Fully Exponential Laplace Approximations to Expectations and Variances of Nonpositive Functions

Abstract

Abstract Tierney and Kadane (1986) presented a simple second-order approximation for posterior expectations of positive functions. They used Laplace's method for asymptotic evaluation of integrals, in which the integrand is written as f(θ)exp(-nh(θ)) and the function h is approximated by a quadratic. The form in which they applied Laplace's method, however, was fully exponential: The integrand was written instead as exp[− nh(θ) + log f(θ)]; this allowed first-order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second-order expansion for the ratio. Other second-order expansions (Hartigan 1965; Johnson 1970; Lindley 1961, 1980; Mosteller and Wallace 1964) require computation of more derivatives of the log-likelihood function. In this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions. To obtain a second-order approximation to an expectation E(g(θ)), we use the fully exponential method to approximate the moment-generating function E(exp(sg(θ))), whose integrand is positive, and then differentiate the result. This method is formally equivalent to that of Lindley and that of Mosteller and Wallace, yet does not require third derivatives of the likelihood function. It is also equivalent to another alternative approach to the approximation of E(g(θ)): We may add a large constant c to g(θ), apply the fully exponential method to E(c + g(θ)), and subtract c; on passing to the limit as c tends to infinity we regain the approximation based on the moment-generating function. Furthermore, the second derivative of the logarithm of the approximation E(exp(sg(θ))), which is an approximate cumulant-generating function, yields a simple second-order approximation to the variance. In deriving these results we omit rigorous justification of formal manipulations, which may be found in Kass, Tierney, and Kadane (in press). Although our point of view is Bayesian, our results have applications to non-Bayesian inference as well (DiCiccio 1986).