Problems
1. Exercise in the text
2. Exercise in the text
3. A common prior you may come across is the Cauchy distribution, particularly as a starting point for modeling regression coefficients or as a hyperprior (in the form of a 'half-Cauchy') for variance parameters.

(a) Plot a standard Cauchy distribution with location 0 and scale 1. Plot it along with a standard Normal distribution and t distributions
with various degrees of freedom. Discuss the relationship between these distributions. What aspect of the Cauchy sets it
apart from Normal or t distributions?

(b) Based on the comparisons made in (a), what makes the standard Cauchy an ideal default (i.e., starting point) prior. Of the distributions in (a), which distribution would you use if you were very confident in your prior information? What about in the case you
are very unconfident?

(c) Based on (b), it should be clear when the Cauchy is useful over the Normal or t distributions. Since the Cauchy distribution
as a prior is more applicable to regression or hierarchical modeling situations, let's instead directly apply the distribution to
some data, using Gibbs to approximate our posterior. Before we do that, though, we have to rewrite the Cauchy in a form that
exploits its relationship with the other distributions mentioned above. Given the Cauchy distribution

(d) Identify the density for the distriubtion p(x|µ) (this may be more apparent before inserting the values for α and β).
(e) Now let's construct a Gibbs sampler for the following model:

where σ2λi is the scaled variance for the Normal distribution and 1σ2 is a noninformative prior placed on µ and σ2 . Write out the
complete joint density p(xi|λi)p(λi)p(µ, σ2).
• HINT: Note the index on λ.

(f) Show that the Gibbs updates below can be obtained from the joint density in (e).

(g) Using the data X provided, construct a Gibbs sampler to approximate the posterior distribution. Begin with a 1000 iteration
burn-in and then run the sampler for an additional 1500 iterations. Your output should consist of an N × 1500 matrix of λi values and two N-vectors for µ and σ2.