Now approximate the integral of the un normalized posterior

Question

Joint posterior modes for hierarchical models:

(a) Show that the posterior density for the coagulation example has a degenerate mode at τ=0 and θ_j =μ for all j.

(b) The rest of this exercise demonstrates that the degenerate mode represents a very small part of the posterior distribution. First estimate an upper bound on the integral of the un normalized posterior density in the neighborhood of the degenerate mode. (Approximate the integrand so that the integral is analytically tractable.)

(c) Now approximate the integral of the un normalized posterior density in the neighborhood of the other mode using the density at the mode and the second derivative matrix of the log posterior density at the mode.

(d) Finally, estimate an upper bound on the posterior mass in the neighborhood of the degenerate mode.