Archive for Bayesian CART

ABC variable selection

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , on July 18, 2018 by xi'an

Prior to the ISBA 2018 meeting, Yi Liu, Veronika Ročková, and Yuexi Wang arXived a paper on relying ABC for finding relevant variables, which is a very original approach in that ABC is not as much the object as it is a tool. And which Veronika considered during her Susie Bayarri lecture at ISBA 2018. In other words, it is not about selecting summary variables for running ABC but quite the opposite, selecting variables in a non-linear model through an ABC step. I was going to separate the two selections into algorithmic and statistical selections, but it is more like projections in the observation and covariate spaces. With ABC still providing an appealing approach to approximate the marginal likelihood. Now, one may wonder at the relevance of ABC for variable selection, aka model choice, given our warning call of a few years ago. But the current paper does not require low-dimension summary statistics, hence avoids the difficulty with the “other” Bayes factor.

In the paper, the authors consider a spike-and… forest prior!, where the Bayesian CART selection of active covariates proceeds through a regression tree, selected covariates appearing in the tree and others not appearing. With a sparsity prior on the tree partitions and this new ABC approach to select the subset of active covariates. A specific feature is in splitting the data, one part to learn about the regression function, simulating from this function and comparing with the remainder of the data. The paper further establishes that ABC Bayesian Forests are consistent for variable selection.

“…we observe a curious empirical connection between π(θ|x,ε), obtained with ABC Bayesian Forests  and rescaled variable importances obtained with Random Forests.”

The difference with our ABC-RF model choice paper is that we select summary statistics [for classification] rather than covariates. For instance, in the current paper, simulation of pseudo-data will depend on the selected subset of covariates, meaning simulating a model index, and then generating the pseudo-data, acceptance being a function of the L² distance between data and pseudo-data. And then relying on all ABC simulations to find which variables are in more often than not to derive the median probability model of Barbieri and Berger (2004). Which does not work very well if implemented naïvely. Because of the immense size of the model space, it is quite hard to find pseudo-data close to actual data, resulting in either very high tolerance or very low acceptance. The authors get over this difficulty by a neat device that reminds me of fractional or intrinsic (pseudo-)Bayes factors in that the dataset is split into two parts, one that learns about the posterior given the model index and another one that simulates from this posterior to compare with the left-over data. Bringing simulations closer to the data. I do not remember seeing this trick before in ABC settings, but it is very neat, assuming the small data posterior can be simulated (which may be a fundamental reason for the trick to remain unused!). Note that the split varies at each iteration, which means there is no impact of ordering the observations.

Bayesian regression trees [seminar]

Posted in pictures, Statistics, University life with tags , , , , , , , , , , on January 26, 2018 by xi'an
During her visit to Paris, Veronika Rockovà (Chicago Booth) will give a talk in ENSAE-CREST on the Saclay Plateau at 2pm. Here is the abstract
Posterior Concentration for Bayesian Regression Trees and Ensembles
(joint with Stephanie van der Pas)Since their inception in the 1980’s, regression trees have been one of the more widely used non-parametric prediction methods. Tree-structured methods yield a histogram reconstruction of the regression surface, where the bins correspond to terminal nodes of recursive partitioning. Trees are powerful, yet  susceptible to over-fitting.  Strategies against overfitting have traditionally relied on  pruning  greedily grown trees. The Bayesian framework offers an alternative remedy against overfitting through priors. Roughly speaking, a good prior  charges smaller trees where overfitting does not occur. While the consistency of random histograms, trees and their ensembles  has been studied quite extensively, the theoretical understanding of the Bayesian counterparts has  been  missing. In this paper, we take a step towards understanding why/when do Bayesian trees and their ensembles not overfit. To address this question, we study the speed at which the posterior concentrates around the true smooth regression function. We propose a spike-and-tree variant of the popular Bayesian CART prior and establish new theoretical results showing that  regression trees (and their ensembles) (a) are capable of recovering smooth regression surfaces, achieving optimal rates up to a log factor, (b) can adapt to the unknown level of smoothness and (c) can perform effective dimension reduction when p>n. These results  provide a piece of missing theoretical evidence explaining why Bayesian trees (and additive variants thereof) have worked so well in practice.