Hidden Markov models (HMMs) are flexible tools for clustering dependent data coming from unknown populations, allowing nonparametric modelling of the population densities. Identifiability fails when the data is in fact independent and identically distributed (i.i.d.), and we study the frontier between learnable and unlearnable two-state nonparametric HMMs. Learning the parameters of the HMM requires solving a nonlinear inverse problem whose difficulty depends not only on the smoothnesses of the populations but also on the distance to the i.i.d. boundary of the parameter set. The latter difficulty is mostly ignored in the literature in favour of assumptions precluding nearly independent data. This is the first work conducting a precise nonasymptotic, nonparametric analysis of the minimax risk taking into account all aspects of the hardness of the problem, in the case of two populations. Our analysis reveals an unexpected interplay between the distance to the i.i.d. boundary and the relative smoothnesses of the two populations: a surprising and intriguing transition occurs in the rate when the two densities have differing smoothnesses. We obtain upper and lower bounds revealing that, close to the i.i.d. boundary, it is possible to "borrow strength" from the estimator of the smoother density to improve the risk of the other.

Membership inference attacks (MIA) can reveal whether a particular data point was part of the training dataset, potentially exposing sensitive information about individuals. This article provides theoretical guarantees by exploring the fundamental statistical limitations associated with MIAs on machine learning models at large. More precisely, we first derive the statistical quantity that governs the effectiveness and success of such attacks. We then theoretically prove that in a non-linear regression setting with overfitting learning procedures, attacks may have a high probability of success. Finally, we investigate several situations for which we provide bounds on this quantity of interest. Interestingly, our findings indicate that discretizing the data might enhance the learning procedure's security. Specifically, it is demonstrated to be limited by a constant, which quantifies the diversity of the underlying data distribution. We illustrate those results through simple simulations.

We consider the problem of state estimation in general state-space models using variational inference. For a generic variational family defined using the same backward decomposition as the actual joint smoothing distribution, we establish under mixing assumptions that the variational approximation of expectations of additive state functionals induces an error which grows at most linearly in the number of observations. This guarantee is consistent with the known upper bounds for the approximation of smoothing distributions using standard Monte Carlo methods. We illustrate our theoretical result with state-of-the art variational solutions based both on the backward parameterization and on alternatives using forward decompositions. This numerical study proposes guidelines for variational inference based on neural networks in state-space models.

We study the frontier between learnable and unlearnable hidden Markov models (HMMs). HMMs are flexible tools for clustering dependent data coming from unknown populations. The model parameters are known to be fully identifiable (up to label-switching) without any modelling assumption on the distributions of the populations as soon as the clusters are distinct and the hidden chain is ergodic with a full rank transition matrix. In the limit as any one of these conditions fails, it becomes impossible in general to identify parameters. For a chain with two hidden states we prove nonasymptotic minimax upper and lower bounds, matching up to constants, which exhibit thresholds at which the parameters become learnable. We also provide an upper bound on the relative entropy rate for parameters in a neighbourhood of the unlearnable region which may have interest in itself.

Given a nonparametric Hidden Markov Model (HMM) with two states, the question of constructing efficient multiple testing procedures is considered, treating the states as unknown null and alternative hypotheses. A procedure is introduced, based on nonparametric empirical Bayes ideas, that controls the False Discovery Rate (FDR) at a user-specified level. Guarantees on power are also provided, in the form of a control of the true positive rate. One of the key steps in the construction requires supremum-norm convergence of preliminary estimators of the emission densities of the HMM. We provide the existence of such estimators, with convergence at the optimal minimax rate, for the case of a HMM with $J\ge 2$ states, which is of independent interest.

We introduce a new general identifiable framework for principled disentanglement referred to as Structured Nonlinear Independent Component Analysis (SNICA). Our contribution is to extend the identifiability theory of deep generative models for a very broad class of structured models. While previous works have shown identifiability for specific classes of time-series models, our theorems extend this to more general temporal structures as well as to models with more complex structures such as spatial dependencies. In particular, we establish the major result that identifiability for this framework holds even in the presence of noise of unknown distribution. Finally, as an example of our framework's flexibility, we introduce the first nonlinear ICA model for time-series that combines the following very useful properties: it accounts for both nonstationarity and autocorrelation in a fully unsupervised setting; performs dimensionality reduction; models hidden states; and enables principled estimation and inference by variational maximum-likelihood.

