5.3 Marginal distributions: Given observations Y1, ..., Y, ~ i.i.d. normal(0, o?) and using the conjugate prior distribution for # and o, derive the formula for p(0 y1, ...,un), the marginal posterior distribution of #, conditional on the data but marginal over o. Check your work by comparing your formula to a Monte Carlo estimate of the marginal distribution, using some values of Y1, .... Ym, po, of, to and ko that you choose. Also derive p(o' (1. .... Un), where a? = 1/o' is the precision. 5.4 Jeffreys' prior: For sampling models expressed in terms of a p-dimensional vector v. Jeffreys' prior (Exercise 3.11) is defined as py() o VI(v)|. where I()) is the determinant of the p x p matrix I() having entries I()kl = -E[O' logp(Yly)/OwOw]. a) Show that Jeffreys' prior for the normal model is py (0, g?) x (2)-3/2. b) Let y = (y1. . .., un) be the observed values of an i.i.d. sample from a normal(0, o') population. Find a probability density py (0, oly) such that pr(0, o' ly) x p,(0,o' )p(y|0, o'). It may be convenient to write this joint density as py(0 0', y) x pa(oly). Can this joint density be considered a posterior density