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The following seminal paper by the great Kolmogorov introduced the important statistical concept of Bayesian Sufficiency.

Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi de Gauss. Bull. Acad. Sci. URSS Ser. Math. 6, 3–32.

This paper is cited in diverse sources such as the standard textbook by Lehmann and Romano, Testing Statistical Hypotheses (2008) (p. 20) and Blackwell and Ramamoorthi's oft-cited paper A Bayes But Not Classically Sufficient Statistic (1982) (p. 1).

Kolmogorov wrote the paper in Russian. Despite its importance, it does not seem to have been translated into any other major European language (see here and here).

Would someone please translate the definition of Bayesian Sufficiency from Kolmogorov's paper, as well as, if possible, any pertinent definitions, theorems and examples?

The concept may not be termed 'Bayesian Sufficiency' in the article. Here's a concise definition of it, excerpted from Blackwell and Ramamoorthi's paper, to help identify the definition in Kolmogorov's paper.

Let $X$ be a random variable whose distribution $P_\theta$ depends on the parameter $\theta$, and let $Y$ be a function of $X$. [...] $Y$ is [Bayes] sufficient if for every prior distribution of $\theta$ the posterior distribution of $\theta$ given $X$ depends on $Y$ only.

More rigorous, measure-theoretic definitions of Bayesian sufficiency may be found here.

I'm particularly interested in finding out which, if any, of the two variations on the concept of Bayesian Sufficiency described in the last link matches Kolmogorov's original definition.

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Evan Aad
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    30 pages...! Too much, and if you're a graduate student this, among other things, is what's expected from you: to invest the time to flirt with a russian girl/boy and have her/him to translate that for us. Seriously, if you don't find a translation and the paper really is important to your research, then you'll have to have some proficiency in the language to at least understand the basics in it. Besides this I'm sure the web already has sites with some more or less reliable translation from several languages... – DonAntonio May 13 '13 at 23:28
  • @DonAntonio: I didn't ask for the entire 30 pages to be translated; just the definition of Bayesian sufficiency. Flirting is my plan B. – Evan Aad May 14 '13 at 06:15
  • Oh, I see @Evan . Do you happen to know where in that long paper it is? Page, No. of theorem or something? – DonAntonio May 14 '13 at 07:32
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    @DonAntonio: No, i don't, handsome. – Evan Aad May 14 '13 at 11:41
  • Hehe...unless you're a girl, @Evan, than handsome is not going to work. :) – DonAntonio May 14 '13 at 11:43
  • Consider Google Translate a girl/boy friend for translation purpose -- although it was a pain to copy and paste paragraphs back and forth... – Quar Dec 11 '19 at 18:57

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I looked through it and it seems like it is mostly talking about confidence intervals estimation and has a (imho) pretty standard definition of sufficient statistics, like (page 6):

We would call statistic $\chi(x_1\dots x_n)$ a sufficient statistic for parameter $\theta_j$ if conditional probability $P(\theta_j | x_1 \dots x_n)$ is fully described by the prior probabilities of $\theta_0 \dots \theta_s$ and the value of $\chi$.

Below this definition he shows what is the sufficient statistic for the bandwidth of the Guassian given samples from it.

Ben Usman
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