## Electronic Journal of Statistics

### Exchangeable Bernoulli random variables and Bayes’ postulate

#### Abstract

We discuss the implications of Bayes’ postulate in the setting of exchangeable Bernoulli random variables. Bayes’ postulate, here, stipulates a uniform distribution on the total number of successes in any number of trials. For an infinite sequence of exchangeable Bernoulli variables the conditions of Bayes’ postulate are equivalent to a uniform (prior) distribution on the underlying mixing variable which necessarily exists by De Finetti’s representation theorem. We show that in the presence of exchangeability, the conditions of Bayes’ postulate are implied by a considerably weaker assumption which only specifies the probability of $n$ successes in $n$ trials, for every $n$. The equivalence of the Bayes’ postulate and the weak assumption holds for both finite and infinite sequences. We also explore characterizations of the joint distribution of finitely many exchangeable Bernoulli variables in terms of probability statements similar to the weak assumption. Finally, we consider extensions of Bayes’ postulate in the framework of exchangeable multinomial trials.

#### Article information

Source
Electron. J. Statist., Volume 7 (2013), 2193-2208.

Dates
First available in Project Euclid: 16 September 2013

https://projecteuclid.org/euclid.ejs/1379336968

Digital Object Identifier
doi:10.1214/13-EJS835

Mathematical Reviews number (MathSciNet)
MR3104917

Zentralblatt MATH identifier
1293.62007

#### Citation

Banerjee, Moulinath; Richardson, Thomas. Exchangeable Bernoulli random variables and Bayes’ postulate. Electron. J. Statist. 7 (2013), 2193--2208. doi:10.1214/13-EJS835. https://projecteuclid.org/euclid.ejs/1379336968

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