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In a time series data generated by Monte Carlo Metropolis algorithm, when is the standard error (correlation between two data points is assumed to be negligible) is higher - when the change in the state of the system is accepted abundantly or when it is rejected most of the times, given that change is a accepted when a generated random number is less than the Boltzmann probability and rejected when it is greater?

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  • $\begingroup$ If the question is not clear, please comment here. $\endgroup$
    – luffy
    Commented Sep 17, 2018 at 21:40
  • $\begingroup$ You can't assume the correlation between two data points is negligible if the proposal is rejected "most of the time(s)", as most of the time two consecutive values will be the same. $\endgroup$
    – jbowman
    Commented Dec 31, 2022 at 22:06

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If I understand your question correctly, you are trying to tune a Metropolis-Hastings algorithm. A standard set-up would be to have a Markov Chain $(X_t)$ targeting distribution $\pi$, with at step $t$

$$\begin{eqnarray} Y_t&=&X_{t-1}+\epsilon_t\quad \epsilon_t\sim\mathcal N(0, \sigma^2)\\ \alpha_t&=&\frac{\pi(Y_t)}{\pi(X_{t-1})}\\ X_t&=&\begin{cases}Y_t \text{ if }\alpha_t>U_t\quad U_t\sim U(0,1)\\X_{t-1} \text{ otherwise.}\end{cases} \end{eqnarray}$$

The choice of $\sigma$ will impact the behaviour that you describe. If $\sigma$ is small, then $Y_t$ is very close to $X_{t-1}$: the value will often be accepted, but the chain will move by small steps (thus the autocorrelation is high). If $\sigma$ is large, then the proposed value will often be rejected (thus the autocorrelation is also high), but when it is accepted, the chain moves by a lot.

There is a sweet spot in the middle, where you make changes that are large enough to make the chain move quite fast, but small enough that you don't reject too often. In a simple case, Gelman, Gilks and Roberts (1997) have shown that this sweet spot is obtained when $E[\alpha_t]=0.234$. In practice, as long as the dimension of your problem is reasonable, an acceptance probability in that area will lead to good results.

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