David Bickel, statistician

For an observed sample of data, the likelihood function specifies the probability or probability density of that observation as a function of the parameter value. Since each sample hypothesis corresponds to a single parameter value, the likelihood of any simple hypothesis is an uncontroversial function of the data and the model. However, there is no standard definition of the likelihood of a composite hypothesis, which instead corresponds to multiple parameter values. Such a definition could be useful not only for quantifying the strength of statistical evidence in favor of composite hypotheses that are faced in both science and law, but also for likelihood-based measures of corroboration and of explanatory power for epistemological research involving Popper’s critical rationalism or recent accounts of inference to the best explanation.
Interpreting the likelihood function under the coherence framework of de Finetti, this paper mathematically formulates the problem by defining the likelihood of a simple or composite hypothesis as a subjective probability of the observed data conditional on the truth of the hypothesis. In the probability theory of this framework, conditional probabilities given a hypothesis or event of probability zero are well defined, even for finite parameter sets. That differs from the familiar probability measures that Kolmogorov introduced for frequency-type probabilities, each of which, in the finite case, can only have zero probability mass if its event cannot occur. (The latter but not the former agrees in spirit with Cournot’s principle that an event of infinitesimally small probability is physically impossible.) Thus, in the de Finetti framework, the likelihood function assigns a conditional probability to each simple hypothesis, whether or not its probability is zero.
When the parameter set is finite, every coherent conditional probability of a sample of discrete data given a composite hypothesis is a weighted arithmetic mean of the conditional probabilities of the simple hypotheses that together constitute the composite hypothesis. In other words, the coherence constraint requires that the likelihood of a composite hypothesis be a linear combination of the likelihoods of its constituent simple hypotheses. Important special cases include the maximum and the minimum of the likelihood over the parameter set. They are made possible in the non-Kolmogorov framework by assigning zero probability to all of the simple hypotheses except those of maximum or minimum likelihood.
The main result of the paper extends this result to infinite parameter sets. In general, the likelihood of a composite hypothesis is a mixture of the likelihoods of its component simple hypotheses.