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I'm reading a blog about Thompson Sampling, and I'm having some trouble understanding some statistical concepts.

I believe I understand when the author says $$ p(\mu_a \mid \mbox{data}_a) = \mbox{Beta}(S_{a,t}+1, F_{a,t}+1) = \frac{S_{a,t}!F_{a,t}!}{S_{a,t} + F_{a,t} + 1} . $$ But in the "Continuous Variables and the Normal Approximation" section the author states $$ p(\mu_a \mid \mbox{data}_a) = \mbox{N}(\mu_{a, t}, \sigma_{a,t}^2/N_{a,t}) . $$ Isn't this N function the notation for a normal distribution? How does one get a distinct probability from a Normal distribution, like the Beta Function, in this case?

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  • $\begingroup$ You can approximate a beta with a normal when the two parameters are large and equal. Some details on the error at John Cook's site. $\endgroup$
    – dimitriy
    Commented Jun 13, 2014 at 18:14
  • $\begingroup$ So what I would be looking for is the max of the normal distribution? $\endgroup$
    – hoshi
    Commented Jun 13, 2014 at 18:16

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The last equality $=\frac{S_{a,t}!F_{a,t}!}{S_{a,t}+F_{a,t}+1}$ in the first equation of your question is not found in the blog article, and is actually not what the author meant.

Instead, the intended meaning is that the conditional distribution of $\mu_a$ conditional on $data_a$ is a beta distribution with these parameters, just as in the latter case, the conditional distribution of $\mu_a$ conditional on $data_a$ is a normal distribution. Both of these are continuous distributions, there should be no conceptual difference.

I'm not sure what you mean by 'getting a distinct probability', but it sounds like you might be confused about the difference betweeen probabilities and probability density functions. It might help to take a look, e.g., at the answers to this frequent question or read some introduction to probability/statistics, if this is indeed the issue here.

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  • $\begingroup$ Ok I think I understand what the author meant now. The author pulls random samples from the distributions, to determine the best choice. $\endgroup$
    – hoshi
    Commented Jun 13, 2014 at 21:28

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