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"# Bayesian Imputation for Missing Values in Discrete Covariates\n",
"Missing data is a very widespread problem in practical applications, both in covariates ('explanatory variables') and outcomes.\n",
"When performing bayesian inference with MCMC, imputing discrete missing values is not possible using Hamiltonian Monte Carlo techniques.\n",
"One way around this problem is to create a new model that enumerates the discrete variables and does inference over the new model, which, for a single discrete variable, is a mixture model. (see e.g. [Stan's user guide on Latent Discrete Parameters](https://mc-stan.org/docs/2_18/stan-users-guide/change-point-section.html))\n",
"Enumerating the discrete latent sites requires some manual math work that can get tedious for complex models.\n",
"Inference by automatic enumeration of discrete variables is implemented in numpyro and allows for a very convenient way of dealing with missing discrete data.\n"