Date: 2016-07-09 04:47 pm (UTC)
ext_116426: (Default)
OK, this is out of my comfort zone, but if I had to solve this problem here's how I'd tackle it.

The set of coins' true probabilities has a mean and variance. For purposes of building a maximum-likelihood model, we need to assume some distribution or family of distributions, so assume a normal distribution. (This step is already sketchy because we know the values are bounded between 0 and 1, and thus the variance is maximized at 1/4, so maybe research would show a better starting point.)

Given model parameters M = ( mean probability of heads, variance ) we can then calculate a likelihood score on your observations. (Normally Theta is used for M in the literature.)

We need to work with density functions f( X | M ). Fortunately your example is discrete, so the probability density function is the same as the probability itself. What that notation means is that given an event X == a measurement of one particular coin, calculate the probability of that outcome given a particular choice of model parameters M.

So, for a trivial example, say M = ( 0, 0 ). Then f( zero heads in 3 samples | M ) = 1.

A less trivial example, say M = ( 0.5, 0 ). Then f( zero heads in 3 samples | M ) = 1/8.

I'm assuming you're going to do this numerically so a Monte Carlo approach may be good enough--- pick a bunch of coin from C at random with distribution M, flip them each the same number times as in X, record the results for many trials, and use that as f( X | M ). (You might want to pick your model for C in such a way as to make calculating f( X | M ) easy.)

Then we define likelihood of the observations as

f( X1, X2, ..., XN | M ) = f( X1 | M ) * ... * f( XN | M )

though often log-likelihood is used which lets you add instead of multiply.

Now that you know how to calculate the likelihood score, you can then find the value of M which maximizes the likelihood, keeping your observations fixed. If f( X | M ) can be represented as a function, it may be possible to come up with a mathematical answer in terms of the X's but in practice I think it will be too complicated to do so.

So you will have a lot of computational work to do trying out various M's to find one which has a high likelihood.

As an example of simplifying the model for your coins, you could assume that they have discrete probabilities, in a simple case the coins have J probability of being all-heads, K probability of being all-tails, and 1-J-K probability of being fair. Then your model M = (J, K) but the formula for f( X | M ) can be expressed simply:

If X has all heads, then f( X | M ) = J * (1-J-K)*(0.5)^( num trials of this coin)
If X has all tails, then f( X | M ) = K * (1-J-K)*(0.5)^(num trials)
If X has a mix, then f(X | M) = (1-J-K)

Unfortunately that approach gets more complicated too if we add another bucket.
Anonymous( )Anonymous This account has disabled anonymous posting.
OpenID( )OpenID You can comment on this post while signed in with an account from many other sites, once you have confirmed your email address. Sign in using OpenID.
Account name:
If you don't have an account you can create one now.
HTML doesn't work in the subject.


Notice: This account is set to log the IP addresses of everyone who comments.
Links will be displayed as unclickable URLs to help prevent spam.


Dvd Avins

October 2016


Style Credit

Expand Cut Tags

No cut tags
Page generated Oct. 18th, 2017 01:57 am
Powered by Dreamwidth Studios