HW5 Problem 3

[Nancy]

Part (d) asks us to find a decomposition of g_i (theta_i) into a
product of unimodal functions.

I simulated some data (Z_ij) from the Rasch Model, with scalar values
for tau_theta, tau_beta, mu, and vector values for theta (index by i),
beta (indexed by j). Then, I wrote a function to complute g_i
(theta_i). For each i, the plot of g_i (theta_i) vs. theta_i appears
to be unimodal.

If g_i (theta_i) is already unimodal, then what is the need for a
decomposition? Or, is my interpretation of the problem wrong? Thanks!

[Will Chang]

If by unimodal, you mean "one hump", then it seems that way to me
too. I see no reason why you can't sample from the slice directly by
guessing at the bounds. But I will humor everyone by breaking
g(theta) up.

Actually, is the point of obtaining B that you can build a piecewise
bounding distribution for the original g, for which it is easy to
calculate the bounds of each slice on the original g?

Or are we just supposed to do slice sampling with multiple auxiliary
variables on the components of g?

WAIT. These two end up being strangely alike ...

[Percy Liang]

> If by unimodal, you mean "one hump", then it seems that way to me
> too. I see no reason why you can't sample from the slice directly by
> guessing at the bounds. But I will humor everyone by breaking
> g(theta) up.
>
> Actually, is the point of obtaining B that you can build a piecewise
> bounding distribution for the original g, for which it is easy to
> calculate the bounds of each slice on the original g?

Unimodal is not a necessary condition for being able to calculate the
bounds of the horizontal slices easily. The unimodal slices g_ij are
easy to bound...whereas g_i might not be.

>
> Or are we just supposed to do slice sampling with multiple auxiliary
> variables on the components of g?

That's exactly what you're supposed to do.

-Percy