7.1 The big idea 1/2

We are going to use a Normal-Normal model:

\[ Y|\mu \sim Norm(\mu, 0.75^2)\]

\[ \mu \sim Norm(0, 1^2) \]

Observed outcome 6.25:

\[ \mu|(Y = 6.25) \sim Norm(4, 0.6^2)\]

Main idea: chain need to spend more time around \(\mu\) value. Remember \(\mu^{i+1}\) is dependant of \(\mu^{i}\).

How are we going to visit every part of the posterior dustribution:

1. step 1 : propose a random location \(\mu'\) (I prefer \(\mu_{proposal}\)) for the nex stop

2. step 2 : Decide whether to:

   • go to the proposed location: \(\mu_{proposal} = \mu^{i+1}\)
   • stay at the current location: \(\mu = \mu^{i+1}\)

Monte Carlo algorithm:

• step 1 propose location: draw \(\mu\) from posterior model with \(pdf \quad f(\mu|y)\)

• step 2 : Go there