I have an estimation problem for a binomial data. I got a sample and from that I can get an estimation. But I also have a kind of prior information about the p. But mind it, this prior is just a single value, not a distribution of p, so I can't use Bayesian. The prior is also an estimation for the same distribution from some other sample, but unfortunately, I dont have the full sample, I just have the value of the estimate. Does anyone has any idea how to include this information in the estimation? One solution is taking AM of the two estimates, but I can't find any theoretic support for that. Does taking simple AM really makes sense?
1 Answer
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The key to this problem is how much you trust the preliminary estimate, i.e. how many record is it worth. Let it be $n$ and let your sample has $m$ records. Then, it is possible to calculate $$\hat{p}_{\rm post}=\frac{n\hat{p}_{\rm pri}+\sum_{i}^{m}x_i}{n+m}$$ where $x_i$ are the realizations from the available sample, $\hat{p}_{\rm pri}$ is the prior estimate and $\hat{p}_{\rm post}$ is the posterior estimate.
The challenge is to determine $n$ which can be generally any positive number. If you have no other information, I would recommend to try different values and inspect the results.
Alternatively, it is possible to treat $n$ as an hyper-parameter in the Bayesian sense. Thus, to assign to have a prior pdf $f(n)$ and to update it by the data using Bayesian learning.
