Importance sampling: probit regression. Refer to Example 4.2.

a. Based on Figure 4.2 and the design rule as it is summarized in the example, which dose k

? is assigned to the next, (n + 1)-st patient?

b. Let θ =

(a, b). Evaluate the marginal posterior distributions h(πk | Dn)

using importance sampling, using a bivariate normal importance function p(θ) = N(m, V), as in the example. Plot h(πk | Dn), k = 1, . . . , K. In the plot indicate the four intervals for under-dosing, target dose, excessive dose, and unacceptable toxicity by vertical dotted lines on the interval boundaries.

c. Let {θ

(m)

; m = 1, . . . , M} denote the importance sample from p(θ). Figure 4.2

(a) shows the importance weights w

(m) ≡ h(θ

(m)

| y)/p(θ

(m)

) for an importance Monte Carlo sample {θ

(m)

; m = 1, . . . , M} of size M = 100.

Redo the importance sampling estimate, now with M = 1000, and plot the histogram of weights. What do you observe?

d. Now consider an alternative bivariate Student t importance function, p2(θ) =

t2(m, V; ν) with ν = 4. Plot a histogram of the importance sampling weights w

(m)

2

= h(θ

(m)

| y)/p2(θ

(m)

) and compare.