This paper considers hidden Markov models where the observations are given as the sum of a latent state which lies in a general state space and some independent noise with unknown distribution. It is shown that these fully nonparametric translation models are identifiable with respect to both the distribution of the latent variables and the distribution of the noise, under mostly a light tail assumption on the latent variables. Two nonparametric estimation methods are proposed and we prove that the corresponding estimators are consistent for the weak convergence topology. These results are illustrated with numerical experiments.

In this paper, we consider the filtering and smoothing recursions in nonparametric finite state space hidden Markov models (HMMs) when the parameters of the model are unknown and replaced by estimators. We provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors. We prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the L1-risk of the estimators. It has been proved very recently that statistical inference for finite state space nonparametric HMMs is possible. We study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the L2-risk of the nonparametric spectral estimators. In the case where the observation space is compact, this provides explicit rates for the filtering and smoothing errors in total variation norm. The performance of the spectral method is assessed with simulated data for both the estimation of the (nonparametric) conditional distribution of the observations and the estimation of the marginal smoothing distributions.

We consider stationary hidden Markov models with finite state space and nonparametric modeling of the emission distributions. It has remained unknown until very recently that such models are identifiable. In this paper, we propose a new penalized least- squares estimator for the emission distributions which is statistically optimal and practically tractable. We prove a non asymptotic oracle inequality for our nonparametric estimator of the emission distributions. A consequence is that this new estimator is rate minimax adaptive up to a logarithmic term. Our methodology is based on projections of the emission distributions onto nested subspaces of increasing complexity. The popular spectral estimators are unable to achieve the optimal rate but may be used as initial points in our procedure. Simulations are given that show the improvement obtained when applying the least-squares minimization consecutively to the spectral estimation.

In this paper, we study the problem of lossless universal source coding for stationary memoryless sources on countably infinite alphabets. This task is generally not achievable without restricting the class of sources over which universality is desired. Building on our prior work, we propose natural families of sources characterized by a common dominating envelope. We particularly emphasize the notion of adaptivity, which is the ability to perform as well as an oracle knowing the envelope, without actually knowing it. This is closely related to the notion of hierarchical universal source coding, but with the important difference that families of envelope classes are not discretely indexed and not necessarily nested. Our contribution is to extend the classes of envelopes over which adaptive universal source coding is possible, namely by including max-stable (heavy-tailed) envelopes which are excellent models in many applications, such as natural language modeling. We derive a minimax lower bound on the redundancy of any code on such envelope classes, including an oracle that knows the envelope. We then propose a constructive code that does not use knowledge of the envelope. The code is computationally efficient and is structured to use an expanding threshold for auto-censoring (ETAC), and we therefore dub it the ETAC-code. We prove that the ETAC-code achieves the lower bound on the minimax redundancy within a factor logarithmic in the sequence length, and can be therefore qualified as a near-adaptive code over families of heavy-tailed envelopes. For finite and light-tailed envelopes, the penalty is even less, and the same code follows closely previous results that explicitly made the light-tailed assumption. Our technical results are founded on methods from regular variation theory and concentration of measure.

This paper sheds light on adaptive coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate (log-concave envelope distributions). We prove that the auto-censuring (AC) code is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on nonasymptotic bounds for maxima of samples from discrete distributions with finite and nondecreasing hazard rate.

Consider an i.i.d. sequence of random variables whose distribution f* lies in one of a nested family of models M_q, q>=1. The smallest index q* such that M_{q*} contains f* is called the model order. We establish strong consistency of the penalized likelihood order estimator in a general setting with penalties of order \eta(q) log log n, where \eta(q) is a dimensional quantity. Moreover, such penalties are shown to be minimal. In contrast to previous work, an a priori upper bound on the model order is not assumed. The results rely on a sharp characterization of the pathwise fluctuations of the generalized likelihood ratio statistic under entropy assumptions on the model classes. Our results are applied to the geometrically complex problem of location mixture order estimation, which is widely used but poorly understood.

