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"... Abstract. We propose a convergence mode for prior distributions which allows a sequence of probability measures to have an improper limiting measure. We define a sequence of vague priors as a sequence of probability measures that converges to a non-informative prior. We consider some cases where vag ..."

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Abstract. We propose a convergence mode for prior distributions which allows a sequence of probability measures to have an improper limiting measure. We define a sequence of vague priors as a sequence of probability measures that converges to a non-informative prior. We consider some cases where vague priors have necessarily large variances and other cases where they have not. We give some con-structions of vague priors that approximate the Haar measures or the Jeffreys priors. Then, we study the consequences of the convergence of prior distributions on the posterior analysis. We also revisit the Jeffreys-Lindley paradox.