## The Annals of Mathematical Statistics

### The Weighted Likelihood Ratio, Sharp Hypotheses about Chances, the Order of a Markov Chain

#### Abstract

The Bayesian theory for testing a sharp hypothesis, defined by fixed values of parameters, is here presented in general terms. Arbitrary positive prior probability is attached to the hypothesis. The ratio of posterior to prior odds for the hypothesis is given by the weighted likelihood ratio, shown here to equal Leonard J. Savage's (1963) ratio of a posterior to a prior density (2.21). This Bayesian approach to hypothesis testing was suggested by Jeffreys (1948), Savage (1959), (1961), Lindley (1961), and Good (1950), (1965), but obscured some what by approximations and unique choices of prior distributions. This Bayesian theory is distinct from that of Lindley (1965) and that of Dickey (1967a). Applications are given to hypotheses about multinomial means, for example, equality of two binomial probabilities. A new test is presented for the order of a finite-state Markov chain.

#### Article information

Source
Ann. Math. Statist., Volume 41, Number 1 (1970), 214-226.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177697203

Digital Object Identifier
doi:10.1214/aoms/1177697203

Mathematical Reviews number (MathSciNet)
MR258187

Zentralblatt MATH identifier
0188.50102

JSTOR