We introduce inference trees (ITs), a new class of inference methods that build on ideas from Monte Carlo tree search to perform adaptive sampling in a manner that balances exploration with exploitation, ensures consistency, and alleviates pathologies in existing adaptive methods. ITs adaptively sample from hierarchical partitions of the parameter space, while simultaneously learning these partitions in an online manner. This enables ITs to not only identify regions of high posterior mass, but also maintain uncertainty estimates to track regions where significant posterior mass may have been missed. ITs can be based on any inference method that provides a consistent estimate of the marginal likelihood. They are particularly effective when combined with sequential Monte Carlo, where they capture long-range dependencies and yield improvements beyond proposal adaptation alone.

Natural gradient descent (NGD) minimises the cost function on a Riemannian manifold whose metric is defined by the Fisher information. In this work, we prove that if the model predictions on the training set approach the true conditional distribution of labels given inputs, then the noise inherent in minibatch gradients causes the stationary distribution of NGD to approach a Bayesian posterior, whose temperature $T \approx \epsilon N/(2B)$ is controlled by the learning rate $\epsilon$, training set size $N$ and batch size $B$. The parameter-dependence of the Fisher metric introduces an implicit prior over the parameters, which we identify as the well-known Jeffreys prior. To support our claims, we show that the distribution of samples from NGD is close to the Laplace approximation to the posterior when $T = 1$. Furthermore, the test loss of ensembles drawn using NGD falls rapidly as we increase the batch size until $B \approx \epsilon N/2$, while above this point the test loss is constant or rises slowly.

Using stochastic gradient search and the optimal filter derivative, it is possible to perform recursive (i.e., online) maximum likelihood estimation in a non-linear state-space model. As the optimal filter and its derivative are analytically intractable for such a model, they need to be approximated numerically. In [Poyiadjis, Doucet and Singh, Biometrika 2018], a recursive maximum likelihood algorithm based on a particle approximation to the optimal filter derivative has been proposed and studied through numerical simulations. Here, this algorithm and its asymptotic behavior are analyzed theoretically. We show that the algorithm accurately estimates maxima to the underlying (average) log-likelihood when the number of particles is sufficiently large. We also derive (relatively) tight bounds on the estimation error. The obtained results hold under (relatively) mild conditions and cover several classes of non-linear state-space models met in practice.

As artificial intelligence is increasingly affecting all parts of society and life, there is growing recognition that human interpretability of machine learning models is important. It is often argued that accuracy or other similar generalization performance metrics must be sacrificed in order to gain interpretability. Such arguments, however, fail to acknowledge that the overall decision-making system is composed of two entities: the learned model and a human who fuses together model outputs with his or her own information. As such, the relevant performance criteria should be for the entire system, not just for the machine learning component. In this work, we characterize the performance of such two-node tandem data fusion systems using the theory of distributed detection. In doing so, we work in the population setting and model interpretable learned models as multi-level quantizers. We prove that under our abstraction, the overall system of a human with an interpretable classifier outperforms one with a black box classifier.

Learning of dynamical systems, or state-space models, is central to many machine learning problems, such as reinforcement learning, sequence modeling, and autonomous systems. Furthermore, state-space models are at the core of recent model developments within the machine learning area, such as Gaussian process state-space models (Frigola et al. 2014a; Mattos et al. 2016; etc.), infinite factorial dynamical models (Gael et al., 2009; Valera et al., 2015), and stochastic recurrent neural networks (Fraccaro et al., 2016, for example). A strategy to learn state-space models, independently suggested by Digalakis et al. (1993) and Ghahramani and Hinton (1996), is the use of the Expectation Maximization (EM, Dempster et al. 1977) method. Even though originally proposed only for maximum likelihood estimation of linear models with Gaussian noise, the strategy can be generalized to the more challenging nonlinear and non-Gaussian cases, as well as the empirical Bayes setting. Many contributions have been made during the last decade, and this paper takes another step along the path towards a more computationally efficient method with a solid theoretical ground for learning of nonlinear dynamical systems.

We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the non-spiked one is asymptotically normal below a certain reconstruction threshold which is not necessarily of a "spectral" nature, and that it is degenerate above. This establishes the maximal region of contiguity between the planted and null models. It is known that this threshold also marks a phase transition for estimating the spike: the latter task is possible above the threshold and impossible below. Therefore, both estimation and detection undergo the same transition in this random matrix model. We also provide further information about the performance of the optimal test. Our proofs are based on Gaussian interpolation methods and a rigorous incarnation of the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington--Kirkpatrick spin-glass model.

