Bayes Calculations from Quantile Implied Likelihood
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A Bayesian model can have a likelihood function that is analytically or computationally intractable, perhaps due to large data sample size or high parameter dimensionality. For such a model, this article introduces a likelihood function that approximates the exact likelihood through its quantile function, and is defined by an asymptotic chi-square distribution based on confidence distribution theory. This Quantile Implied Likelihood (QIL) gives rise to an approximate posterior distribution, which can be estimated either by maximizing the penalized log-likelihood, or by a standard adaptive Metropolis or importance sampling algorithm. The QIL approach to Bayesian Computation is illustrated through the Bayesian modeling and analysis of simulated and real data sets having sample sizes that reach the millions. Models include the Student's t, g-and-h, and g-and-k distributions; the Bayesian logit regression model; and a novel high-dimensional Bayesian nonparametric model for distributions under unknown stochastic precedence order-constraints.
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