We propose a general class of flexible models for longitudinal data with special emphasis on discrete-time survival data. The model is a finite mixture model where the subjects are allowed to move between components through time. The time-varying probabilities of component memberships are modeled as a function of subject-specific time-varying covariates. This allows for interesting within-subject dynamics and manageable computations even with a large number of subjects. Each parameter in the component densities and in the mixing function is connected to its own set of covariates through a link function. The models are estimated using a Bayesian approach via a highly efficient Markov Chain Monte Carlo (MCMC) algorithm with tailored proposals and variable selection in all sets of covariates. The focus of the paper is on models for discrete-time survival data with an application to bankruptcy prediction for Swedish firms, using both exponential and Weibull mixture components. The dynamic mixture-of-experts models are shown to have an interesting interpretation and to dramatically improve the out-of-sample predictive density forecasts compared to models with time-invariant mixture probabilities.

The computing time for Markov Chain Monte Carlo (MCMC) algorithms can be prohibitively large for data sets with many observations, especially when the data density for each observation is costly to evaluate. We propose a framework where the likelihood function is estimated from a random subset of the data, resulting in substantially fewer density evaluations. The data subsets are selected using an efficient Probability Proportional-to-Size (PPS) sampling scheme, where the inclusion probability of an observation is proportional to an approximation of its contribution to the log-likelihood function. Three broad classes of approximations are presented. The proposed algorithm is shown to sample from a distribution that is within O(m^-1/2) of the true posterior, where m is the subsample size. Moreover, the constant in the  O(m^-1/2) error bound of the likelihood is shown to be small and the approximation error is demonstrated to be negligible even for a small m in our applications. We propose a simple way to adaptively choose the sample size m during the MCMC to optimize sampling efficiency for a fixed computational budget. The method is applied to a bivariate probit model on a data set with half a million observations, and on a Weibull regression model with random effects for discrete-time survival data.

We propose a generic Markov Chain Monte Carlo (MCMC) algorithm to speed up computations for data sets with many observations. A key feature of our approach is the use of the highly efficient difference estimator from the survey sampling literature to estimate the log-likelihood accurately using only a small fraction of the data. Our algorithm improves on the O(n) complexity of regular MCMC by operating over local data clusters instead of the full sample when computing the likelihood. The likelihood estimate is used in a Pseudo-marginal framework to sample from a perturbed posterior which is within O(m^-1/2) of the true posterior, where m is the subsample size. The method is applied to a logistic regression model to predict firm bankruptcy for a large data set. We document a significant speed up in comparison to the standard MCMC on the full data set.

The complexity of Markov Chain Monte Carlo (MCMC) algorithms arises from the requirement of a likelihood evaluation for the full data set in each iteration. Payne and Mallick (2014) propose to speed up the Metropolis-Hastings algorithm by a delayed acceptance approach where the acceptance decision proceeds in two stages. In the first stage, an estimate of the likelihood based on a random subsample determines if it is likely that the draw will be accepted and, if so, the second stage uses the full data likelihood to decide upon final acceptance. Evaluating the full data likelihood is thus avoided for draws that are unlikely to be accepted. We propose a more precise likelihood estimator which incorporates auxiliary information about the full data likelihood while only operating on a sparse set of the data. It is proved that the resulting delayed acceptance MCMC is asymptotically more efficient compared to that of Payne and Mallick (2014). Furthermore, we adapt the method to handle data sets that are too large to fit in Random-Access Memory (RAM). This adaptation results in an algorithm that samples from an approximate posterior with well studied theoretical properties in the literature.