What question did this study set out to answer?

The aim is to present a Bayesian approach to testing sharp hypotheses using weighted likelihood ratios.

February 1, 1970Open Access

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

JDJames DickeyAugusta University BLBennet P. LientzUniversity of Southern California

Key Points

The aim is to present a Bayesian approach to testing sharp hypotheses using weighted likelihood ratios.
Examined fixed parameter hypotheses with arbitrary prior probabilities.
Calculated the ratio of posterior to prior odds using the weighted likelihood ratio.
Applied the methodology to multinomial means and evaluated binomial probabilities.
The weighted likelihood ratio was shown to equal Savage's posterior to prior density ratio (2.21).
Introduced a new test for determining the order of a finite-state 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.

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