Question

can someone explains to me in details. i provided the answers yet i dont understand it

realization of a random sample X1, X2, X3 from an N(u,a2) distri 2. Suppose that xi,x2, x3 are a bution with o known. If we decide to use Bayesian inference to obtain a reliable estimate for the parameter , and use 1 h(H) e 2 the prior distribution of the parameter i. Write down an expression for the posterior distribution (on-1) as of You do not need to evaluate any integral.) (3 marks) 2. Solution (3 marks) The prior distribution is 1 h(H) V27T (On-1) The likelihood is 2 1 1 1 f (1, 2, 3) e a27T av27 av27T Therefore, the posterior distribution of u is f(x1, x2,, n4)h(12) SO(2, ,an|4)h(4)du () 2 14*x __e()* av27T 2 1( 27T 1 X e av27 av2

Answer 1

Let's jump straight away to the very last expression, I am assuming this is the part where there is a loss of clarity.

We know that we are provided with the density functions for $\mu, X_i$, this implies that these random variables are continuous.

Now, in the last expression we are provided with the following:

f(x1, 2, ) h(u) P(42,3) f2,3h()du

The above expression of posterior distribution for $\mu$ is of the form

Pr(BA) Pr(A) Pr(AB)=, Pr(B|A) Pr(A)

Which I can rewrite it as denominator is merely a normalising constant that would make the whole expression as valid probability.

Pr(AB) Pr(B|A) Pr(A) x constant

I can further, rewrite the above expression as ,

Pr(AB) Pr(BA Pr(A)

Now if I think about our original expression for the posterior distribution of $\mu$,

• 
  Pr(AB
  is nothing but the posterior probability i.e.
  p(u1, 2, 3
  ,
• 
  Pr(BA
  is nothing but the joint probability i.e.
  2 f(x14)xf(2)xf (x34) f(1, 2, 34) X e V2T av2T V 1 V2π
  
• And finally, $\Pr(A)$ is the prior distribution of $\mu$ i.e.
  h (μ) V2π (or-1)