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Bayesian regression tree ensembles that adapt to smoothness and sparsity
Ensembles of decision trees are a useful tool for obtaining flexible estimates of regression functions. Examples of these methods include gradient-boosted decision trees, random forests and Bayesian classification and regression trees. Two potential shortcomings of tree ensembles are their lack of smoothness and their vulnerability to the curse of dimensionality. We show that these issues can be overcome by instead considering sparsity inducing soft decision trees in which the decisions are treated as probabilistic. We implement this in the context of the Bayesian additive regression trees framework and illustrate its promising performance through testing on benchmark data sets. We provide strong theoretical support for our methodology by showing that the posterior distribution concentrates at the minimax rate (up to a logarithmic factor) for sparse functions and functions with additive structures in the high dimensional regime where the dimensionality of the covariate space is allowed to grow nearly exponentially in the sample size. Our method also adapts to the unknown smoothness and sparsity levels, and can be implemented by making minimal modifications to existing Bayesian additive regression tree algorithms.
Journal Article
Bayesian statistics the fun way : understanding statistics and probability with Star Wars, LEGO, and Rubber Ducks
\"An introduction to Bayesian statistics with simple and pop culture-based explanations. Topics covered include measuring your own uncertainty in a belief, applying Bayes' theorem, and calculating distributions\"-- Provided by publisher.
Book
Model misspecification in approximate Bayesian computation
We analyse the behaviour of approximate Bayesian computation (ABC) when the model generating the simulated data differs from the actual data-generating process, i.e. when the data simulator in ABC is misspecified.We demonstrate both theoretically and in simple, but practically relevant, examples that when the model is misspecified different versions of ABC can yield substantially different results. Our theoretical results demonstrate that even though the model is misspecified, under regularity conditions, the accept–reject ABC approach concentrates posterior mass on an appropriately defined pseudotrue parameter value. However, under model misspecification the ABC posterior does not yield credible sets with valid frequentist coverage and has non-standard asymptotic behaviour. In addition, we examine the theoretical behaviour of the popular local regression adjustment to ABC under model misspecification and demonstrate that this approach concentrates posterior mass on a pseudotrue value that is completely different from accept–reject ABC. Using our theoretical results, we suggest two approaches to diagnose model misspecification in ABC. All theoretical results and diagnostics are illustrated in a simple running example.
Journal Article
Bayesian inference on complicated data
Due to great applications in various fields, such as social science, biomedicine, genomics, and signal processing, and the improvement of computing ability, Bayesian inference has made substantial developments for analyzing complicated data. This book introduces key ideas of Bayesian sampling methods, Bayesian estimation, and selection of the prior.
Book
A general framework for updating belief distributions
We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.
Journal Article
Mapping local and global variability in plant trait distributions
Our ability to understand and predict the response of ecosystems to a changing environment depends on quantifying vegetation functional diversity. However, representing this diversity at the global scale is challenging. Typically, in Earth system models, characterization of plant diversity has been limited to grouping related species into plant functional types (PFTs), with all trait variation in a PFT collapsed into a single mean value that is applied globally. Using the largest global plant trait database and state of the art Bayesian modeling, we created fine-grained global maps of plant trait distributions that can be applied to Earth system models. Focusing on a set of plant traits closely coupled to photosynthesis and foliar respiration-specific leaf area (SLA) and dry mass-based concentrations of leaf nitrogen (N-m) and phosphorus (P-m), we characterize how traits vary within and among over 50,000 similar to 50 x 50-km cells across the entire vegetated land surface. We do this in several ways-without defining the PFT of each grid cell and using 4 or 14 PFTs; each model's predictions are evaluated against out-of-sample data. This endeavor advances prior trait mapping by generating global maps that preserve variability across scales by using modern Bayesian spatial statistical modeling in combination with a database over three times larger than that in previous analyses. Our maps reveal that the most diverse grid cells possess trait variability close to the range of global PFT means.
Journal Article
The doomsday calculation : how an equation that predicts the future is transforming everything we know about life and the universe
\"Thomas Bayes devised a theorem that allowed him to assign probabilities to events that had never happened before, and it is being used in contemporary thought to answer questions about human existence.\" --Provided by publisher.
Book
Variational Inference: A Review for Statisticians
One of the core problems of modern statistics is to approximate difficult-to-compute probability densities. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation involving the posterior density. In this article, we review variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI has been used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling. The idea behind VI is to first posit a family of densities and then to find a member of that family which is close to the target density. Closeness is measured by Kullback-Leibler divergence. We review the ideas behind mean-field variational inference, discuss the special case of VI applied to exponential family models, present a full example with a Bayesian mixture of Gaussians, and derive a variant that uses stochastic optimization to scale up to massive data. We discuss modern research in VI and highlight important open problems. VI is powerful, but it is not yet well understood. Our hope in writing this article is to catalyze statistical research on this class of algorithms. Supplementary materials for this article are available online.
Journal Article