models often rely on the forward-backward sampler. This makes them computationally slow as the length of the time series increases, motivating the development of sub-sampling-based approaches. These approximate the full posterior by using small random subsequences of the data at each MCMC iteration within stochastic gradient MCMC. In the presence of imbalanced data resulting from rare latent states, subsequences often exclude rare latent state data, leading to inaccurate inference and prediction/detection of rare events. We propose a targeted subsampling (TASS) approach that over-samples observations corresponding to rare latent states when calculating the stochastic gradient of parameters associated with them. TASS uses an initial clustering of the data to construct subsequence weights that reduce the variance in gradient estimation. This leads to improved sampling efficiency, in particular in settings where the rare latent states correspond to extreme observations. We demonstrate substantial gains in predictive and inferential accuracy on real and synthetic examples.

An intriguing new class of piecewise deterministic Markov processes (PDMPs) has recently been proposed as an alternative to Markov chain Monte Carlo (MCMC). We propose a new class of PDMPs termed Gibbs zig-zag samplers, which allow parameters to be updated in blocks with a zig-zag sampler applied to certain parameters and traditional MCMC-style updates to others. We demonstrate the flexibility of this framework on posterior sampling for logistic models with shrinkage priors for high-dimensional regression and random effects, and provide conditions for geometric ergodicity and the validity of a central limit theorem.

It is common to be interested in rankings or order relationships among entities. In complex settings where one does not directly measure a univariate statistic upon which to base ranks, such inferences typically rely on statistical models having entity-specific parameters. These can be treated as random effects in hierarchical models characterizing variation among the entities. In this paper, we are particularly interested in the problem of ranking basketball players in terms of their contribution to team performance. Using data from the United States National Basketball Association (NBA), we find that many players have similar latent ability levels, making any single estimated ranking highly misleading. The current literature fails to provide summaries of order relationships that adequately account for uncertainty. Motivated by this, we propose a Bayesian strategy for characterizing uncertainty in inferences on order relationships among players and lineups. Our approach adapts to scenarios in which uncertainty in ordering is high by producing more conservative results that improve interpretability. This is achieved through a reward function within a decision theoretic framework. We apply our approach to data from the 2009-10 NBA season.

Mixture Markov Model (MMM) is a widely used tool to cluster sequences of events coming from a finite state-space. However the MMM likelihood being multi-modal, the challenge remains in its maximization. Although Expectation-Maximization (EM) algorithm remains one of the most popular ways to estimate the MMM parameters, however convergence of EM algorithm is not always guaranteed. Given the computational challenges in maximizing the mixture likelihood on the constrained parameter space, we develop a pattern search-based global optimization technique which can optimize any objective function on a collection of simplexes, which is eventually used to maximize MMM likelihood. This is shown to outperform other related global optimization techniques. In simulation experiments, the proposed method is shown to outperform the expectation-maximization (EM) algorithm in the context of MMM estimation performance. The proposed method is applied to cluster Multiple sclerosis (MS) patients based on their treatment sequences of disease-modifying therapies (DMTs). We also propose a novel method to cluster people with MS based on DMT prescriptions and associated clinical features (covariates) using MMM with covariates. Based on the analysis, we divided MS patients into 3 clusters. Further cluster-specific summaries of relevant covariates indicate patient differences among the clusters.

Summary Sequential Monte Carlo methods are typically not straightforward to implement on parallel architectures. This is because standard resampling schemes involve communication between all particles. The $$\alpha$$-sequential Monte Carlo method was proposed recently as a potential solution to this that limits communication between particles. This limited communication is controlled through a sequence of stochastic matrices known as $$\alpha$$ matrices. We study the influence of the communication structure on the convergence and stability properties of the resulting algorithms. In particular, we quantitatively show that the mixing properties of the $$\alpha$$ matrices play an important role in the stability properties of the algorithm. Moreover, we prove that one can ensure good mixing properties by using randomized communication structures where each particle only communicates with a few neighbouring particles. The resulting algorithms converge at the usual Monte Carlo rate. This leads to efficient versions of distributed sequential Monte Carlo.

Bayesian computational algorithms tend to scale poorly as data size increases. This has motivated divide-and-conquer-based approaches for scalable inference. These divide the data into subsets, perform inference for each subset in parallel, and then combine these inferences. While appealing theoretical properties and practical performance have been demonstrated for independent observations, scalable inference for dependent data remains challenging. In this work, we study the problem of Bayesian inference from very long time series. The literature in this area focuses mainly on approximate approaches that usually lack rigorous theoretical guarantees and may provide arbitrarily poor accuracy in practice. We propose a simple and scalable divide-and-conquer method, and provide accuracy guarantees. Numerical simulations and real data applications demonstrate the effectiveness of our approach.

