Bayesian Inference

[1807.04920] On the Complexity of Iterative Tropical Computation with Applications to Markov Decision Processes

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Quantification under prior probability shift: the ratio estimator and its extensions

The quantification problem consists of determining the prevalence of a given label in a target population. However, one often has access to the labels in a sample from the training population but not in the target population. A common assumption in this situation is that of prior probability shift, that is, once the labels are known, the distribution of the features is the same in the training and target populations. In this paper, we derive a new lower bound for the risk of the quantification problem under the prior shift assumption. Complementing this lower bound, we present a new approximately minimax class of estimators, ratio estimators, which generalize several previous proposals in the literature. Using a weaker version of the prior shift assumption, which can be tested, we show that ratio estimators can be used to build confidence intervals for the quantification problem. We also extend the ratio estimator so that it can: (i) incorporate labels from the target population, when they are available and (ii) estimate how the prevalence of positive labels varies according to a function of certain covariates.

Constraint-based Causal Discovery for Non-Linear Structural Causal Models with Cycles and Latent Confounders

We address the problem of causal discovery from data, making use of the recently proposed causal modeling framework of modular structural causal models (mSCM) to handle cycles, latent confounders and non-linearities. We introduce {\sigma}-connection graphs ({\sigma}-CG), a new class of mixed graphs (containing undirected, bidirected and directed edges) with additional structure, and extend the concept of {\sigma}-separation, the appropriate generalization of the well-known notion of d-separation in this setting, to apply to {\sigma}-CGs. We prove the closedness of {\sigma}-separation under marginalisation and conditioning and exploit this to implement a test of {\sigma}-separation on a {\sigma}-CG. This then leads us to the first causal discovery algorithm that can handle non-linear functional relations, latent confounders, cyclic causal relationships, and data from different (stochastic) perfect interventions. As a proof of concept, we show on synthetic data how well the algorithm recovers features of the causal graph of modular structural causal models.

Sampling and Inference for Beta Neutral-to-the-Left Models of Sparse Networks

Empirical evidence suggests that heavy-tailed degree distributions occurring in many real networks are well-approximated by power laws with exponents $\eta$ that may take values either less than and greater than two. Models based on various forms of exchangeability are able to capture power laws with $\eta < 2$, and admit tractable inference algorithms; we draw on previous results to show that $\eta > 2$ cannot be generated by the forms of exchangeability used in existing random graph models. Preferential attachment models generate power law exponents greater than two, but have been of limited use as statistical models due to the inherent difficulty of performing inference in non-exchangeable models. Motivated by this gap, we design and implement inference algorithms for a recently proposed class of models that generates $\eta$ of all possible values. We show that although they are not exchangeable, these models have probabilistic structure amenable to inference. Our methods make a large class of previously intractable models useful for statistical inference.

BALSON: Bayesian Least Squares Optimization with Nonnegative L1-Norm Constraint

A Bayesian approach termed BAyesian Least Squares Optimization with Nonnegative L1-norm constraint (BALSON) is proposed. The error distribution of data fitting is described by Gaussian likelihood. The parameter distribution is assumed to be a Dirichlet distribution. With the Bayes rule, searching for the optimal parameters is equivalent to finding the mode of the posterior distribution. In order to explicitly characterize the nonnegative L1-norm constraint of the parameters, we further approximate the true posterior distribution by a Dirichlet distribution. We estimate the statistics of the approximating Dirichlet posterior distribution by sampling methods. Four sampling methods have been introduced. With the estimated posterior distributions, the original parameters can be effectively reconstructed in polynomial fitting problems, and the BALSON framework is found to perform better than conventional methods.

Bayesian Neural Networks: Bayes' Theorem Applied to Deep Learning

The article was written by Amber Zhou, a Financial Analyst at I Know First. Deep learning has become a buzzward in recent years. In fact, it has once gained much attention and excitements under the name neural networks early back in 1980's. However due to the lack of sufficient compute power and training examples, it gradually experienced a depression in the following decade. As we are entering the Era of Big Data in light of the explosion of computer power, deep learning has recently seen a revival.

Bayesian Learning for Machine Learning: Part 1 - Introduction to Bayesian Learning - DZone AI

In this article, I will provide a basic introduction to Bayesian learning and explore topics such as frequentist statistics, the drawbacks of the frequentist method, Bayes's theorem (introduced with an example), and the differences between the frequentist and Bayesian methods using the coin flip experiment as the example. To begin, let's try to answer this question: what is the frequentist method? When we flip a coin, there are two possible outcomes -- heads or tails. Of course, there is a third rare possibility where the coin balances on its edge without falling onto either side, which we assume is not a possible outcome of the coin flip for our discussion. We conduct a series of coin flips and record our observations i.e. the number of the heads (or tails) observed for a certain number of coin flips. In this experiment, we are trying to determine the fairness of the coin, using the number of heads (or tails) that we observe.

Is The Variational Bayesian Method The Most Difficult Machine Learning Technique?

Data scientist Stefano Cosentino observed in a post that the Bayesian approach leans more towards the distributions associated with each parameter. For instance, he writes that the two parameters depicted below, as shown by the Gaussian curves after a trained Bayesian network has converged. Hence the Bayesian approach, where the parameters are unknown quantities can be considered as random variables. University of Buffalo's paper defines the Bayesian approach to uncertainty, which treats all uncertain quantities as random variables and uses the laws of probability to manipulate those uncertain quantities. Hence, the right Bayesian approach integrates over all uncertain quantities rather than optimise them, states the paper.

A comparative study of artificial intelligence and human doctors for the purpose of triage and diagnosis

Online symptom checkers have significant potential to improve patient care, however their reliability and accuracy remain variable. We hypothesised that an artificial intelligence (AI) powered triage and diagnostic system would compare favourably with human doctors with respect to triage and diagnostic accuracy. We performed a prospective validation study of the accuracy and safety of an AI powered triage and diagnostic system. Identical cases were evaluated by both an AI system and human doctors. Differential diagnoses and triage outcomes were evaluated by an independent judge, who was blinded from knowing the source (AI system or human doctor) of the outcomes. Independently of these cases, vignettes from publicly available resources were also assessed to provide a benchmark to previous studies and the diagnostic component of the MRCGP exam. Overall we found that the Babylon AI powered Triage and Diagnostic System was able to identify the condition modelled by a clinical vignette with accuracy comparable to human doctors (in terms of precision and recall). In addition, we found that the triage advice recommended by the AI System was, on average, safer than that of human doctors, when compared to the ranges of acceptable triage provided by independent expert judges, with only a minimal reduction in appropriateness.

Computational Cognitive Science lab: Reading list on Bayesian methods

This list is intended to introduce some of the tools of Bayesian statistics and machine learning that can be useful to computational research in cognitive science. The first section mentions several useful general references, and the others provide supplementary readings on specific topics. If you would like to suggest some additions to the list, contact Tom Griffiths.