I'm doing a PhD in probability theory, focusing mostly on mixing times. It's a pure maths PhD, considering precise models and showing rigorous mixing results. I'm also interested in stuff like machine learning. A brief overview of some google results suggests that MCMC is used quite a lot in these fields, however any references that I could find are of the following (approximate) form:
we want to simulate a complicated distribution; write a piece of code to sample this and run it 1,000,000 times.
The questions aren't about using rigorous probabilistic methods to find the mixing time. Maybe this is because in general people doing MC/AI aren't so interested in this? I don't know, so I thought I'd ask here..
Specifically, I'm asking for references to work in AI/ML that uses rigorous probabilistic methods to shows bounds on mixing times (runtimes for MCMC algorithms).
As mentioned above, it's possible that such a reference doesn't exist. If this is the case, any suggestions of ways that mixing times could be applied to AI/ML would be equally desirable. (Just because there isn't work in an area yet doesn't mean it can't be started!)