  The prior distribution on parameters of a likelihood is the usual starting
point for Bayesian uncertainty quantification. In this paper, we present a
different perspective. Given a finite data sample $Y_{1:n}$ of size $n$ from an
infinite population, we focus on the missing $Y_{n+1:\infty}$ as the source of
statistical uncertainty, with the parameter of interest being known precisely
given $Y_{1:\infty}$. We argue that the foundation of Bayesian inference is to
assign a predictive distribution on $Y_{n+1:\infty}$ conditional on $Y_{1:n}$,
which then induces a distribution on the parameter of interest. Demonstrating
an application of martingales, Doob shows that choosing the Bayesian predictive
distribution returns the conventional posterior as the distribution of the
parameter. Taking this as our cue, we relax the predictive machine, avoiding
the need for the predictive to be derived solely from the usual prior to
posterior to predictive density formula. We introduce the martingale posterior
distribution, which returns Bayesian uncertainty directly on any statistic of
interest without the need for the likelihood and prior, and this distribution
can be sampled through a computational scheme we name predictive resampling. To
that end, we introduce new predictive methodologies for multivariate density
estimation, regression and classification that build upon recent work on
bivariate copulas.
