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High dimensional probabilistic models are used for many modern scientific and engineering data analysis tasks. Interpreting neural spike trains, compressing video, identifying features in DNA microarrays, and recognizing particles in high energy physics all rely upon the ability to find and model complex structure in a high dimensional space. Despite their great promise, high dimensional probabilistic models are frequently computationally intractable to work with in practice. In this thesis I develop solutions to overcome this intractability, primarily in the context of energy based models.A common cause of intractability is that model distributions cannot be analytically normalized. Probabilities can only be computed up to a constant, making training exceedingly difficult. To solve this problem I propose `minimum probability flow learning', a variational technique for parameter estimation in such models. The utility of this training technique is demonstrated in the case of an Ising model, a Hopfield auto-associative memory, an independent component analysis model of natural images, and a deep belief network.A second common difficulty in training probabilistic models arises when the parameter space is ill-conditioned. This makes gradient descent optimization slow and impractical, but can be alleviated using the natural gradient. I show here that the natural gradient can be related to signal whitening, and provide specific prescriptions for applying it to learning problems.It is also difficult to evaluate the performance of models that cannot be analytically normalized, providing a particular challenge to hypothesis testing and model comparison. To overcome this, I introduce a method termed `Hamiltonian annealed importance sampling,' which more efficiently estimates the normalization constant of non-analytically-normalizable models. This method is then used to calculate and compare the log likelihoods of several state of the art probabilistic models of natural image patches. Finally, many tasks performed with a trained probabilistic model (for instance, image denoising or inpainting and speech recognition) involve generating samples from the model distribution, which is typically a very computationally expensive process. I introduce a modification to Hamiltonian Monte Carlo sampling that reduces the tendency of sampling trajectories to double back on themselves, and enables statistically independent samples to be generated more rapidly.Taken together, it is my hope that these contributions will help scientists and engineers to build and manipulate probabilistic models.

We consider estimating the marginal likelihood in settings with independent and identically distributed (i.i.d.) data. We propose estimating the predictive distributions in a sequential factorization of the marginal likelihood in such settings by using stochastic gradient Markov Chain Monte Carlo techniques. This approach is far more efficient than traditional marginal likelihood estimation techniques such as nested sampling and annealed importance sampling due to its use of mini-batches to approximate the likelihood. Stability of the estimates is provided by an adaptive annealing schedule. The resulting stochastic gradient annealed importance sampling (SGAIS) technique, which is the key contribution of our paper, enables us to estimate the marginal likelihood of a number of models considerably faster than traditional approaches, with no noticeable loss of accuracy. An important benefit of our approach is that the marginal likelihood is calculated in an online fashion as data becomes available, allowing the estimates to be used for applications such as online weighted model combination.

In this paper, we investigate restricted Boltzmann machines (RBMs) from the exponential family perspective, en-abling the visible units to follow any suitable distributions from the exponential family. We derive a unified view to compute the free energy function for exponential family RBMs (exp-RBMs). Based on that, annealed important sampling (AIS) is generalized to the entire exponential family, allowing for estimating the log-partition function and log-likelihood. Our experiments on a document processing task demonstrate that the generalized free energy functions and AIS estimation perform well in helping capture useful knowledge from the data; the estimated log-partition functions are stable. The appropriate instances of exp-RBMs can generate novel and meaningful samples and can be applied to classification tasks., 2018 International Joint Conference on Neural Networks (IJCNN), July 7-13, 2018, Rio de Janeiro, Brazil, Series: Proceedings of International Joint Conference on Neural Networks

We present an original simulation-based method to estimate likelihood ratios efficiently for general state-space models. Our method relies on a novel use of the conditional Sequential Monte Carlo (cSMC) algorithm introduced in \citet{Andrieu_et_al_2010} and presents several practical advantages over standard approaches. The ratio is estimated using a unique source of randomness instead of estimating separately the two likelihood terms involved. Beyond the benefits in terms of variance reduction one may expect in general from this type of approach, an important point here is that the variance of this estimator decreases as the distance between the likelihood parameters decreases. We show how this can be exploited in the context of Monte Carlo Markov chain (MCMC) algorithms, leading to the development of a new class of exact-approximate MCMC methods to perform Bayesian static parameter inference in state-space models. We show through simulations that, in contrast to the Particle Marginal Metropolis-Hastings (PMMH) algorithm of Andrieu_et_al_2010, the computational effort required by this novel MCMC scheme scales very favourably for large data sets.

Sequential Monte Carlo methods, also known as particle methods, are a popular set of techniques for approximating high-dimensional probability distributions and their normalizing constants. These methods have found numerous applications in statistics and related fields; for example, for inference in nonlinear non-Gaussian state space models, and in complex static models. Like many Monte Carlo sampling schemes, they rely on proposal distributions which crucially impact their performance. We introduce here a class of controlled sequential Monte Carlo algorithms, where the proposal distributions are determined by approximating the solution to an associated optimal control problem using an iterative scheme. This method builds upon a number of existing algorithms in econometrics, physics and statistics for inference in state space models, and generalizes these methods so as to accommodate complex static models. We provide a theoretical analysis concerning the fluctuation and stability of this methodology that also provides insight into the properties of related algorithms. We demonstrate significant gains over state-of-the-art methods at a fixed computational complexity on a variety of applications.

Sequential Monte Carlo (SMC) methods are widely used for filtering purposes of non-linear economic or financial models. Nevertheless the SMC scope encompasses wider applications such as estimating static model parameters so much that it is becoming a serious alternative to Markov- Chain Monte-Carlo (MCMC) methods. Not only SMC algorithms draw posterior distributions of static or dynamic parameters but additionally provide an estimate of the normalizing constant. The tempered and time (TNT) algorithm, developed in the paper, combines (off-line) tempered SMC inference with on-line SMC inference for estimating many slightly different distributions. The method encompasses the Iterated Batch Importance Sampling (IBIS) algorithm and more generally the Resample Move (RM) algorithm. Besides the number of particles, the TNT algorithm self-adjusts its calibrated parameters and relies on a new MCMC kernel that allows for particle interactions. The algorithm is well suited for efficiently back-testing models. We conclude by comparing in-sample and out-of-sample performances of complex volatility models.

We present methods based on Metropolis-coupled Markov chain Monte Carlo (MC3) and annealed importance sampling (AIS) for estimating the posterior distribution of Bayesian networks. The methods draw samples from an appropriate distribution of partial orders on the nodes, continued by sampling directed acyclic graphs (DAGs) conditionally on the sampled partial orders. We show that the computations needed for the sampling algorithms are feasible as long as the encountered partial orders have relatively few down-sets. While the algorithms assume suitable modularity properties of the priors, arbitrary priors can be handled by dividing the importance weight of each sampled DAG by the number of topological sorts it has - we give a practical dynamic programming algorithm to compute these numbers. Our empirical results demonstrate that the presented partial-order-based samplers are superior to previous Markov chain Monte Carlo methods, which sample DAGs either directly or via linear orders on the nodes. The results also suggest that the convergence rate of the estimators based on AIS are competitive to those of MC3. Thus AIS is the preferred method, as it enables easier large-scale parallelization and, in addition, supplies good probabilistic lower bound guarantees for the marginal likelihood of the model.