A continuous-time Markov process (CTMP) is a collection of variables indexed by a continuous quantity, time. It obeys the Markov property that the distribution over a future variable is independent of past variables given the state at the present time. We introduce continuous-time Markov process representations and algorithms for filtering, smoothing, expected sufficient statistics calculations, and model estimation, assuming no prior knowledge of continuous-time processes but some basic knowledge of probability and statistics. We begin by describing "flat" or unstructured Markov processes and then move to structured Markov processes (those arising from state spaces consisting of assignments to variables) including Kronecker, decision-diagram, and continuous-time Bayesian network representations. We provide the first connection between decision-diagrams and continuous-time Bayesian networks.

Model selection is indispensable to high-dimensional sparse modeling in selecting the best set of covariates among a sequence of candidate models. Most existing work assumes implicitly that the model is correctly specified or of fixed dimensions. Yet model misspecification and high dimensionality are common in real applications. In this paper, we investigate two classical Kullback-Leibler divergence and Bayesian principles of model selection in the setting of high-dimensional misspecified models. Asymptotic expansions of these principles reveal that the effect of model misspecification is crucial and should be taken into account, leading to the generalized AIC and generalized BIC in high dimensions. With a natural choice of prior probabilities, we suggest the generalized BIC with prior probability which involves a logarithmic factor of the dimensionality in penalizing model complexity. We further establish the consistency of the covariance contrast matrix estimator in a general setting. Our results and new method are supported by numerical studies.

This paper considers statistical estimation problems where the probability distribution of the observed random variable is invariant with respect to actions of a finite topological group. It is shown that any such distribution must satisfy a restricted finite mixture representation. When specialized to the case of distributions over the sphere that are invariant to the actions of a finite spherical symmetry group $\mathcal G$, a group-invariant extension of the Von Mises Fisher (VMF) distribution is obtained. The $\mathcal G$-invariant VMF is parameterized by location and scale parameters that specify the distribution's mean orientation and its concentration about the mean, respectively. Using the restricted finite mixture representation these parameters can be estimated using an Expectation Maximization (EM) maximum likelihood (ML) estimation algorithm. This is illustrated for the problem of mean crystal orientation estimation under the spherically symmetric group associated with the crystal form, e.g., cubic or octahedral or hexahedral. Simulations and experiments establish the advantages of the extended VMF EM-ML estimator for data acquired by Electron Backscatter Diffraction (EBSD) microscopy of a polycrystalline Nickel alloy sample.

Principal Component Analysis (PCA) has wide applications in machine learning, text mining and computer vision. Classical PCA based on a Gaussian noise model is fragile to noise of large magnitude. Laplace noise assumption based PCA methods cannot deal with dense noise effectively. In this paper, we propose Cauchy Principal Component Analysis (Cauchy PCA), a very simple yet effective PCA method which is robust to various types of noise. We utilize Cauchy distribution to model noise and derive Cauchy PCA under the maximum likelihood estimation (MLE) framework with low rank constraint. Our method can robustly estimate the low rank matrix regardless of whether noise is large or small, dense or sparse. We analyze the robustness of Cauchy PCA from a robust statistics view and present an efficient singular value projection optimization method. Experimental results on both simulated data and real applications demonstrate the robustness of Cauchy PCA to various noise patterns.

We propose the supervised hierarchical Dirichlet process (sHDP), a nonparametric generative model for the joint distribution of a group of observations and a response variable directly associated with that whole group. We compare the sHDP with another leading method for regression on grouped data, the supervised latent Dirichlet allocation (sLDA) model. We evaluate our method on two real-world classification problems and two real-world regression problems. Bayesian nonparametric regression models based on the Dirichlet process, such as the Dirichlet process-generalised linear models (DP-GLM) have previously been explored; these models allow flexibility in modelling nonlinear relationships. However, until now, Hierarchical Dirichlet Process (HDP) mixtures have not seen significant use in supervised problems with grouped data since a straightforward application of the HDP on the grouped data results in learnt clusters that are not predictive of the responses. The sHDP solves this problem by allowing for clusters to be learnt jointly from the group structure and from the label assigned to each group.

Markov Chain Monte Carlo (MCMC) algorithms are a workhorse of probabilistic modeling and inference, but are difficult to debug, and are prone to silent failure if implemented naïvely. We outline several strategies for testing the correctness of MCMC algorithms. Specifically, we advocate writing code in a modular way, where conditional probability calculations are kept separate from the logic of the sampler. We discuss strategies for both unit testing and integration testing. As a running example, we show how a Python implementation of Gibbs sampling for a mixture of Gaussians model can be tested.

Bayesian methods suffer from the problem of how to specify prior beliefs. One interesting idea is to consider worst-case priors. This requires solving a stochastic zero-sum game. In this paper, we extend well-known results from bandit theory in order to discover minimax-Bayes policies and discuss when they are practical.

This work presents a technique for statistically modeling errors introduced by reduced-order models. The method employs Gaussian-process regression to construct a mapping from a small number of computationally inexpensive `error indicators' to a distribution over the true error. The variance of this distribution can be interpreted as the (epistemic) uncertainty introduced by the reduced-order model. To model normed errors, the method employs existing rigorous error bounds and residual norms as indicators; numerical experiments show that the method leads to a near-optimal expected effectivity in contrast to typical error bounds. To model errors in general outputs, the method uses dual-weighted residuals---which are amenable to uncertainty control---as indicators. Experiments illustrate that correcting the reduced-order-model output with this surrogate can improve prediction accuracy by an order of magnitude; this contrasts with existing `multifidelity correction' approaches, which often fail for reduced-order models and suffer from the curse of dimensionality. The proposed error surrogates also lead to a notion of `probabilistic rigor', i.e., the surrogate bounds the error with specified probability.

We propose a Bayesian framework of Gaussian process in order to extend Fisher's discriminant to classify functional data such as spectra and images. The probability structure for our extended Fisher's discriminant is explicitly formulated, and we utilize the smoothness assumptions of functional data as prior probabilities. Existing methods which directly employ the smoothness assumption of functional data can be shown as special cases within this framework given corresponding priors while their estimates of the unknowns are one-step approximations to the proposed MAP estimates. Empirical results on various simulation studies and different real applications show that the proposed method significantly outperforms the other Fisher's discriminant methods for functional data.

In many domains, scientists build complex simulators of natural phenomena that encode their hypotheses about the underlying processes. These simulators can be deterministic or stochastic, fast or slow, constrained or unconstrained, and so on. Optimizing the simulators with respect to a set of parameter values is common practice, resulting in a single parameter setting that minimizes an objective subject to constraints. We propose a post optimization posterior analysis that computes and visualizes all the models that can generate equally good or better simulation results, subject to constraints. These optimization posteriors are desirable for a number of reasons among which easy interpretability, automatic parameter sensitivity and correlation analysis and posterior predictive analysis. We develop a new sampling framework based on approximate Bayesian computation (ABC) with one-sided kernels. In collaboration with two groups of scientists we applied POPE to two important biological simulators: a fast and stochastic simulator of stem-cell cycling and a slow and deterministic simulator of tumor growth patterns.