both the maximum likelihood and the Bayes methods.

In this paper, we provide more evidence for the contention that logical consequence should be understood in normative terms. Hartry Field and John MacFarlane covered the classical case. We extend their work, examining what it means for an agent to be obliged to infer a conclusion when faced with uncertain information or reasoning within a non-monotonic, defeasible, logical framework (which allows e. g. for inference to be drawn from premises considered true unless evidence to the contrary is presented).

CMF is a technique for simultaneously learning low-rank representations based on a collection of matrices with shared entities. A typical example is the joint modeling of user-item, item-property, and user-feature matrices in a recommender system. The key idea in CMF is that the embeddings are shared across the matrices, which enables transferring information between them. The existing solutions, however, break down when the individual matrices have low-rank structure not shared with others. In this work we present a novel CMF solution that allows each of the matrices to have a separate low-rank structure that is independent of the other matrices, as well as structures that are shared only by a subset of them. We compare MAP and variational Bayesian solutions based on alternating optimization algorithms and show that the model automatically infers the nature of each factor using group-wise sparsity. Our approach supports in a principled way continuous, binary and count observations and is efficient for sparse matrices involving missing data. We illustrate the solution on a number of examples, focusing in particular on an interesting use-case of augmented multi-view learning.