Estimation of the number of trials n made after having observed s successes with probability p | Vose Software

# Estimation of the number of trials n made after having observed s successes with probability p

### The problem

Consider the situation where we have observed s successes and know the probability of success p, but would like to know how many trials were actually done to have observed those successes. We wish to estimate a value that is fixed, so we require a distribution that represents our uncertainty about what the true value is. There are two possible situations: we either know that the trials stopped on the sth success or we do not.

##### Results

If we know that the trials stopped on the sth success, we can model our uncertainty about the true value of n as:

n = NegBin(s,p) + s

If, on the other hand, we do not know that the last trial was a success (though it could have been), then our uncertainty about n is modelled as:

n = NegBin(s+1,p) + s

Both of these formulae result from a Bayesian analysis with Uniform priors for n.

##### Derivations

Let x be the number of failures that were carried out before the sth success. We will use a uniform prior for x, i.e. p(x) = c, and, from the binomial distribution, the likelihood function is the probability that at the (s+x-1)th trial there had been (s-1) successes and then the (s+x)th trial was a success, which is just the Negative Binomial probability mass function:

Since we are using a uniform prior (assuming no prior knowledge), and the equation for l(X|x) comes directly from a distribution (so it must sum to unity) we can dispense with the formality of normalizing the posterior distribution to one, and observe:

i.e. that x = NegBin(s,p).

In the second case, we do not know that the last trial was a success, only that in however many trials were completed, there were just s successes. We have the same Uniform prior for the number of failures, but our likelihood function is just the binomial probability mass function, i.e.:

Since this does not have the form of a probability mass function of a distribution, we need to complete the Bayesian analysis, so:

The denominator sum turns out to equal 1/p. This can be easily seen by substituting s = a-1, which gives:

If the exponent for p were equal to a instead of (a-1), we would have the probability mass function of the Negative Binomial distribution, which would then sum to unity, so our denominator must sum to 1/p.

The posterior distribution then reduces to:

which is just a NegBin(s+1,p) distribution.