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Non-informative priors via sieves and packing numbers. (English) Zbl 0904.62007

Panchapakesan, S. (ed.) et al., Advances in statistical decision. Theory and applications. Basel: Birkhäuser. Statistics for Industry and Technology. 119-132 (1997).
A method for the construction of a non-informative prior is considered. Let \(K\) be a compact metric space with a metric \(\rho.\) Let \(D(\varepsilon,K)\) be the cardinality of an \(\varepsilon\)-net with the maximum possible cardinality. Let \(X_{i}\)’s be i.i.d. with density \(f(\cdot,\theta)\) (with respect to a \(\sigma\)-finite measure \(\nu\)), where \(\theta\in\Theta\) and \(\Theta\) is an open subset of \(R^{d}.\) It is assumed that there exists \(\Psi(\cdot,\theta)\in (L^2(v))^{d}\) such that for any compact \(K\subset\Theta.\) \[ \sup\limits_{\theta\in K}\int\bigl| f^{1\over 2} (x,\theta+h)-f^{1\over 2}(x,\theta)-h^{T}\Psi(x,\theta) \bigr|^2 \nu(dx)=\text{o}(\| h\|^2), \quad\text{as}\quad \| h\|\to 0. \] The next relation defines the Fisher information \(I(\theta)=4\int\Psi(x,\theta)(\Psi(x,\theta))^{T} \nu(dx).\) It is assumed that \(I(\theta)\) is positive definite and the map \(\theta\to I(\theta)\) is continuous. The following result is the main theorem of this paper:
If on every compact subset \(K\subset \Theta\) \[ \inf\biggl\{\int\bigl( f^{1\over 2}(x,\theta_1)- f^{1\over 2}(x,\theta_2)\bigr)^2 \nu(dx)\colon \theta_1, \theta_2 \in K, \|\theta_1-\theta_2\|\geq\varepsilon \biggr\} >0, \] then for all \(Q\subset K\) with vol\((\partial Q)=0\) the equality holds: \[ \lim\limits_{\varepsilon \to 0}{D(\varepsilon,Q)\over D(\varepsilon,K)}= \int\limits_{Q}(\det I(\theta))^{1\over 2} d\theta \biggl(\int\limits_{K}(\det I(\theta))^{1\over 2} d\theta \biggr)^{-1}. \] The authors prove that \(\mu(Q)=\int_{Q} (\det I(\theta))^{1\over 2} d\theta\) is a measure (Jeffrey’s measure). A certain class of infinite dimensional families is considered, too.
For the entire collection see [Zbl 0990.62506].

MSC:

62C10 Bayesian problems; characterization of Bayes procedures
62B99 Sufficiency and information
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