# Metropolis Algorithm

I am currently doing my statistics thesis on modelling football data which requires quite a great knowledge of Bayesian Theory especially MCMC methods. However I have some minor problems regarding the Metropolis algorithm:

• Any distribution can be attributed to the proposal density function?

• Also, the initial values given to the parameters (i.e. the first step of the Metropolis algorithm) are random or there must be some thinking behind the choice of values?

Thank you

• If you're just using a metropolis algorithm, you need a symmetric proposal distribution, where "symmetric" in this context means $p(a|b) = p(b|a)$ (i.e. the probability of moving from $a$ to $b$ is the same as the probability of moving from $b$ to $a$). If you want to use an asymmetric proposal, you need to use a metropolis-hastings algorithm, which is basically the exact same thing except that the acceptance ratio is multiplied by a factor that corrects for the asymmetry of the proposal distribution.