We introduce Sequential Neural Posterior Score Estimation (SNPSE) and Sequential Neural Likelihood Score Estimation (SNLSE), two new score-based methods for Bayesian inference in simulator-based models. Our methods, inspired by the success of score-based methods in generative modelling, leverage conditional score-based diffusion models to generate samples from the posterior distribution of interest. These models can be trained using one of two possible objective functions, one of which approximates the score of the intractable likelihood, while the other directly estimates the score of the posterior. We embed these models into a sequential training procedure, which guides simulations using the current approximation of the posterior at the observation of interest, thereby reducing the simulation cost. We validate our methods, as well as their amortised, non-sequential variants, on several numerical examples, demonstrating comparable or superior performance to existing state-of-the-art methods such as Sequential Neural Posterior Estimation (SNPE) and Sequential Neural Likelihood Estimation (SNLE).

Graphical models are an important tool in exploring relationships between variables in complex, multivariate data. Methods for learning such graphical models are well developed in the case where all variables are either continuous or discrete, including in high-dimensions. However, in many applications data span variables of different types (e.g. continuous, count, binary, ordinal, etc.), whose principled joint analysis is nontrivial. Latent Gaussian copula models, in which all variables are modeled as transformations of underlying jointly Gaussian variables, represent a useful approach. Recent advances have shown how the binary-continuous case can be tackled, but the general mixed variable type regime remains challenging. In this work, we make the simple yet useful observation that classical ideas concerning polychoric and polyserial correlations can be leveraged in a latent Gaussian copula framework. Building on this observation we propose flexible and scalable methodology for data with variables of entirely general mixed type. We study the key properties of the approaches theoretically and empirically, via extensive simulations as well an illustrative application to data from the UK Biobank concerning COVID-19 risk factors.

Decomposition of the evidence lower bound (ELBO) objective of VAE used for density estimation revealed the deficiency of VAE for representation learning and suggested ways to improve the model. In this paper, we investigate whether we can get similar insights by decomposing the ELBO for semi-supervised classification using VAE model. Specifically, we show that mutual information between input and class labels decreases during maximization of ELBO objective. We propose a method to address this issue. We also enforce cluster assumption to aid in classification. Experiments on a diverse datasets verify that our method can be used to improve the classification performance of existing VAE based semi-supervised models. Experiments also show that, this can be achieved without sacrificing the generative power of the model.

Markov Chain Monte Carlo (MCMC) methods are the workhorse of Bayesian computation when closed formulas for estimators or probability distributions are not available. For this reason they have been central to the development and success of high-dimensional Bayesian statistics in the last decades, where one attempts to generate samples from some posterior distribution Π( |data) arising from a prior Π on D-dimensional Euclidean space and the observed data vector. MCMC methods tend to perform well in a large variety of problems, are very flexible and user-friendly, and enjoy many theoretical guarantees. Under mild assumptions, they are known to converge to their stationary'target' distributions as a consequence of the ergodic theorem, albeit perhaps at a slow speed, requiring a large number of iterations to provide numerically accurate algorithms. When the target distribution is log-concave, MCMC algorithms are known to mix rapidly, even in high dimensions.

Nowadays, machine learning models and applications have become increasingly pervasive. With this rapid increase in the development and employment of machine learning models, a concern regarding privacy has risen. Thus, there is a legitimate need to protect the data from leaking and from any attacks. One of the strongest and most prevalent privacy models that can be used to protect machine learning models from any attacks and vulnerabilities is differential privacy (DP). DP is strict and rigid definition of privacy, where it can guarantee that an adversary is not capable to reliably predict if a specific participant is included in the dataset or not. It works by injecting a noise to the data whether to the inputs, the outputs, the ground truth labels, the objective functions, or even to the gradients to alleviate the privacy issue and protect the data. To this end, this survey paper presents different differentially private machine learning algorithms categorized into two main categories (traditional machine learning models vs. deep learning models). Moreover, future research directions for differential privacy with machine learning algorithms are outlined.

Due to the noises in crowdsourced labels, label aggregation (LA) has emerged as a standard procedure to post-process crowdsourced labels. LA methods estimate true labels from crowdsourced labels by modeling worker qualities. Most existing LA methods are iterative in nature. They need to traverse all the crowdsourced labels multiple times in order to jointly and iteratively update true labels and worker qualities until convergence. Consequently, these methods have high space and time complexities. In this paper, we treat LA as a dynamic system and model it as a Dynamic Bayesian network. From the dynamic model we derive two light-weight algorithms, LA\textsuperscript{onepass} and LA\textsuperscript{twopass}, which can effectively and efficiently estimate worker qualities and true labels by traversing all the labels at most twice. Due to the dynamic nature, the proposed algorithms can also estimate true labels online without re-visiting historical data. We theoretically prove the convergence property of the proposed algorithms, and bound the error of estimated worker qualities. We also analyze the space and time complexities of the proposed algorithms and show that they are equivalent to those of majority voting. Experiments conducted on 20 real-world datasets demonstrate that the proposed algorithms can effectively and efficiently aggregate labels in both offline and online settings even if they traverse all the labels at most twice.

