The holy trinity when it comes to being Bayesian. I will provide my experience in using the first two packages and my high level opinion of the third (haven't used it in practice). Of course then there is the mad men (old professors who are becoming irrelevant) who actually do their own Gibbs sampling. You specify the generative model for the data. You feed in the data as observations and then it samples from the posterior of the data for you.

The superior performance of ensemble methods with infinite models are well known. Most of these methods are based on optimization problems in infinite-dimensional spaces with some regularization, for instance, boosting methods and convex neural networks use $L^1$-regularization with the non-negative constraint. However, due to the difficulty of handling $L^1$-regularization, these problems require early stopping or a rough approximation to solve it inexactly. In this paper, we propose a new ensemble learning method that performs in a space of probability measures, that is, our method can handle the $L^1$-constraint and the non-negative constraint in a rigorous way. Such an optimization is realized by proposing a general purpose stochastic optimization method for learning probability measures via parameterization using transport maps on base models. As a result of running the method, a transport map to output an infinite ensemble is obtained, which forms a residual-type network. From the perspective of functional gradient methods, we give a convergence rate as fast as that of a stochastic optimization method for finite dimensional nonconvex problems. Moreover, we show an interior optimality property of a local optimality condition used in our analysis.

Markov models, or probabilistic graphical models, explicitly establish a correspondence between statistical independence in a probability distribution and certain separation criteria holding in a graph. They were originated at the interface between statistics, where Markov random fields were predominant [Darroch et al., 1980], and artificial intelligence, with a focus on Bayesian networks [Pearl, 1985, 1986]. These two models are now considered the traditional ones, but still are widely applied and nowadays there is a significant amount of research devoted to them [Daly et al., 2011, Uhler, 2012]. They both share the modelling of conditional independences: Bayesian networks relate them with acyclic directed graphs, whereas in Markov fields they are associated with undirected graphs. However, the models they represent are only equivalent under additional assumptions on the respective graphs.

Title: How proper are Bayesian models in the astronomical literature? Which authors of this paper are endorsers? Disable MathJax (What is MathJax?)

Efficient Monte Carlo inference often requires manual construction of model-specific proposals. We propose an approach to automated proposal construction by training neural networks to provide fast approximations to block Gibbs conditionals. The learned proposals generalize to occurrences of common structural motifs both within a given model and across different models, allowing for the construction of a library of learned inference primitives that can accelerate inference on unseen models with no model-specific training required. We explore several applications including open-universe Gaussian mixture models, in which our learned proposals outperform a hand-tuned sampler, and a real-world named entity recognition task, in which our sampler's ability to escape local modes yields higher final F1 scores than single-site Gibbs.

Probabilistic (or Bayesian) modeling and learning offers interesting possibilities for systematic representation of uncertainty using probability theory. However, probabilistic learning often leads to computationally challenging problems. Some problems of this type that were previously intractable can now be solved on standard personal computers thanks to recent advances in Monte Carlo methods. In particular, for learning of unknown parameters in nonlinear state-space models, methods based on the particle filter (a Monte Carlo method) have proven very useful. A notoriously challenging problem, however, still occurs when the observations in the state-space model are highly informative, i.e. when there is very little or no measurement noise present, relative to the amount of process noise. The particle filter will then struggle in estimating one of the basic components for probabilistic learning, namely the likelihood $p($data$|$parameters$)$. To this end we suggest an algorithm which initially assumes that there is substantial amount of artificial measurement noise present. The variance of this noise is sequentially decreased in an adaptive fashion such that we, in the end, recover the original problem or possibly a very close approximation of it. The main component in our algorithm is a sequential Monte Carlo (SMC) sampler, which gives our proposed method a clear resemblance to the SMC^2 method. Another natural link is also made to the ideas underlying the approximate Bayesian computation (ABC). We illustrate it with numerical examples, and in particular show promising results for a challenging Wiener-Hammerstein benchmark problem.

Abstract--Manufacturers of safety-critical systems must make the case that their product is sufficiently safe for public deployment. Much of this case often relies upon critical event outcomes from real-world testing, requiring manufacturers to be strategic about how they allocate testing resources in order to maximize their chances of demonstrating system safety. This work frames the partially observable and belief-dependent problem of test scheduling as a Markov decision process, which can be solved efficiently to yield closed-loop manufacturer testing policies. By solving for policies over a wide range of problem formulations, we are able to provide high-level guidance for manufacturers and regulators on issues relating to the testing of safety-critical systems. This guidance spans an array of topics, including circumstances under which manufacturers should continue testing despite observed incidents, when manufacturers should test aggressively, and when regulators should increase or reduce the real-world testing requirements for an autonomous vehicle. I. INTRODUCTION Confidence must be established in safety-critical systems such as autonomous vehicles prior to their widespread release. Establishing confidence is difficult because the space of driving scenarios is vast and accidents are rare. Automotive manufacturers can build confidence by conducting test drives on public roadways and make the safety case based on the frequency of observed hazardous events like disengagements and traffic accidents. Each manufacturer must devise a testing strategy capable of providing sufficient evidence that their system is safe enough for widespread adoption. Real-world testing that is too aggressive may yield hazardous events that diminish confidence in system safety. However, a manufacturer that is reluctant to test their product may forfeit opportunities to identify and address shortcomings, and may ultimately not be able to compete in the market. The fundamental tension between the desire to thoroughly test a product and the urgency to forego further testing in favor of bringing the product to market is not unique to the automotive industry.

In this post, we consider different approaches for time series modeling. The forecasting approaches using linear models, ARIMA alpgorithm, XGBoost machine learning algorithm are described. Results of different model combinations are shown. For probabilistic modeling the approaches using copulas and Bayesian inference are considered. Time series analysis, especially forecasting, is an important problem of modern predictive analytics.

In this video, Data Scientist Aaron Kramer explains what Bayesian inference is and how it's useful for business applications li...

We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running computations, and its own probabilities. We show that our algorithm, an instance of what we call a logical inductor, satisfies a number of intuitive desiderata, including: (1) it learns to predict patterns of truth and falsehood in logical statements, often long before having the resources to evaluate the statements, so long as the patterns can be written down in polynomial time; (2) it learns to use appropriate statistical summaries to predict sequences of statements whose truth values appear pseudorandom; and (3) it learns to have accurate beliefs about its own current beliefs, in a manner that avoids the standard paradoxes of self-reference. For example, if a given computer program only ever produces outputs in a certain range, a logical inductor learns this fact in a timely manner; and if late digits in the decimal expansion of $\pi$ are difficult to predict, then a logical inductor learns to assign $\approx 10\%$ probability to "the $n$th digit of $\pi$ is a 7" for large $n$. Logical inductors also learn to trust their future beliefs more than their current beliefs, and their beliefs are coherent in the limit (whenever $\phi \implies \psi$, $\mathbb{P}_\infty(\phi) \le \mathbb{P}_\infty(\psi)$, and so on); and logical inductors strictly dominate the universal semimeasure in the limit. These properties and many others all follow from a single logical induction criterion, which is motivated by a series of stock trading analogies. Roughly speaking, each logical sentence $\phi$ is associated with a stock that is worth \$1 per share if [...]