  In statistical practice, a realistic Bayesian model for a given data set can
be defined by a likelihood function that is analytically or computationally
intractable, due to large data sample size, high parameter dimensionality, or
complex likelihood functional form. This in turn poses challenges to the
computation and inference of the posterior distribution of the model
parameters. For such a model, a tractable likelihood function is introduced
which approximates the exact likelihood through its quantile function. It is
defined by an asymptotic chi-square confidence distribution for a pivotal
quantity, which is generated by the asymptotic normal distribution of the
sample quantiles given model parameters. This Quantile Implied Likelihood (QIL)
gives rise to an approximate posterior distribution which can be estimated by
using penalized log-likelihood maximization or any suitable Monte Carlo
algorithm. The QIL approach to Bayesian Computation is illustrated through the
Bayesian analysis of simulated and real data sets having sample sizes that
reach the millions. The analyses involve various models for univariate or
multivariate iid or non-iid data, with low or high parameter dimensionality,
many of which are defined by intractable likelihoods. The probability models
include the Student's t, g-and-h, and g-and-k distributions; the Bayesian logit
regression model with many covariates; exponential random graph model, a
doubly-intractable model for networks; the multivariate skew normal model, for
robust inference of the inverse-covariance matrix when it is large relative to
the sample size; and the Wallenius distribution model.
