The Ising model is a celebrated example of a Markov random field, introduced in statistical physics to model ferromagnetism. This is a discrete exponential family with binary outcomes, where the sufficient statistic involves a quadratic term designed to capture correlations arising from pairwise interactions. However, in many situations the dependencies in a network arise not just from pairs, but from peer-group effects. A convenient mathematical framework for capturing higher-order dependencies, is the $p$-tensor Ising model, where the sufficient statistic consists of a multilinear polynomial of degree $p$. This thesis develops a framework for statistical inference of the natural parameters in $p$-tensor Ising models. We begin with the Curie-Weiss Ising model, where we unearth various non-standard phenomena in the asymptotics of the maximum-likelihood (ML) estimates of the parameters, such as the presence of a critical curve in the interior of the parameter space on which these estimates have a limiting mixture distribution, and a surprising superefficiency phenomenon at the boundary point(s) of this curve. ML estimation fails in more general $p$-tensor Ising models due to the presence of a computationally intractable normalizing constant. To overcome this issue, we use the popular maximum pseudo-likelihood (MPL) method, which avoids computing the inexplicit normalizing constant based on conditional distributions. We derive general conditions under which the MPL estimate is $\sqrt{N}$-consistent, where $N$ is the size of the underlying network. Finally, we consider a more general Ising model, which incorporates high-dimensional covariates at the nodes of the network, that can also be viewed as a logistic regression model with dependent observations. In this model, we show that the parameters can be estimated consistently under sparsity assumptions on the true covariate vector.
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We study a variation of Bayesian M-ary hypothesis testing in which the test outputs a list of L candidates out of the M possible upon processing the observation. We study the minimum error probability of list hypothesis testing, where an error is defined as the event where the true hypothesis is not in the list output by the test. We derive two exact expressions of the minimum probability or error. The first is expressed as the error probability of a certain non-Bayesian binary hypothesis test, and is reminiscent of the meta-converse bound. The second, is expressed as the tail probability of the likelihood ratio between the two distributions involved in the aforementioned non-Bayesian binary hypothesis test.
The autoregressive process is one of the fundamental and most important models that analyze a time series. Theoretical results and practical tools for fitting an autoregressive process with i.i.d. innovations are well-established. However, when the innovations are white noise but not i.i.d., those tools fail to generate a consistent confidence interval for the autoregressive coefficients. Focus on an autoregressive process with \textit{dependent} and \textit{non-stationary} innovations, this paper provides a consistent result and a Gaussian approximation theorem for the Yule-Walker estimator. Moreover, it introduces the second order wild bootstrap that constructs a consistent confidence interval for the estimator. Numerical experiments confirm the validity of the proposed algorithm with different kinds of white noise innovations. Meanwhile, the classical method(e.g., AR(Sieve) bootstrap) fails to generate a correct confidence interval when the innovations are dependent. According to Kreiss et al. \cite{10.1214/11-AOS900} and the Wold decomposition, assuming a real-life time series satisfies an autoregressive process is reasonable. However, innovations in that process are more likely to be white noises instead of i.i.d.. Therefore, our method should provide a practical tool that handles real-life problems.
We focus on the problem of manifold estimation: given a set of observations sampled close to some unknown submanifold $M$, one wants to recover information about the geometry of $M$. Minimax estimators which have been proposed so far all depend crucially on the a priori knowledge of some parameters quantifying the underlying distribution generating the sample (such as bounds on its density), whereas those quantities will be unknown in practice. Our contribution to the matter is twofold: first, we introduce a one-parameter family of manifold estimators $(\hat{M}_t)_{t\geq 0}$ based on a localized version of convex hulls, and show that for some choice of $t$, the corresponding estimator is minimax on the class of models of $C^2$ manifolds introduced in [Genovese et al., Manifold estimation and singular deconvolution under Hausdorff loss]. Second, we propose a completely data-driven selection procedure for the parameter $t$, leading to a minimax adaptive manifold estimator on this class of models. This selection procedure actually allows us to recover the Hausdorff distance between the set of observations and $M$, and can therefore be used as a scale parameter in other settings, such as tangent space estimation.
In this paper several related estimation problems are addressed from a Bayesian point of view and optimal estimators are obtained for each of them when some natural loss functions are considered. Namely, we are interested in estimating a regression curve. Simultaneously, the estimation problems of a conditional distribution function, or a conditional density, or even the conditional distribution itself, are considered. All these problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, parametric and non-parametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function comonly used in estimating a real function of the unknown parameter. The cornerstone of the mentioned Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate these results.
The Glivenko-Cantelli theorem states that the empirical distribution function converges uniformly almost surely to the theoretical distribution for a random variable $X \in \mathbb{R}$. This is an important result because it establishes the fact that sampling does capture the dispersion measure the distribution function $F$ imposes. In essence, sampling permits one to learn and infer the behavior of $F$ by only looking at observations from $X$. The probabilities that are inferred from samples $\mathbf{X}$ will become more precise as the sample size increases and more data becomes available. Therefore, it is valid to study distributions via samples. The proof present here is constructive, meaning that the result is derived directly from the fact that the empirical distribution function converges pointwise almost surely to the theoretical distribution. The work includes a proof of this preliminary statement and attempts to motivate the intuition one gets from sampling techniques when studying the regions in which a model concentrates probability. The sets where dispersion is described with precision by the empirical distribution function will eventually cover the entire sample space.
We develop a prior probability model for temporal Poisson process intensities through structured mixtures of Erlang densities with common scale parameter, mixing on the integer shape parameters. The mixture weights are constructed through increments of a cumulative intensity function which is modeled nonparametrically with a gamma process prior. Such model specification provides a novel extension of Erlang mixtures for density estimation to the intensity estimation setting. The prior model structure supports general shapes for the point process intensity function, and it also enables effective handling of the Poisson process likelihood normalizing term resulting in efficient posterior simulation. The Erlang mixture modeling approach is further elaborated to develop an inference method for spatial Poisson processes. The methodology is examined relative to existing Bayesian nonparametric modeling approaches, including empirical comparison with Gaussian process prior based models, and is illustrated with synthetic and real data examples.
Phase estimation is a quantum algorithm for measuring the eigenvalues of a Hamiltonian. We propose and rigorously analyse a randomized phase estimation algorithm with two distinctive features. First, our algorithm has complexity independent of the number of terms L in the Hamiltonian. Second, unlike previous L-independent approaches, such as those based on qDRIFT, all sources of error in our algorithm can be suppressed by collecting more data samples, without increasing the circuit depth.
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs.
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed implementation schemes behind solving {high-dimensional} ODEs with a probabilistic numerical algorithm. This has not been possible before due to matrix-matrix operations in each solver step, but is crucial for scientifically relevant problems -- most importantly, the solution of discretised {partial} differential equations. In a nutshell, efficient high-dimensional probabilistic ODE solutions build either on independence assumptions or on Kronecker structure in the prior model. We evaluate the resulting efficiency on a range of problems, including the probabilistic numerical simulation of a differential equation with millions of dimensions.
Amortized inference has led to efficient approximate inference for large datasets. The quality of posterior inference is largely determined by two factors: a) the ability of the variational distribution to model the true posterior and b) the capacity of the recognition network to generalize inference over all datapoints. We analyze approximate inference in variational autoencoders in terms of these factors. We find that suboptimal inference is often due to amortizing inference rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation.
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