Comparison of classical and Bayesian estimates of probability p in a binomial process


We compare the classical statistics estimate for p with the Bayesian estimate using three versions of an uninformed prior:

Beta(0,0) which does not exist;

Beta(0.5,0.5) which has a peak at zero and one; and

Beta(1,1) which is a Uniform distribution

 

When s is small (or close to n by reflection) the classical and Bayesian with Beta(0,0) prior give the widest distribution. The Bayesian with Beta(1,1) prior gives an estimate closer to 0.5, the Bayesian with Beta(0,0) prior gives an estimate the furthest away from 0.5, and the classical and Bayesian with Beta(0.5,0.5) prior lie in between.

The classical and Bayesian with Beta(0.5,0.5) prior give very similar results for n>9 and 0<s<n. All methods tend to the same result as n gets large, and tend more quickly to the same result as s approaches n/2. The Bayesian method with Beta(0,0) prior only works for 0<s<n.

It is interesting to see from density plots where these four techniques place their emphasis:

When s=1 and n=2 the Bayesian inference Beta(0,0) prior and the classical method have Uniform(0,1) distributions for p: in other words, there appears to be no information contained in the data except to say that 0<p<1. A Bayesian has already stated that the probability exists (and therefore lies within this range) whilst the classical statistics result must first determine that the probability is not either zero or one.

When s=1 the classical and Bayesian with Beta(0,0) prior results in a mode at p=0 for any n. whilst the Bayesian with Beta(1,1) prior gives a mode at p=1/n, which is more intuitive.

The methods give the following means and modes:

 

Mode

Mean

Classical

» (s-0.6)/(n-1.2)

(s+0.5)/(n+1)

Beta(0,0)

(s-1)/(n-2)

s/n

Beta(0.5,0.5)

(s-0.5)/(n-1)

(s+0.5)/(n+1)

Beta(1,1)

s/n

(s+1)/(n+2)

 

The formulae for the modes have some parameter restrictions.

At Vose Software we use Bayesian inference with a Beta(1,1) prior when we feel we know that there is a stochastic process, and the classical result when we do not. We have found use of the other priors difficult to justify.

See Also

 

ModelRisk

Monte Carlo simulation in Excel. Learn more

Tamara

Adding risk and uncertainty to your project schedule. Learn more

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