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In the PRML chapter 11, The Metropolis-Hasting algorithm,

For a sampler with Gaussian distribution as proposal distribution.

The original distribution is correlated multivariate Gaussian distribution, with standard deviations $\sigma_{min}$ and $\sigma_{max}$

The transition probability is shown as below: $q(z'|z) = \mathcal{N}(z, \rho^2I)$

Because the proposed distribution is symmetric, so $q(z'|z) = q(z|z')$

Using Metropolis-hastings algorithm, the accept rate is

$A(z'|z)= min \left(1, \frac{\pi(z')q(z|z')}{\pi(z)q(z'|z)} \right)$

After cancel out the transition probability, we have

$A(z'|z)= min \left(1, \frac{\pi(z')}{\pi(z)} \right)$

It shown that the accept rate depends on the location of two points in the original distribution.

But I don't know how to include $\sigma_{max}$ or $\sigma_{min}$ in this equation.

And according to the article, the number of steps seperating states that are independent is order of $(\sigma_{max}/\sigma_{min})^2 $

How to generate this conclusion?

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Kelvin_neverKnow
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  • 1. The link to "the article" is missing. 2. The connection between $\rho$ and $(\sigma_\min,\sigma_\max)$ is not explicited. – Xi'an Dec 30 '20 at 08:04
  • @Xi'an, thank you, I just updated the missing information. $\rho$ is the standard deviation of proposed transition probability. I don't know why choosing $\sigma_{min}$ as $\rho$ will help with accept rate – Kelvin_neverKnow Dec 30 '20 at 09:35

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