In this paper we present a comprehensive view of prominent causal discovery algorithms, categorized into two main categories (1) assuming acyclic and no latent variables, and (2) allowing both cycles and latent variables, along with experimental results comparing them from three perspectives: (a) structural accuracy, (b) standard predictive accuracy, and (c) accuracy of counterfactual inference. For (b) and (c) we train causal Bayesian networks with structures as predicted by each causal discovery technique to carry out counterfactual or standard predictive inference. We compare causal algorithms on two pub- licly available and one simulated datasets having different sample sizes: small, medium and large. Experiments show that structural accuracy of a technique does not necessarily correlate with higher accuracy of inferencing tasks. Fur- ther, surveyed structure learning algorithms do not perform well in terms of structural accuracy in case of datasets having large number of variables.

Minimum energy paths for transitions such as atomic and/or spin rearrangements in thermalized systems are the transition paths of largest statistical weight. Such paths are frequently calculated using the nudged elastic band method, where an initial path is iteratively shifted to the nearest minimum energy path. The computational effort can be large, especially when ab initio or electron density functional calculations are used to evaluate the energy and atomic forces. Here, we show how the number of such evaluations can be reduced by an order of magnitude using a Gaussian process regression approach where an approximate energy surface is generated and refined in each iteration. When the goal is to evaluate the transition rate within harmonic transition state theory, the evaluation of the Hessian matrix at the initial and final state minima can be carried out beforehand and used as input in the minimum energy path calculation, thereby improving stability and reducing the number of iterations needed for convergence. A Gaussian process model also provides an uncertainty estimate for the approximate energy surface, and this can be used to focus the calculations on the lesser-known part of the path, thereby reducing the number of needed energy and force evaluations to a half in the present calculations. The methodology is illustrated using the two-dimensional M\"uller-Brown potential surface and performance assessed on an established benchmark involving 13 rearrangement transitions of a heptamer island on a solid surface.

Sampling is a fundamental topic in graph signal processing, having found applications in estimation, clustering, and video compression. In contrast to traditional signal processing, the irregularity of the signal domain makes selecting a sampling set non-trivial and hard to analyze. Indeed, though conditions for graph signal interpolation from noiseless samples exist, they do not lead to a unique sampling set. The presence of noise makes choosing among these sampling sets a hard combinatorial problem. Although greedy sampling schemes are commonly used in practice, they have no performance guarantee. This work takes a twofold approach to address this issue. First, universal performance bounds are derived for the Bayesian estimation of graph signals from noisy samples. In contrast to currently available bounds, they are not restricted to specific sampling schemes and hold for any sampling sets. Second, this paper provides near-optimal guarantees for greedy sampling by introducing the concept of approximate submodularity and updating the classical greedy bound. It then provides explicit bounds on the approximate supermodularity of the interpolation mean-square error showing that it can be optimized with worst-case guarantees using greedy search even though it is not supermodular. Simulations illustrate the derived bound for different graph models and show an application of graph signal sampling to reduce the complexity of kernel principal component analysis.

Looking at the picture above you may have easily figured out \(P(\text{red} \text{yellow})\) by thinking "This is easy! There are 6 yellow pegs, 4 of them are over red so the probability of being over a red block if I'm on a yellow one is 4/6". If you did follow this line of thinking congratulations, you just independently discovered Bayes' Theorem! Of course mathematical language is extremely concise, and human intuition is able to easily jump steps in its reasoning process; getting from our intuition to Bayes' Theorem will require a bit of work. Let's begin formalizing this intuition by coming up with a way to calculate "there are 6 yellow pegs."

Flow cytometry is a high-throughput technology used to quantify multiple surface and intracellular markers at the level of a single cell. This enables to identify cell sub-types, and to determine their relative proportions. Improvements of this technology allow to describe millions of individual cells from a blood sample using multiple markers. This results in high-dimensional datasets, whose manual analysis is highly time-consuming and poorly reproducible. While several methods have been developed to perform automatic recognition of cell populations, most of them treat and analyze each sample independently. However, in practice, individual samples are rarely independent (e.g. longitudinal studies). Here, we propose to use a Bayesian nonparametric approach with Dirichlet process mixture (DPM) of multivariate skew $t$-distributions to perform model based clustering of flow-cytometry data. DPM models directly estimate the number of cell populations from the data, avoiding model selection issues, and skew $t$-distributions provides robustness to outliers and non-elliptical shape of cell populations. To accommodate repeated measurements, we propose a sequential strategy relying on a parametric approximation of the posterior. We illustrate the good performance of our method on simulated data, on an experimental benchmark dataset, and on new longitudinal data from the DALIA-1 trial which evaluates a therapeutic vaccine against HIV. On the benchmark dataset, the sequential strategy outperforms all other methods evaluated, and similarly, leads to improved performance on the DALIA-1 data. We have made the method available for the community in the R package NPflow.

