A note on quasi-likelihood for exponential families

Maximum likelihood estimation for exponential families depends exclusively on the first two moments of the data. Recognizing this, Wedderburn [1974. Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika 61, 439-447] proposed estimating regression parameters based on a quasi-likelihood function requiring only the relationship between the mean and variance. We extend quasi-likelihood to situations in which there exists vague prior information on the mean parameters. It is shown when data are exponential family with quadratic variance functions, maximum a posteriori inference under a conjugate prior relies solely on two moments of the data and the prior distribution. This result suggests a Bayesian analog of quasi-likelihood for which only two moments of the data and two moments of the prior need be specified.