We develop a Bayesian tree ensemble model to estimate heterogeneous treatment effects in censored survival data with high-dimensional covariates. Instead of imposing sparsity through the tree structure, we place a horseshoe prior directly on the step heights to achieve adaptive global-local shrinkage. This strategy allows flexible regularisation and reduces noise. We develop a reversible jump Gibbs sampler to accommodate the non-conjugate horseshoe prior within the tree ensemble framework. We show through extensive simulations that the method accurately estimates treatment effects in high-dimensional covariate spaces, at various sparsity levels, and under non-linear treatment effect functions. We further illustrate the practical utility of the proposed approach by a re-analysis of pancreatic ductal adenocarcinoma (PDAC) survival data from The Cancer Genome Atlas.

We consider learning the structures of Gaussian latent tree models with vector observations when a subset of them are arbitrarily corrupted. First, we present the sample complexities of Recursive Grouping (RG) and Chow-Liu Recursive Grouping (CLRG) without the assumption that the effective depth is bounded in the number of observed nodes, significantly generalizing the results in Choi et al. (2011). We show that Chow-Liu initialization in CLRG greatly reduces the sample complexity of RG from being exponential in the diameter of the tree to only logarithmic in the diameter for the hidden Markov model (HMM).

We propose stepwise variational inference (VI) with vine copulas: a universal VI procedure that combines vine copulas with a novel stepwise estimation procedure of the variational parameters. Vine copulas consist of a nested sequence of trees built from copulas, where more complex latent dependence can be modeled with increasing number of trees. We propose to estimate the vine copula approximate posterior in a stepwise fashion, tree by tree along the vine structure. Further, we show that the usual backward Kullback-Leibler divergence cannot recover the correct parameters in the vine copula model, thus the evidence lower bound is defined based on the Rényi divergence. Finally, an intuitive stopping criterion for adding further trees to the vine eliminates the need to pre-define a complexity parameter of the variational distribution, as required for most other approaches. Thus, our method interpolates between mean-field VI (MFVI) and full latent dependence. In many applications, in particular sparse Gaussian processes, our method is parsimonious with parameters, while outperforming MFVI.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Mean-field variational inference (VI) is computationally scalable, but its highly-demanding independence requirement hinders it from being applied to wider scenarios. Although many VI methods that take correlation into account have been proposed, these methods generally are not scalable enough to capture the correlation among data instances, which often arises in applications with graph-structured data or explicit constraints. In this paper, we developed the Tree-structured Variational Inference (TreeVI), which uses a tree structure to capture the correlation of latent variables in the posterior distribution. We show that samples from the tree-structured posterior can be reparameterized efficiently and parallelly, making its training cost just 2 or 3 times that of VI under the mean-field assumption. To capture correlation with more complicated structure, the TreeVI is further extended to the multiple-tree case. Furthermore, we show that the underlying tree structure can be automatically learned from training data. With experiments on synthetic datasets, constrained clustering, user matching and link prediction, we demonstrate that the TreeVI is superior in capturing instance-level correlation in posteriors and enhancing the performance of downstream applications.

TreeV AE hierarchically divides samples according to their intrinsic characteristics, shedding light on hidden structures in the data.