This paper describes universal lossless coding strategies for compressing sources on countably infinite alphabets. Classes of memoryless sources defined by an envelope condition on the marginal distribution provide benchmarks for coding techniques originating from the theory of universal coding over finite alphabets. We prove general upper bounds on minimax regret and lower bounds on minimax redundancy for such source classes. The general upper bounds emphasize the role of the normalized maximum likelihood (NML) codes with respect to minimax regret in the infinite alphabet context. Lower bounds are derived by tailoring sharp bounds on the redundancy of Krichevsky-Trofimov coders for sources over finite alphabets. Up to logarithmic (resp., constant) factors the bounds are matching for source classes defined by algebraically declining (resp., exponentially vanishing) envelopes. Effective and (almost) adaptive coding techniques are described for the collection of source classes defined by algebraically vanishing envelopes. Those results extend our knowledge concerning universal coding to contexts where the key tools from parametric inference are known to fail.

This paper is concerned with error exponents in testing problems raised by autoregressive (AR) modeling. The tests to be considered are variants of generalized likelihood ratio testing corresponding to traditional approaches to autoregressive moving-average (ARMA) modeling estimation. In several related problems, such as Markov order or hidden Markov model order estimation, optimal error exponents have been determined thanks to large deviations theory. AR order testing is specially challenging since the natural tests rely on quadratic forms of Gaussian processes. In sharp contrast with empirical measures of Markov chains, the large deviation principles (LDPs) satisfied by Gaussian quadratic forms do not always admit an information-theoretic representation. Despite this impediment, we prove the existence of nontrivial error exponents for Gaussian AR order testing. And furthermore, we exhibit situations where the exponents are optimal. These results are obtained by showing that the log-likelihood process indexed by AR models of a given order satisfy an LDP upper bound with a weakened information-theoretic representation.

We consider the estimation of the number of hidden states (the order) of a discrete-time finite-alphabet hidden Markov model (HMM). The estimators we investigate are related to code-based order estimators: penalized maximum-likelihood (ML) estimators and penalized versions of the mixture estimator introduced by Liu and Narayan (1994). We prove strong consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works. We prove a version of Stein's lemma for HMM order estimation and derive an upper bound on underestimation exponents. Then we prove that this upper bound can be achieved by the penalized ML estimator and by the penalized mixture estimator. The proof of the latter result gets around the elusive nature of the ML in HMM by resorting to large-deviation techniques for empirical processes. Finally, we prove that for any consistent HMM order estimator, for most HMM, the overestimation exponent is null.

In generalized moment problems (signed) measures are searched to fit given observations, or continuous functions are searched to fit given constraints. Known convex methods for solving such problems, and their stochastic interpretations via maximum entropy on the mean (MEM) and in a Bayesian sense are reviewed, with some improvements on previous results. Then the MEM and Bayesian approaches are extended to default models with a dependence structure, yielding new families of solutions. One family involves a transfer kernel, and allows using prior information such as modality, convexity, or Sobolev norms. Another family of solutions with possibly nonconvex criteria, is arrived at using default models with exchangeable random variables. The main technical tools are convex analysis and large deviations theory.

We propose a new method for the blind deconvolution of a discrete linear system perturbed with additive noise. The method comes from a characterization of discrete variables when perturbed with additive noise with unknown variance together with a characterization of this variance through Hankel matrix. Based on this probabilistic description, an estimator is proposed for the inverse system and the variance of the noise. These estimators are shown to be consistent under weak assumptions, whatever the signal-to-noise ratio is. In particular, the input signal needs not be independently distributed. Numerical examples demonstrate the effectiveness of the method, even when nonstationary signals are used as inputs.

An approximate maximum likelihood method for blind source separation and deconvolution of noisy signal is proposed. This technique relies upon a data augmentation scheme, where the (unobserved) input are viewed as the missing data. In the technique described, the input signal distribution is modeled by a mixture of Gaussian distributions, enabling the use of explicit formula for computing the posterior density and conditional expectation and thus avoiding Monte-Carlo integrations. Because this technique is able to capture some salient features of the input signal distribution, it performs generally much better than third-order or fourth-order cumulant based techniques.

A common approach to blind deconvolution of Bernoulli-Gaussian processes consists of performing both signal restoration and hyperparameter identification through maximization of a single generalized likelihood criterion. It is shown on a simple example that the resulting hyperparameter estimates may not converge toward any meaningful value. Therefore, other more reliable approaches should be adopted whenever possible.>