Classification with high-dimensional data is of widespread interest and often involves dealing with imbalanced data. Bayesian classification approaches are hampered by the fact that current Markov chain Monte Carlo algorithms for posterior computation become inefficient as the number [Formula: see text] of predictors or the number [Formula: see text] of subjects to classify gets large, because of the increasing computational time per step and worsening mixing rates. One strategy is to employ a gradient-based sampler to improve mixing while using data subsamples to reduce the per-step computational complexity. However, the usual subsampling breaks down when applied to imbalanced data. Instead, we generalize piecewise-deterministic Markov chain Monte Carlo algorithms to include importance-weighted and mini-batch subsampling. These maintain the correct stationary distribution with arbitrarily small subsamples and substantially outperform current competitors. We provide theoretical support for the proposed approach and demonstrate its performance gains in simulated data examples and an application to cancer data.

In conducting non-linear dimensionality reduction and feature learning, it is common to suppose that the data lie near a lower-dimensional manifold. A class of model-based approaches for such problems includes latent variables in an unknown non-linear regression function; this includes Gaussian process latent variable models and variational auto-encoders (VAEs) as special cases. VAEs are artificial neural networks (ANNs) that employ approximations to make computation tractable; however, current implementations lack adequate uncertainty quantification in estimating the parameters, predictive densities, and lower-dimensional subspace, and can be unstable and lack interpretability in practice. We attempt to solve these problems by deploying Markov chain Monte Carlo sampling algorithms (MCMC) for Bayesian inference in ANN models with latent variables. We address issues of identifiability by imposing constraints on the ANN parameters as well as by using anchor points. This is demonstrated on simulated and real data examples. We find that current MCMC sampling schemes face fundamental challenges in neural networks involving latent variables, motivating new research directions.

Particle filters are a powerful and flexible tool for performing inference on state-space models. They involve a collection of samples evolving over time through a combination of sampling and re-sampling steps. The re-sampling step is necessary to ensure that weight degeneracy is avoided. In several situations of statistical interest, it is important to be able to compare the estimates produced by two different particle filters; consequently, being able to efficiently couple two particle filter trajectories is often of paramount importance. In this text, we propose several ways to do so. In particular, we leverage ideas from the optimal transportation literature. In general, though computing the optimal transport map is extremely computationally expensive, to deal with this, we introduce computationally tractable approximations to optimal transport couplings. We demonstrate that our resulting algorithms for coupling two particle filter trajectories often perform orders of magnitude more efficiently than more standard approaches.

Markov chain Monte Carlo (MCMC) algorithms for hidden Markov models often rely on the forward-backward sampler. This makes them computationally slow as the length of the time series increases, motivating the development of sub-sampling-based approaches. These approximate the full posterior by using small random subsequences of the data at each MCMC iteration within stochastic gradient MCMC. In the presence of imbalanced data resulting from rare latent states, subsequences often exclude rare latent state data, leading to inaccurate inference and prediction/detection of rare events. We propose a targeted sub-sampling (TASS) approach that over-samples observations corresponding to rare latent states when calculating the stochastic gradient of parameters associated with them. TASS uses an initial clustering of the data to construct subsequence weights that reduce the variance in gradient estimation. This leads to improved sampling efficiency, in particular in settings where the rare latent states correspond to extreme observations. We demonstrate substantial gains in predictive and inferential accuracy on real and synthetic examples.

Estimation of parameters that obey specific constraints is crucial in statistics and machine learning; for example, when parameters are required to satisfy boundedness, monotonicity, or linear inequalities. Traditional approaches impose these constraints via constraint-specific transformations or sampling approaches, or by truncating the posterior distribution. Such methods often result in computational challenges, limited flexibility, and a lack of generality. We propose a generalized framework for constrained Bayesian inference by projecting the unconstrained posterior distribution into the space of the parameter constraints, providing a computationally efficient and easily implementable solution for a large class of problems. We rigorously establish the theoretical foundations of the projected posterior distribution, as well as providing asymptotic results for posterior consistency, posterior contraction, and optimal coverage properties. Our methodology is validated through both theoretical arguments and practical applications.

Pricing options is an important problem in financial engineering. In many scenarios of practical interest, financial option prices associated to an underlying asset reduces to computing an expectation w.r.t.~a diffusion process. In general, these expectations cannot be calculated analytically, and one way to approximate these quantities is via the Monte Carlo method; Monte Carlo methods have been used to price options since at least the 1970's. It has been seen in Del Moral, P. \& Shevchenko, P.V. (2014) `Valuation of barrier options using Sequential Monte Carlo' and Jasra, A. \& Del Moral, P. (2011) `Sequential Monte Carlo for option pricing' that Sequential Monte Carlo (SMC) methods are a natural tool to apply in this context and can vastly improve over standard Monte Carlo. In this article, in a similar spirit to Del Moral, P. \& Shevchenko, P.V. (2014) `Valuation of barrier options using sequential Monte Carlo' and Jasra, A. \& Del Moral, P. (2011) `Sequential Monte Carlo for option pricing' we show that one can achieve significant gains by using SMC methods by constructing a sequence of artificial target densities over time. In particular, we approximate the optimal importance sampling distribution in the SMC algorithm by using a sequence of weighting functions. This is demonstrated on two examples, barrier options and target accrual redemption notes (TARN's). We also provide a proof of unbiasedness of our SMC estimate.