Bayesian Analysis of the Two Parameter Gamma

DistributionBy C. Yeshwanth and Mohit Gupta

The Gamma distribution is characterized by the following probability density function

θ is the scale parameter of the distribution and

α is the shape parameter

The Gamma Distribution

An inference method that uses Bayes’ rule to update prior beliefs based on data

Allows a-priori information about the parameters to be used in the analysis method

A posterior distribution over the hypothesis space is obtained using the Bayesian update rule

Conjugate priors are priors which result in a posterior distribution belonging to the same family after the Bayesian update

The Bayesian Analysis Method

A conjugate prior for the Gamma Distribution

and are the shape and scale parameters of the Gamma Distribution

n’, v’, s’ and p’ can be interpreted as encoding prior information about the parameters

Conjugate Priors

The posterior distribution after the Bayesian update is proportional to

Where v’’ = v’ + n, p’’ = p’p, s’’ =s’ + s, and n’’ = n’+n with p = and s =

The posterior thus belongs to the same family as the prior

Conjugate Priors

The marginal distribution over is proportional to

Where = =

The Method

The first 4 moments of are computed numerically using the distribution provided above as a closed form solution is not known

The marginal distribution of given is observed to be a Gamma distribution

The moments of are computed from the moments of alpha already computed

The distributions are then fitted with curves from the Pearson Distribution and confidence intervals are extracted

The Method

The Pearson distribution is a family of continuous probability distributions

These are used to fit a distribution given the mean, the variance, the skewness and the kurtosis of the distribution.

There are many families included in Pearson curves like Beta distribution, gamma distribution and Inverse gamma distribution

Pearson Distribution

Depending on the first four moments, it is decided that which Pearson curve will best fit the given data

After fixing the type of the distribution, the parameters of that distribution are calculated to obtain the exact distribution.

Pearson Distribution

An alternative conjugate prior for the parameters is given by

The approach described for this prior involves numerically integrating over for each value of

The approach described before avoids extra computation by providing approximations to the required integrals

Alternative Approaches

We did not use the alternative for 2 reasons The family of densities described by the

alternative approaches is a subset of the families described in the first approach

The computational costs of using the second approach are too high

Alternative Approaches

Number of Samples Estimate of Alpha Estimate of Theta

100 4.1809 4.1210

1000 4.8559 4.9054

100000 5.0165 5.0164

Experiments

The actual values used for the shape and scale parameters were 5 and 5 respectively

Actual Value of Shape

Estimated Value of Shape

Estimated Value of Scale

1 1.0072 5.0441

5 4.9795 4.9925

10 9.9898 4.9976

20 20.9499 5.2362

Experiments

The number of samples was fixed to 100000 when performing this estimation and the scale parameter was fixed to 5

Actual Value of Scale

Estimated Value of Shape

Estimated Value of Scale

1 4.9847 0.9995

5 5.0148 5.0145

10 5.0249 10.0654

20 4.9761 19.8601

Experiments

The number of samples was fixed to 100000 when performing this estimation and the shape parameter was fixed to 5

Errors in Confidence Interval

The accuracy of the point estimates increases with increasing the number of samples

Credible confidence intervals are difficult to extract from the raw distributions especially for the scale parameter

We attempted to fit a Pearson Curve to the distributions to extract the confidence interval

This is because of the skewed nature of the Pearson estimate of the posterior density

Conclusions

Thank You

“Bayesian Analysis of the Two-Parameter Gamma Distribution” by Robert B. Miller

“Conjugate classes for gamma distributions” by Eivind Damsleth.

http://en.wikipedia.org/wiki/Pearson_distribution

http://en.wikipedia.org/wiki/Gamma_distribution

http://en.wikipedia.org/wiki/Cumulants http://www.mathworks.in/matlabcentral/

fileexchange/26516-pearspdf

References

where , are the first four cumulants of

Appendix