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Locally uniform prior distributions. (English) Zbl 0853.62008

Summary: Suppose that \(X_\sigma \mid \theta \sim N (\theta, \sigma^2)\) and that \(\sigma \to 0\). For which prior distributions on \(\theta\) is the posterior distribution of \(\theta\) given \(X_\sigma\) asymptotically \(N(X_\sigma, \sigma^2)\) when in fact \(X_\sigma \sim N (\theta_0, \sigma^2)\)? It is well known that the stated convergence occurs when \(\theta\) has a prior density that is positive and continuous at \(\theta_0\).
It turns out that the necessary and sufficient conditions for convergence allow a wider class of prior distributions – the locally uniform and tail-bounded prior distributions. This class includes certain discrete prior distributions that may be used to reproduce minimum description length approaches to estimation and model selection.

MSC:

62A01 Foundations and philosophical topics in statistics
62F15 Bayesian inference
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References:

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