This course is about the discussion of sampling and exploring data, as well as basic probability theory and Bayes' rule. A variety of exploratory data analysis techniques will be covered, including numeric summary statistics and basic data visualization. The concepts and techniques you will find in this course will serve as building blocks for the inference and modeling courses in the Specialization.

Equipping artificial agents with useful exploration mechanisms remains a challenge to this day. Humans, on the other hand, seem to manage the trade-off between exploration and exploitation effortlessly. In the present article, we put forward the hypothesis that they accomplish this by making optimal use of limited computational resources. We study this hypothesis by meta-learning reinforcement learning algorithms that sacrifice performance for a shorter description length (defined as the number of bits required to implement the given algorithm). The emerging class of models captures human exploration behavior better than previously considered approaches, such as Boltzmann exploration, upper confidence bound algorithms, and Thompson sampling. We additionally demonstrate that changing the description length in our class of models produces the intended effects: reducing description length captures the behavior of brain-lesioned patients while increasing it mirrors cognitive development during adolescence.

The last decade's research in artificial intelligence had a significant impact on the advance of autonomous driving. Yet, safety remains a major concern when it comes to deploying such systems in high-risk environments. The objective of this thesis is to develop methodological tools which provide reliable uncertainty estimates for deep neural networks. First, we introduce a new criterion to reliably estimate model confidence: the true class probability (TCP). We show that TCP offers better properties for failure prediction than current uncertainty measures. Since the true class is by essence unknown at test time, we propose to learn TCP criterion from data with an auxiliary model, introducing a specific learning scheme adapted to this context. The relevance of the proposed approach is validated on image classification and semantic segmentation datasets. Then, we extend our learned confidence approach to the task of domain adaptation where it improves the selection of pseudo-labels in self-training methods. Finally, we tackle the challenge of jointly detecting misclassification and out-of-distributions samples by introducing a new uncertainty measure based on evidential models and defined on the simplex.

We present a Bayesian graph neural network (BGNN) that can estimate the weak lensing convergence ($\kappa$) from photometric measurements of galaxies along a given line of sight. The method is of particular interest in strong gravitational time delay cosmography (TDC), where characterizing the "external convergence" ($\kappa_{\rm ext}$) from the lens environment and line of sight is necessary for precise inference of the Hubble constant ($H_0$). Starting from a large-scale simulation with a $\kappa$ resolution of $\sim$1$'$, we introduce fluctuations on galaxy-galaxy lensing scales of $\sim$1$''$ and extract random sightlines to train our BGNN. We then evaluate the model on test sets with varying degrees of overlap with the training distribution. For each test set of 1,000 sightlines, the BGNN infers the individual $\kappa$ posteriors, which we combine in a hierarchical Bayesian model to yield constraints on the hyperparameters governing the population. For a test field well sampled by the training set, the BGNN recovers the population mean of $\kappa$ precisely and without bias, resulting in a contribution to the $H_0$ error budget well under 1\%. In the tails of the training set with sparse samples, the BGNN, which can ingest all available information about each sightline, extracts more $\kappa$ signal compared to a simplified version of the traditional method based on matching galaxy number counts, which is limited by sample variance. Our hierarchical inference pipeline using BGNNs promises to improve the $\kappa_{\rm ext}$ characterization for precision TDC. The implementation of our pipeline is available as a public Python package, Node to Joy.

Given a Lipschitz or smooth convex function $\, f:K \to \mathbb{R}$ for a bounded polytope $K \subseteq \mathbb{R}^d$ defined by $m$ inequalities, we consider the problem of sampling from the log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to $K$. Interest in this problem derives from its applications to Bayesian inference and differentially private learning. Our main result is a generalization of the Dikin walk Markov chain to this setting that requires at most $O((md + d L^2 R^2) \times md^{\omega-1}) \log(\frac{w}{\delta}))$ arithmetic operations to sample from $\pi$ within error $\delta>0$ in the total variation distance from a $w$-warm start. Here $L$ is the Lipschitz-constant of $f$, $K$ is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega$ is the matrix-multiplication constant. Our algorithm improves on the running time of prior works for a range of parameter settings important for the aforementioned learning applications. Technically, we depart from previous Dikin walks by adding a "soft-threshold" regularizer derived from the Lipschitz or smoothness properties of $f$ to the log-barrier function for $K$ that allows our version of the Dikin walk to propose updates that have a high Metropolis acceptance ratio for $f$, while at the same time remaining inside the polytope $K$.

