We investigate both the theoretical and algorithmic aspects of likelihood-based methods for recovering a complex-valued signal from multiple sets of measurements, referred to as looks, affected by speckle (multiplicative) noise. Our theoretical contributions include establishing the first existing theoretical upper bound on the Mean Squared Error (MSE) of the maximum likelihood estimator under the deep image prior hypothesis. Our theoretical results capture the dependence of MSE upon the number of parameters in the deep image prior, the number of looks, the signal dimension, and the number of measurements per look. On the algorithmic side, we introduce the concept of bagged Deep Image Priors (Bagged-DIP) and integrate them with projected gradient descent. Furthermore, we show how employing Newton-Schulz algorithm for calculating matrix inverses within the iterations of PGD reduces the computational complexity of the algorithm. We will show that this method achieves the state-of-the-art performance.

The leading strategy for analyzing unstructured data uses two steps. First, latent variables of economic interest are estimated with an upstream information retrieval model. Second, the estimates are treated as "data" in a downstream econometric model. We establish theoretical arguments for why this two-step strategy leads to biased inference in empirically plausible settings. More constructively, we propose a one-step strategy for valid inference that uses the upstream and downstream models jointly. The one-step strategy (i) substantially reduces bias in simulations; (ii) has quantitatively important effects in a leading application using CEO time-use data; and (iii) can be readily adapted by applied researchers.

The pursuit of direct model equation discovery has been an ongoing and significant area of interest in scientific machine learning. The popular sparse identification of nonlinear dynamics (SINDy) framework [1] offers a promising approach to extract parsimonious equations directly from data. SINDy's promotion of parsimony by sparse regression allows for the identification of an interpretable model that balances accuracy with generalizability, while its simplicity leads to a relatively efficient and fast learning process compared to other machine learning techniques. The framework has been successfully applied in a variety of applications, such as model idenficiation in plasma physics [2], control engineering [3, 4], biological transport problems [5], socio-cognitive systems [6], epidemiology [7, 8] and turbulence modelling [9]. Furthermore, its remarkable extendibility has attracted a range of modifications, including the adaptation to discover partial differential equations [10], the extension to libraries of rational functions [11], the integration of ensembling techniques to improve data efficiency [12] and the use of weak formulations [13, 14] to avoid noise amplification when computing derivatives from discrete data. One major difficulty in using scientific machine learning methods in fields such as biophysics, ecology, and microbiology, is that measured data from these fields is often noisy and scarce.

Stein variational gradient descent (SVGD) is a prominent particle-based variational inference method used for sampling a target distribution. SVGD has attracted interest for application in machine-learning techniques such as Bayesian inference. In this paper, we propose novel trainable algorithms that incorporate a deep-learning technique called deep unfolding,into SVGD. This approach facilitates the learning of the internal parameters of SVGD, thereby accelerating its convergence speed. To evaluate the proposed trainable SVGD algorithms, we conducted numerical simulations of three tasks: sampling a one-dimensional Gaussian mixture, performing Bayesian logistic regression, and learning Bayesian neural networks. The results show that our proposed algorithms exhibit faster convergence than the conventional variants of SVGD.

A universal feature of human societies is the adoption of systems of rules and norms in the service of cooperative ends. How can we build learning agents that do the same, so that they may flexibly cooperate with the human institutions they are embedded in? We hypothesize that agents can achieve this by assuming there exists a shared set of norms that most others comply with while pursuing their individual desires, even if they do not know the exact content of those norms. By assuming shared norms, a newly introduced agent can infer the norms of an existing population from observations of compliance and violation. Furthermore, groups of agents can converge to a shared set of norms, even if they initially diverge in their beliefs about what the norms are. This in turn enables the stability of the normative system: since agents can bootstrap common knowledge of the norms, this leads the norms to be widely adhered to, enabling new entrants to rapidly learn those norms. We formalize this framework in the context of Markov games and demonstrate its operation in a multi-agent environment via approximately Bayesian rule induction of obligative and prohibitive norms. Using our approach, agents are able to rapidly learn and sustain a variety of cooperative institutions, including resource management norms and compensation for pro-social labor, promoting collective welfare while still allowing agents to act in their own interests.

