The inverse Gaussian distribution has the density a. A candidate for a Metropolis–Hastings...

The inverse Gaussian distribution has the density

a. A candidate for a Metropolis–Hastings algorithm targeting f is the G(α, β) distribution. Show that

b. After maximizing in x, the goal would be to minimize the bound on f /g over (α, β) for fixed (θ1, θ2). This is impossible analytically, but for chosen values of (θ1, θ2) we can plot this function of (α, β). Do so using for instance persp. Do any patterns emerge?

c. The mean of the inverse Gaussian distribution is so taking α = will make the means of the candidate and target coincide. For θ1 = θ2, match means and find an “optimal” candidate in terms of the acceptance rate.