We consider the problem of recovering a $d-$dimensional manifold $\mathcal{M} \subset \mathbb{R}^n$ when provided with noiseless samples from $\mathcal{M}$. There are many algorithms (e.g., Isomap) that are used in practice to fit manifolds and thus reduce the dimensionality of a given data set. Ideally, the estimate $\mathcal{M}_\mathrm{put}$ of $\mathcal{M}$ should be an actual manifold of a certain smoothness; furthermore, $\mathcal{M}_\mathrm{put}$ should be arbitrarily close to $\mathcal{M}$ in Hausdorff distance given a large enough sample. Generally speaking, existing manifold learning algorithms do not meet these criteria. Fefferman, Mitter, and Narayanan (2016) have developed an algorithm whose output is provably a manifold. The key idea is to define an approximate squared-distance function (asdf) to $\mathcal{M}$. Then, $\mathcal{M}_\mathrm{put}$ is given by the set of points where the gradient of the asdf is orthogonal to the subspace spanned by the largest $n - d$ eigenvectors of the Hessian of the asdf. As long as the asdf meets certain regularity conditions, $\mathcal{M}_\mathrm{put}$ is a manifold that is arbitrarily close in Hausdorff distance to $\mathcal{M}$. In this paper, we define two asdfs that can be calculated from the data and show that they meet the required regularity conditions. The first asdf is based on kernel density estimation, and the second is based on estimation of tangent spaces using local principal components analysis.

Curating labeled training data has become the primary bottleneck in machine learning. Recent frameworks address this bottleneck with generative models to synthesize labels at scale from weak supervision sources. The generative model's dependency structure directly affects the quality of the estimated labels, but selecting a structure automatically without any labeled data is a distinct challenge. We propose a structure estimation method that maximizes the $\ell_1$-regularized marginal pseudolikelihood of the observed data. Our analysis shows that the amount of unlabeled data required to identify the true structure scales sublinearly in the number of possible dependencies for a broad class of models. Simulations show that our method is 100$\times$ faster than a maximum likelihood approach and selects $1/4$ as many extraneous dependencies. We also show that our method provides an average of 1.5 F1 points of improvement over existing, user-developed information extraction applications on real-world data such as PubMed journal abstracts.

As reviewed by [Owhadi and Scovel], the field of statistical decision theory introduced by Wald, building on a game theoretic foundation developed by von Neumann and Morgenstern, provides links between Bayesian and frequentist statistical philosophies through the concepts of decision rules, admissibility, and risk functions amongst others. Moreover, a recent thrust of research motivated by machine learning has put much emphasis on prediction problems for which Bayesian methodology has been widely used. The purpose of this note is to demonstrate that classic decision theoretic results can be simply applied to the analysis of prediction problems. In fact, both [Berger] and [Robert] remark upon the ease of applying statistical decision theory within the context of prediction, however, no explicit result is stated in either work; the contribution of this note, therefore, is to highlight a simple way in which the results of statistical decision theory might apply to prediction problems. To the author's knowledge the most similar lines of thought appear in work by [Nayak and El-Baz], where the loss function depends on the underlying parameter (in contrast to what follows).

Social dilemmas have been regarded as the essence of evolution game theory, in which the prisoner's dilemma game is the most famous metaphor for the problem of cooperation. Recent findings revealed people's behavior violated the Sure Thing Principle in such games. Classic probability methodologies have difficulty explaining the underlying mechanisms of people's behavior. In this paper, a novel quantum-like Bayesian Network was proposed to accommodate the paradoxical phenomenon. The special network can take interference into consideration, which is likely to be an efficient way to describe the underlying mechanism. With the assistance of belief entropy, named as Deng entropy, the paper proposes Belief Distance to render the model practical. Tested with empirical data, the proposed model is proved to be predictable and effective.

We propose a new framework for Hamiltonian Monte Carlo (HMC) on truncated probability distributions with smooth underlying density functions. Traditional HMC requires computing the gradient of potential function associated with the target distribution, and therefore does not perform its full power on truncated distributions due to lack of continuity and differentiability. In our framework, we introduce a sharp sigmoid factor in the density function to approximate the probability drop at the truncation boundary. The target potential function is approximated by a new potential which smoothly extends to the entire sample space. HMC is then performed on the approximate potential. While our method is easy to implement and applies to a wide range of problems, it also achieves comparable computational efficiency on various sampling tasks compared to other baseline methods. RBHMC also gives rise to a new approach for Bayesian inference on constrained spaces.