# Are there frequentist approaches to Thompson Sampling?

What is the theoretical reason why Thompson Sampling needs to involve posterior distributions? Why can we not sample over predictive distributions? (or is the issue that predictive frequentist distributions are difficult to obtain?)

• What's really the difference between the distribution collected as updated priors / posteriors vs "frequentist" distribution? The way I see it, it's the difference between an online mean ($\mu\leftarrow\mu+(x_t-\mu)/t$) vs a batch mean ($\mu=\frac1n\sum_{t=1}^nx_t$), they give the same answer. – Kris Feb 1 '19 at 3:50
• From what I can tell, the idea is that posterior distribution gives you probability matching, but it has been difficult for me to find literature on why or why not the predictive distribution gives you the same. What's the difference in posterior distributions vs predictive distributions? Outside of the base trivial case as you have mentioned? They can differ quite a bit. – Jenny Yang Feb 2 '19 at 5:28
• Great question. Nice answer below by JP, and the whole thing as been viewed 158 times in 8 months. I just don't get this site. – Ben Ogorek Sep 15 '19 at 1:17