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# Rates of Convergence of Posterior Distributions

**Xiaotong Shen and Larry Wasserman**

### Abstract:

*We compute the rate at which the posterior
distribution concentrates around the true parameter value.
The spaces we work in are quite general and include
infinite dimensional cases.
The rates are driven by two quantities:
the size of the space, as measure by
metric entropy or bracketing entropy, and
the degree to which the prior concentrates in a small
ball around the true parameter.
We apply the results to several examples.
In some cases,
natural priors give sub-optimal
rates of convergence and better rates can be obtained
by using sieve-based priors.
*
AMS 1990 classification: Primary, 62A15, Secondary: 62E20, 62G15.

*
** Keywords:* Bayesian inference,
asymptotic inference, non-parametric and semi-parametric models,
sieves.

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