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