Getting the intrinsic variance for estimators

[Alex Reeves]

Dear all, 

probably a very basic question but I am fairly new to Emcee. In the documentation, there is a long discussion about the autocorrelation time and how to estimate it.The variance in our integral from the MCMC is given as: sigma**2=(tau_f/N) * VAR(f(theta)). The documentation then goes on to state something along the lines of "if we can estimate tau_f then we can know how many samples we need for sub percent precision". My questions- how do we know what VAR(f(theta)) is?- for example for a 1-D posterior for a parameter in a model where I believe the required f(theta) would be a delta function for each value of the parameter to map out the 1D posterior distribution. 

In more general terms I would like to get a better handle on when I can know when my chains are sufficiently converged and that I can trust the posteriors- is it enough to just have a stable estimate of tau_f (i.e. have a large enough N that the N = 100tau_f criterion is met) or do I also need to know something about the variance for the particular f(theta) I am choosing? 

Thank you very much in advance, 

Alex Reeves 

[Dan Foreman-Mackey]

Hi Alex,

The Var(f(theta)) discussed in the docs is actually the posterior variance for the function f(theta). One way to think about this is that your sampling error is a relative error wrt to precision with which you can measure f(theta).

When looking at convergence, I generally don't explicitly consider Var(f) (in part because any estimator of Var(f) will *also* have sampling uncertainty). If you follow a rule of thumb like N > 100 tau, you'll generally have a large enough number of effective samples that the sampling error is negligible compared to the posterior variance, so that's the main convergence criterion I would consider. But, if you wanted to make a statement about the magnitude of the Monte Carlo error then you'd want to look at Var(f) more explicitly.

Hope this helps!

Dan

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Dan Foreman-Mackey

Associate Research Scientist

Flatiron Institute

https://dfm.io