Question 1

Recall the toy "rabbit holes" problem covered in lecture 5 ("Poisson process" in two regions).

• Data:
  • One patch of land of 1 acre has $x_1 = 12$ rabbit holes.
  • A second, disjoint patch of 1 acre has $x_2 = 32$ holes.
• Question: do the rabbits prefer one patch over the other?

The first model is given by:

\begin{eqnarray} \alpha &\sim \textrm{exp}(0.1) \\ x_i \mid \alpha &\sim \textrm{Poi}(\alpha), \end{eqnarray}

and the second model is given by:

\begin{eqnarray} \beta_i &\sim \textrm{exp}(0.1) \\ x_i \mid \beta_i &\sim \textrm{Poi}(\beta_i). \end{eqnarray}

Analyse this problem with two different Bayesian model comparison methods. Compare the efficiency of the two approximations.

Question 2

Consider a target distribution given by a product of $n$ normal distribution with mean zero and variance one. Consider (1) an SIS algorithm, (2) an SMC algorithm, both using a normal proposal with variance $1.2$.

• Using the asymptotic results shown in class, derive an approximation of $\textrm{Var}(\hat Z_n)/Z_n^2$ for (1) and (2).
• How many particles are required to obtain a relative variance of $0.01$?

Question 3

Write down the general form of the reversible jump acceptance ratio for the non-conjugate DP sampler covered in lecture 12 and 13.

Question 4 (optional)

Complete the missing parts of the code covered in the tutorial. See Lab 1 under the Activities tab.