Adaptive Bayesian estimation of conditional discrete-continuous distributions with an application to stock market trading activity

Adaptive Bayesian Estimation of Discrete‐Continuous Distributions Under Smoothness and Sparsity

“…However, when the conditional distribution is smoother, then it could be beneficial to estimate the conditional distribution directly. In an ongoing work, Norets and Pelenis (forthcoming), we pursue an extension of our posterior contraction results to conditional distribution models based on covariate dependent mixtures; the extension is similar to work by Norets and Pati (2017) on continuous distributions.…”

“…Our conjecture is that, at least without considerable modifications, this kernel estimator is unlikely to deliver the adaptive optimal estimation rates that are established for mixture models in the following section; and, perhaps, that is why the kernel estimator is outperformed by the mixture model in our simulations. A few other applications and favorable comparisons of a fixed m mixture model with standard parametric and nonparametric alternatives can be found in Norets and Pelenis (2012).…”

“…Some results on frequentist asymptotic properties of the posterior distribution in such models have also been established. Norets and Pelenis (2012) obtained approximation results in Kullback-Leibler distance and weak posterior consistency for mixture models with a prior on the number of mixture components. DeYoreo and Kottas (2017) established weak posterior consistency for Dirichlet process mixtures.…”

“…MCMC algorithm for model estimation and proof outlines are given in the appendices. Auxiliary results and proof details are delegated to the Supplemental Material (Norets and Pelenis (2022)).…”

“…From the computational perspective, discrete variables can be easily accommodated through the use of continuous latent variables in Bayesian MCMC estimation (Albert and Chib (1993)). In nonparametric modeling of discrete-continuous data by mixtures, latent variables were used by Canale and Dunson (2011) and Norets and Pelenis (2012) among others. Some results on frequentist asymptotic properties of the posterior distribution in such models have also been established.…”