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In this paper, we present a novel methodology to perform Bayesian inference for Cox processes in which the intensity function is driven by a diffusion process. The novelty lies in the fact that no discretization error is involved, despite the non-tractability of both the likelihood function and the transition density of the diffusion. The methodology is based on an MCMC algorithm and its exactness is built on retrospective sampling techniques. The efficiency of the methodology is investigated in some simulated examples and its applicability is illustrated in some real data analyzes.

Dynamic treatment regimes (DTRs) consist of a sequence of decision rules, one per stage of intervention, that finds effective treatments for individual patients according to patient information history. DTRs can be estimated from models which include the interaction between treatment and a small number of covariates which are often chosen a priori. However, with increasingly large and complex data being collected, it is difficult to know which prognostic factors might be relevant in the treatment rule. Therefore, a more data-driven approach of selecting these covariates might improve the estimated decision rules and simplify models to make them easier to interpret. We propose a variable selection method for DTR estimation using penalized dynamic weighted least squares. Our method has the strong heredity property, that is, an interaction term can be included in the model only if the corresponding main terms have also been selected. Through simulations, we show our method has both the double robustness property and the oracle property, and the newly proposed methods compare favorably with other variable selection approaches.

The stable under iterated tessellation (STIT) process is a stochastic process that produces a recursive partition of space with cut directions drawn independently from a distribution over the sphere. The case of random axis-aligned cuts is known as the Mondrian process. Random forests and Laplace kernel approximations built from the Mondrian process have led to efficient online learning methods and Bayesian optimization. In this work, we utilize tools from stochastic geometry to resolve some fundamental questions concerning STIT processes in machine learning. First, we show that a STIT process with cut directions drawn from a discrete distribution can be efficiently simulated by lifting to a higher dimensional axis-aligned Mondrian process. Second, we characterize all possible kernels that stationary STIT processes and their mixtures can approximate. We also give a uniform convergence rate for the approximation error of the STIT kernels to the targeted kernels, generalizing the work of [3] for the Mondrian case. Third, we obtain consistency results for STIT forests in density estimation and regression. Finally, we give a formula for the density estimator arising from an infinite STIT random 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.

Neighbor Embedding (NE) aims to preserve pairwise similarities between data items and has been shown to yield an effective principle for data visualization. However, even the best existing NE methods such as Stochastic Neighbor Embedding (SNE) may leave large-scale patterns hidden, for example, clusters, despite strong signals being present in the data. To address this, we propose a new cluster visualization method based on the Neighbor Embedding principle. We first present a family of Neighbor Embedding methods which generalizes SNE by using non-normalized Kullback-Leibler divergence with a scale parameter. In this family, much better cluster visualizations often appear with a parameter value different from the one corresponding to SNE. We also develop an efficient software which employs asynchronous stochastic block coordinate descent to optimize the new family of objective functions. Our experimental results demonstrate that the method consistently and substantially improves visualization of data clusters compared with the state-of-the-art NE approaches.

Stochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton-Jacobi-Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ($ n\geq 3$). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme. We show that matrices resulting from spatial discretization and temporal discretization are M--matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided.

Models with intractable normalizing functions have numerous applications ranging from network models to image analysis to spatial point processes. Because the normalizing constants are functions of the parameters of interest, standard Markov chain Monte Carlo cannot be used for Bayesian inference for these models. A number of algorithms have been developed for such models. Some have the posterior distribution as the asymptotic distribution. Other "asymptotically inexact" algorithms do not possess this property. There is limited guidance for evaluating approximations based on these algorithms, and hence it is very hard to tune them. We propose two new diagnostics that address these problems for intractable normalizing function models. Our first diagnostic, inspired by the second Bartlett identity, applies in principle to any asymptotically exact or inexact algorithm. We develop an approximate version of this new diagnostic that is applicable to intractable normalizing function problems. Our second diagnostic is a Monte Carlo approximation to a kernel Stein discrepancy-based diagnostic introduced by Gorham and Mackey (2017). We provide theoretical justification for our methods. We apply our diagnostics to several algorithms in the context of challenging simulated and real data examples, including an Ising model, an exponential random graph model, and a Markov point process.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.

The process of translation is ambiguous, in that there are typically many valid trans- lations for a given sentence. This gives rise to significant variation in parallel cor- pora, however, most current models of machine translation do not account for this variation, instead treating the prob- lem as a deterministic process. To this end, we present a deep generative model of machine translation which incorporates a chain of latent variables, in order to ac- count for local lexical and syntactic varia- tion in parallel corpora. We provide an in- depth analysis of the pitfalls encountered in variational inference for training deep generative models. Experiments on sev- eral different language pairs demonstrate that the model consistently improves over strong baselines.

Dynamic topic models (DTMs) model the evolution of prevalent themes in literature, online media, and other forms of text over time. DTMs assume that word co-occurrence statistics change continuously and therefore impose continuous stochastic process priors on their model parameters. These dynamical priors make inference much harder than in regular topic models, and also limit scalability. In this paper, we present several new results around DTMs. First, we extend the class of tractable priors from Wiener processes to the generic class of Gaussian processes (GPs). This allows us to explore topics that develop smoothly over time, that have a long-term memory or are temporally concentrated (for event detection). Second, we show how to perform scalable approximate inference in these models based on ideas around stochastic variational inference and sparse Gaussian processes. This way we can train a rich family of DTMs to massive data. Our experiments on several large-scale datasets show that our generalized model allows us to find interesting patterns that were not accessible by previous approaches.

In this paper, we develop the continuous time dynamic topic model (cDTM). The cDTM is a dynamic topic model that uses Brownian motion to model the latent topics through a sequential collection of documents, where a "topic" is a pattern of word use that we expect to evolve over the course of the collection. We derive an efficient variational approximate inference algorithm that takes advantage of the sparsity of observations in text, a property that lets us easily handle many time points. In contrast to the cDTM, the original discrete-time dynamic topic model (dDTM) requires that time be discretized. Moreover, the complexity of variational inference for the dDTM grows quickly as time granularity increases, a drawback which limits fine-grained discretization. We demonstrate the cDTM on two news corpora, reporting both predictive perplexity and the novel task of time stamp prediction.