 Marginal likelihood from a nuisance-free statistic

If a statistic T has marginal density depending on a parameter of interest but not on a nuisance parameter, that density gives a likelihood for the interest parameter. It is obtained by integrating out other observed coordinates. In contrast, profile likelihood maximizes the full likelihood over the nuisance parameter. This sampling-statistic construction is different from Bayesian model evidence, which integrates parameters against a prior. No prior is required for a nuisance-free sampling marginal.