Hybrid Bayesian networks (HBN) contain complex conditional probabilistic distributions (CPD) specified as partitioned expressions over discrete and continuous variables. The size of these CPDs grows exponentially with the number of parent nodes when using discrete inference, resulting in significant inefficiency. Normally, an effective way to reduce the CPD size is to use a binary factorization (BF) algorithm to decompose the statistical or arithmetic functions in the CPD by factorizing the number of connected parent nodes to sets of size two. However, the BF algorithm was not designed to handle partitioned expressions. Hence, we propose a new algorithm called stacking factorization (SF) to decompose the partitioned expressions. The SF algorithm creates intermediate nodes to incrementally reconstruct the densities in the original partitioned expression, allowing no more than two continuous parent nodes to be connected to each child node in the resulting HBN. SF can be either used independently or combined with the BF algorithm. We show that the SF+BF algorithm significantly reduces the CPD size and contributes to lowering the tree-width of a model, thus improving efficiency.

Bayesian causal inference, i.e., inferring a posterior over causal models for the use in downstream causal reasoning tasks, poses a hard computational inference problem that is little explored in literature. In this work, we combine techniques from order-based MCMC structure learning with recent advances in gradient-based graph learning into an effective Bayesian causal inference framework. Specifically, we decompose the problem of inferring the causal structure into (i) inferring a topological order over variables and (ii) inferring the parent sets for each variable. When limiting the number of parents per variable, we can exactly marginalise over the parent sets in polynomial time. We further use Gaussian processes to model the unknown causal mechanisms, which also allows their exact marginalisation. This introduces a Rao-Blackwellization scheme, where all components are eliminated from the model, except for the causal order, for which we learn a distribution via gradient-based optimisation. The combination of Rao-Blackwellization with our sequential inference procedure for causal orders yields state-of-the-art on linear and non-linear additive noise benchmarks with scale-free and Erdos-Renyi graph structures.

Most leading implementations of black-box variational inference (BBVI) are based on optimizing a stochastic evidence lower bound (ELBO). But such approaches to BBVI often converge slowly due to the high variance of their gradient estimates. In this work, we propose batch and match (BaM), an alternative approach to BBVI based on a score-based divergence. Notably, this score-based divergence can be optimized by a closed-form proximal update for Gaussian variational families with full covariance matrices. We analyze the convergence of BaM when the target distribution is Gaussian, and we prove that in the limit of infinite batch size the variational parameter updates converge exponentially quickly to the target mean and covariance. We also evaluate the performance of BaM on Gaussian and non-Gaussian target distributions that arise from posterior inference in hierarchical and deep generative models. In these experiments, we find that BaM typically converges in fewer (and sometimes significantly fewer) gradient evaluations than leading implementations of BBVI based on ELBO maximization.

In interactive systems, actions are often correlated, presenting an opportunity for more sample-efficient off-policy evaluation (OPE) and learning (OPL) in large action spaces. We introduce a unified Bayesian framework to capture these correlations through structured and informative priors. In this framework, we propose sDM, a generic Bayesian approach designed for OPE and OPL, grounded in both algorithmic and theoretical foundations. Notably, sDM leverages action correlations without compromising computational efficiency. Moreover, inspired by online Bayesian bandits, we introduce Bayesian metrics that assess the average performance of algorithms across multiple problem instances, deviating from the conventional worst-case assessments. We analyze sDM in OPE and OPL, highlighting the benefits of leveraging action correlations. Empirical evidence showcases the strong performance of sDM.

We introduce a novel Information Criterion (IC), termed Learning under Singularity (LS), designed to enhance the functionality of the Widely Applicable Bayes Information Criterion (WBIC) and the Singular Bayesian Information Criterion (sBIC). LS is effective without regularity constraints and demonstrates stability. Watanabe defined a statistical model or a learning machine as regular if the mapping from a parameter to a probability distribution is one-to-one and its Fisher information matrix is positive definite. In contrast, models not meeting these conditions are termed singular. Over the past decade, several information criteria for singular cases have been proposed, including WBIC and sBIC. WBIC is applicable in non-regular scenarios but faces challenges with large sample sizes and redundant estimation of known learning coefficients. Conversely, sBIC is limited in its broader application due to its dependence on maximum likelihood estimates. LS addresses these limitations by enhancing the utility of both WBIC and sBIC. It incorporates the empirical loss from the Widely Applicable Information Criterion (WAIC) to represent the goodness of fit to the statistical model, along with a penalty term similar to that of sBIC. This approach offers a flexible and robust method for model selection, free from regularity constraints.