Prior estimation method from point estimate
In order to use a point estimate as a prior in a beta distribution, you need to convert it to
b values. (In SkateAnalytics, these are successes or failures.) I figured out that given a point estimate like 0.6, can turn this into a fraction and then solve for
b using the numerator and denominator:
a = numerator,
b = denominator - numerator. (I thought of this from the probability formula, which is
a / (a + b).)
0.6 converts to a fraction pretty easily, but many other values don't, such as 0.7981662. Thankfully, there is a way to use programming to convert numbers into fractions and get their numerators and denominators, which we can then plug into a beta distribution. (I have found a way to do this in R.)
However, this can lead to really large values for the sum of
b (collectively called
N) if the original point estimate has more than one one number after the decimal. Since we're using this to establish a prior to then build a posterior distribution, this large N causes problems for us if the point estimate that we've been given isn't as reliable or credible (imagine if it was truly a random number generated between 0 and 1). The large N contributes "a lot" of information, which can swamp our actual data.
Before making the fraction form of the point estimate, we can first round it to the nearest
1 / N, with this
N being the number of observations that we're OK with contributing to our calculation of the posterior distribution. (I realized that
1 / N is the general form that we need after thinking about this problem for a long while as rounding to the nearest 0.1, as I felt that 10 implied observations seems OK and would likely make it easier for the computer to compute the fractional form of the result.)
How large is a "really large"
N? I'm not sure, but that's something that I think we can test with simulations.