####

#### The idea

The results for the Binomial and Negative Binomial distribu=
tions are both modeling *randomness*: that is to say th=
at they are returning probability distributions of possible future outcomes=
. At times, however, we are looking back at the results of a binomial proce=
ss and wish to determine one of the parameters. For example, we may have ob=
served *n* trials of which *s* were successes and from that i=
nformation would like to estimate *p*. This binomial probability is =
a fundamental property of the stochastic system and can never be observed, =
but we can become progressively more certain about its true value by collec=
ting data.

#### modeling the uncertainty about p

##### Bayesian statistics

If we have observed s su=
ccesses in n random trials, a Bayesia=
n analysis gives the conveniently simple result:

p=3DBeta(s+a, n-s+b, 1)<=
/p>

where a Beta(a,b,1) prio=
r is assumed.

The Beta(1,1,1) is a Uni=
form(0,1) distribution - often considered appropriate as an uninformed prior, in which case we have:

p=3DBeta(s+1, n-s+1,1)

The Beta distribution can be used this way because it i=
s a conjugate distribution to the binomial likeliho=
od function.

##### Classical statistics

Three methods for the es=
timation of p are discussed here.

##### Comparison between estimation methods<=
/h5>

A comparison of the clas=
sical and Bayesian methods of estimating p is provided here.