# By C. Yeshwanth and Mohit Gupta.  An inference method that uses Bayes’ rule to update prior beliefs based on data  Allows a-priori information about.

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By C. Yeshwanth and Mohit Gupta

 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 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

 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.

 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

Number of SamplesEstimate of AlphaEstimate of Theta 1004.18094.1210 10004.85594.9054 1000005.01655.0164 The actual values used for the shape and scale parameters were 5 and 5 respectively

Actual Value of ShapeEstimated Value of Shape Estimated Value of Scale 11.00725.0441 54.97954.9925 109.98984.9976 2020.94995.2362 The number of samples was fixed to 100000 when performing this estimation and the scale parameter was fixed to 5

Actual Value of ScaleEstimated Value of Shape Estimated Value of Scale 14.98470.9995 55.01485.0145 105.024910.0654 204.976119.8601 The number of samples was fixed to 100000 when performing this estimation and the shape parameter was fixed to 5

 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

 “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_distrib ution  http://en.wikipedia.org/wiki/Gamma_distribu tion  http://en.wikipedia.org/wiki/Cumulants  http://www.mathworks.in/matlabcentral/filee xchange/26516-pearspdf

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