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by yummyfajitas 4273 days ago
E(final conversion rate if run another time)... Both of those have a closed form.

I'm curious - where can I learn more?

Lets say you have a free plan, basic plan, and enterprise plan...A dirichlet prior would be the natural thing to use here, IMO.

This would be handled via Dirichlet, and then the results multiplied by their LTV. I thought you were referring to multiple variants - i.e., landing page A, landing page B, landing page C.
1 comments
jules 4272 days ago
Suppose you have Beta(a1,b2) and Beta(a2,b2) at the current step. The expected conversion rates are:

    M(a,b) = a/(a+b)

    E1 = M(a1,b1)
    E2 = M(a2,b2)
If we stop now the expected conversion rate is E = max(E1,E2).

If we continue for another timestep with option 1 then the question is whether that can make us switch from 1 to 2 or from 2 to 1 or not. If it can't then the expected conversion rate is the same whether or not we execute one more step. Lets assume without loss of generality that option 2 is currently winning, but if option 1 gets another conversion then 1 is winning. So the new expected conversion rate is:

    E' = int(p_1(r)*(r*r + (1-r)*E2)), r=0..1)
where p_1 is the probability density of Beta(a1,b1). All the moments of the beta distribution have a closed form, so E' also has a closed form.

You could generalize this to running it for n more times instead of one more time, you'd get an expression of the form:

   E = int(p_1(r1)*p_2(r2)*polynomial(r1,r2))
I suspect that also has a closed form but I'm not sure at first glance.
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