We are seeking to characterize the performance and potential bottlenecks of the latest fast MCMC samplers. I see that Stan is currently using Intel TBB to parallelize the no-U-turn sampler (NUTS) across multiple chains. Do you know of any research attempted to parallelize each sampler itself within one chain. Our group at Google has been very interested in using parallel compute in HMC variants (including NUTS), particularly on accelerators (e.g., GPUs). We've been working in the deep-learning-oriented autodiff accelerator software frameworks TensorFlow and JAX, both of which are supported by our TensorFlow Probability library.

Statistical analysis of tensor-valued data has largely used the tensor-variate normal (TVN) distribution that may be inadequate when data comes from distributions with heavier or lighter tails. We study a general family of elliptically contoured (EC) tensor-variate distributions and derive its characterizations, moments, marginal and conditional distributions, and the EC Wishart distribution. We describe procedures for maximum likelihood estimation from data that are (1) uncorrelated draws from an EC distribution, (2) from a scale mixture of the TVN distribution, and (3) from an underlying but unknown EC distribution, where we extend Tyler's robust estimator. A detailed simulation study highlights the benefits of choosing an EC distribution over the TVN for heavier-tailed data. We develop tensor-variate classification rules using discriminant analysis and EC errors and show that they better predict cats and dogs from images in the Animal Faces-HQ dataset than the TVN-based rules. A novel tensor-on-tensor regression and tensor-variate analysis of variance (TANOVA) framework under EC errors is also demonstrated to better characterize gender, age and ethnic origin than the usual TVN-based TANOVA in the celebrated Labeled Faces of the Wild dataset.

Characterizing the differential excision of mRNA is critical for understanding the functional complexity of a cell or tissue, from normal developmental processes to disease pathogenesis. Most transcript reconstruction methods infer full-length transcripts from high-throughput sequencing data. However, this is a challenging task due to incomplete annotations and the differential expression of transcripts across cell-types, tissues, and experimental conditions. Several recent methods circumvent these difficulties by considering local splicing events, but these methods lose transcript-level splicing information and may conflate transcripts. We develop the first probabilistic model that reconciles the transcript and local splicing perspectives. First, we formalize the sequence of mRNA excisions (SME) reconstruction problem, which aims to assemble variable-length sequences of mRNA excisions from RNA-sequencing data. We then present a novel hierarchical Bayesian admixture model for the Reconstruction of Excised mRNA (BREM). BREM interpolates between local splicing events and full-length transcripts and thus focuses only on SMEs that have high posterior probability. We develop posterior inference algorithms based on Gibbs sampling and local search of independent sets and characterize differential SME usage using generalized linear models based on converged BREM model parameters. We show that BREM achieves higher F1 score for reconstruction tasks and improved accuracy and sensitivity in differential splicing when compared with four state-of-the-art transcript and local splicing methods on simulated data. Lastly, we evaluate BREM on both bulk and scRNA sequencing data based on transcript reconstruction, novelty of transcripts produced, model sensitivity to hyperparameters, and a functional analysis of differentially expressed SMEs, demonstrating that BREM captures relevant biological signal.

This abstracts the core ideas beyond logic programming as such to encompass frameworks from probabilistic databases, probabilistic finite model theory and discrete lifted Bayesian networks. To demonstrate the usefulness of such a general approach, we completely characterise the projective families of distributions representable in the generalised distribution semantics and we demonstrate both that large classes of interesting projective families cannot be represented in a generalised distribution semantics and that already a very limited fragment of logic programming (acyclic determinate logic programs) in the determinsitic part suffices to represent all those projective families that are representable in the generalised distribution semantics at all.

Mathematical proof aims to deliver confident conclusions, but a very similar process of deduction can be used to make uncertain estimates that are open to revision. A key ingredient in such reasoning is the use of a "default" estimate of $\mathbb{E}[XY] = \mathbb{E}[X] \mathbb{E}[Y]$ in the absence of any specific information about the correlation between $X$ and $Y$, which we call *the presumption of independence*. Reasoning based on this heuristic is commonplace, intuitively compelling, and often quite successful -- but completely informal. In this paper we introduce the concept of a heuristic estimator as a potential formalization of this type of defeasible reasoning. We introduce a set of intuitively desirable coherence properties for heuristic estimators that are not satisfied by any existing candidates. Then we present our main open problem: is there a heuristic estimator that formalizes intuitively valid applications of the presumption of independence without also accepting spurious arguments?