# Finding a Scalable Approach to a modified coin-flip problem using MLE model

and thank you in advance for your help!

I am interested in applying MLE to estimate parameters in a "modified" coin flip model, but have been having difficulties scaling the solution.

The problem can be formulated as follows:

• Let there exist $I$ independent "biased" coins, where coin $i \leq I$ has a probability $p_i$ of landing heads.
• You are not told the probabilities, $p_i; i = 1, 2, ..., I$. However, you are given the opportunity to infer their values from a series of $I$ experiments (same as the number of coins).
• In experiment $i \leq I$, coins $1, 2, ..., i$ are each tossed 10 times. So for example, in experiment 1, the first coin is tossed. In experiment 2, the first coin and second coin are tossed, and so on. Let $Y_i$ be the random variable indicating number of heads counted in experiment $i$. Note that $Y_i \sim \sum^i_{j=1} \mathrm{Binomial}(n=10, p_j)$.
• Given a set of data, $\mathbf{y}$, what is the maximum likelihood estimate of the probabilities $p_i; i = 1, 2, ..., I$?

I put together R code to solve this numerically by optimizing the vector $\mathbf{p} = (p_1, p_2..., p_i)$ to maximize the log likelihood function. The log likelihood function was determined by using the sum of independent binomials $P(Y + Z = j) = \sum_{i=0}^j P(Y = i) P(Z = j-i)$ as presented by Butler and Stephens (1993).

This solution worked very well for a 3-coin problem. However, it suffered from major scale problems when increasing the number of coins to, say 10, as this represented a $10^7$ times increase in the number of $\mathbf{p}$ vectors to optimize over.

I also attempted to formulate this as an EM problem, but it's not a typical mixture model and was difficult to find a model with a closed form solution.

My questions are...

1. Is there an algorithm to solve this MLE problem that will scale reasonably well? I've read about Monte Carlo EM, for example. Would an approach like that be applicable here?
2. Are there any "known" solutions to this class of problem? I don't have an extensive background in statistics, so I may be missing an obvious previously described method.