We study optimization methods for solving the maximum likelihood formulation of independent component analysis (ICA). We consider both the the problem constrained to white signals and the unconstrained problem. The Hessian of the objective function is costly to compute, which renders Newton's method impractical for large data sets. Many algorithms proposed in the literature can be rewritten as quasi-Newton methods, for which the Hessian approximation is cheap to compute. These algorithms are very fast on simulated data where the linear mixture assumption really holds. However, on real signals, we observe that their rate of convergence can be severely impaired. In this paper, we investigate the origins of this behavior, and show that the recently proposed Preconditioned ICA for Real Data (Picard) algorithm overcomes this issue on both constrained and unconstrained problems.

Given $n$ samples from a population of individuals belonging to different types with unknown proportions, how do we estimate the probability of discovering a new type at the $(n+1)$-th draw? This is a classical problem in statistics, commonly referred to as the missing mass estimation problem. Recent results by Ohannessian and Dahleh \citet{Oha12} and Mossel and Ohannessian \citet{Mos15} showed: i) the impossibility of estimating (learning) the missing mass without imposing further structural assumptions on the type proportions; ii) the consistency of the Good-Turing estimator for the missing mass under the assumption that the tail of the type proportions decays to zero as a regularly varying function with parameter $\alpha\in(0,1)$. In this paper we rely on tools from Bayesian nonparametrics to provide an alternative, and simpler, proof of the impossibility of a distribution-free estimation of the missing mass. Up to our knowledge, the use of Bayesian ideas to study large sample asymptotics for the missing mass is new, and it could be of independent interest. Still relying on Bayesian nonparametric tools, we then show that under regularly varying type proportions the convergence rate of the Good-Turing estimator is the best rate that any estimator can achieve, up to a slowly varying function, and that minimax rate must be at least $n^{-\alpha/2}$. We conclude with a discussion of our results, and by conjecturing that the Good-Turing estimator is an rate optimal minimax estimator under regularly varying type proportions.

Given independent samples generated from the joint distribution $p(\mathbf{x},\mathbf{y},\mathbf{z})$, we study the problem of Conditional Independence (CI-Testing), i.e., whether the joint equals the CI distribution $p^{CI}(\mathbf{x},\mathbf{y},\mathbf{z})= p(\mathbf{z}) p(\mathbf{y}|\mathbf{z})p(\mathbf{x}|\mathbf{z})$ or not. We cast this problem under the purview of the proposed, provable meta-algorithm, "Mimic and Classify", which is realized in two-steps: (a) Mimic the CI distribution close enough to recover the support, and (b) Classify to distinguish the joint and the CI distribution. Thus, as long as we have a good generative model and a good classifier, we potentially have a sound CI Tester. With this modular paradigm, CI Testing becomes amiable to be handled by state-of-the-art, both generative and classification methods from the modern advances in Deep Learning, which in general can handle issues related to curse of dimensionality and operation in small sample regime. We show intensive numerical experiments on synthetic and real datasets where new mimic methods such conditional GANs, Regression with Neural Nets, outperform the current best CI Testing performance in the literature. Our theoretical results provide analysis on the estimation of null distribution as well as allow for general measures, i.e., when either some of the random variables are discrete and some are continuous or when one or more of them are discrete-continuous mixtures.

The broad set of deep generative models (DGMs) has achieved remarkable advances. However, it is often difficult to incorporate rich structured domain knowledge with the end-to-end DGMs. Posterior regularization (PR) offers a principled framework to impose structured constraints on probabilistic models, but has limited applicability to the diverse DGMs that can lack a Bayesian formulation or even explicit density evaluation. PR also requires constraints to be fully specified {\it a priori}, which is impractical or suboptimal for complex knowledge with learnable uncertain parts. In this paper, we establish mathematical correspondence between PR and reinforcement learning (RL), and, based on the connection, expand PR to learn constraints as the extrinsic reward in RL. The resulting algorithm is model-agnostic to apply to any DGMs, and is flexible to adapt arbitrary constraints with the model jointly. Experiments on human image generation and templated sentence generation show models with learned knowledge constraints by our algorithm greatly improve over base generative models.