1. We want to know the posterior distribution $P(\theta)$ and where modes are, this is the goal.
2. But we cannot calculate $P(\theta)$ analytically, this is the problem.
3. However we can build a Markov Chain.
4. Sampling from the Markov Chain builds the histogram that approximates $P(\theta)$, this is the solution.

[![enter image description here][1]][1]


Surprisingly:

* Prior can be an arbitrary normal distribution, although the choice impacts.
* Proposal can be arbitrary normal distribution whose mean is current $\theta $.

The questions and confusions may be:

1. How come arbitrary normal distribution can be used as the prior?
2. How come $q(\theta_i | \theta_j) * min(1, \frac { prior(\theta_j * L(E|\theta_j) * q(\theta_j | \theta_i)} { prior(\theta_j * L(E|\theta_j) * q(\theta_i | \theta_j)} ) $ can be equivalent with $P(\theta_j|\theta_i)$?

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/pTTV1.jpg
  [2]: https://i.sstatic.net/RIzto.jpg