Mutual Information (MI) and Conditional Mutual Information (CMI) are multi-purpose tools from information theory that are able to naturally measure the statistical dependencies between random variables, thus they are usually of central interest in several statistical and machine learning tasks, such as conditional independence testing and representation learning. However, estimating CMI, or even MI, is infamously challenging due the intractable formulation. In this study, we introduce DINE (Diffeomorphic Information Neural Estimator)-a novel approach for estimating CMI of continuous random variables, inspired by the invariance of CMI over diffeomorphic maps. We show that the variables of interest can be replaced with appropriate surrogates that follow simpler distributions, allowing the CMI to be efficiently evaluated via analytical solutions. Additionally, we demonstrate the quality of the proposed estimator in comparison with state-of-the-arts in three important tasks, including estimating MI, CMI, as well as its application in conditional independence testing. The empirical evaluations show that DINE consistently outperforms competitors in all tasks and is able to adapt very well to complex and high-dimensional relationships.

Parameter estimation in statistics and system identification relies on data that may contain sensitive information. To protect this sensitive information, the notion of \emph{differential privacy} (DP) has been proposed, which enforces confidentiality by introducing randomization in the estimates. Standard algorithms for differentially private estimation are based on adding an appropriate amount of noise to the output of a traditional point estimation method. This leads to an accuracy-privacy trade off, as adding more noise reduces the accuracy while increasing privacy. In this paper, we propose a new Unified Bayes Private Point (UBaPP) approach to Bayes point estimation of the unknown parameters of a data generating mechanism under a DP constraint, that achieves a better accuracy-privacy trade off than traditional approaches. We verify the performance of our approach on a simple numerical example.

Recent progress in autoencoder-based sparse identification of nonlinear dynamics (SINDy) under $\ell_1$ constraints allows joint discoveries of governing equations and latent coordinate systems from spatio-temporal data, including simulated video frames. However, it is challenging for $\ell_1$-based sparse inference to perform correct identification for real data due to the noisy measurements and often limited sample sizes. To address the data-driven discovery of physics in the low-data and high-noise regimes, we propose Bayesian SINDy autoencoders, which incorporate a hierarchical Bayesian sparsifying prior: Spike-and-slab Gaussian Lasso. Bayesian SINDy autoencoder enables the joint discovery of governing equations and coordinate systems with a theoretically guaranteed uncertainty estimate. To resolve the challenging computational tractability of the Bayesian hierarchical setting, we adapt an adaptive empirical Bayesian method with Stochatic gradient Langevin dynamics (SGLD) which gives a computationally tractable way of Bayesian posterior sampling within our framework. Bayesian SINDy autoencoder achieves better physics discovery with lower data and fewer training epochs, along with valid uncertainty quantification suggested by the experimental studies. The Bayesian SINDy autoencoder can be applied to real video data, with accurate physics discovery which correctly identifies the governing equation and provides a close estimate for standard physics constants like gravity $g$, for example, in videos of a pendulum.

A key design constraint when implementing Monte Carlo and variational inference algorithms is that it must be possible to cheaply and exactly evaluate the marginal densities of proposal distributions and variational families. This takes many interesting proposals off the table, such as those based on involved simulations or stochastic optimization. This paper broadens the design space, by presenting a framework for applying Monte Carlo and variational inference algorithms when proposal densities cannot be exactly evaluated. Our framework, recursive auxiliary-variable inference (RAVI), instead approximates the necessary densities using meta-inference: an additional layer of Monte Carlo or variational inference, that targets the proposal, rather than the model. RAVI generalizes and unifies several existing methods for inference with expressive approximating families, which we show correspond to specific choices of meta-inference algorithm, and provides new theory for analyzing their bias and variance. We illustrate RAVI's design framework and theorems by using them to analyze and improve upon Salimans et al.'s Markov Chain Variational Inference, and to design a novel sampler for Dirichlet process mixtures, achieving state-of-the-art results on a standard benchmark dataset from astronomy and on a challenging datacleaning task with Medicare hospital data.