Approximation of Posterior Means and Variances of the Digitised Normal Distribution using Continuous Normal Approximation

Ware, Robert and Lad, Frank (2003) Approximation of Posterior Means and Variances of the Digitised Normal Distribution using Continuous Normal Approximation Christchurch, New Zealand: Department of Mathematics and Statistics, University of Canterbury


Author Ware, Robert
Lad, Frank
Title of report Approximation of Posterior Means and Variances of the Digitised Normal Distribution using Continuous Normal Approximation
Publication date 2003
Publisher Department of Mathematics and Statistics, University of Canterbury
Place of publication Christchurch, New Zealand
Total pages 43
Language eng
Subjects 1117 Public Health and Health Services
Abstract/Summary All Statistics measurements which represent the values of useful unknown quantities have a realm that is both finite and discrete. Thus our uncertainties about any measurement can be represented by discrete probability. Nonetheless, common statistical practice treats probability distributions as representable by continuous densities or mixture densities. Many statistical problems involve the analysis of sequences of observations that the researcher regards exchangeably. Often we wish to find a joint probability mass function over X1, X2 , . . . , Xn with interim interest in the sequence of updated probability mass functions f ( xi+1 | Xi = xi ) for i = 1 , 2 , . . . , n - 1. We investigate how well continuous conjugate theory can approximate real discrete mass functions in various measurement settings. Interest centres on approximating digital Normal mass functions and digital parametric mixtures with continuous Mixture Normal and Normal-Gamma Mixture Normal distributions for such items as E ( Xi+1 | Xi = xi ) and V ( Xi+1 | Xi = xi ). Digital mass functions are generated by specifying a finite realm of measurements for a quantity of interest, finding a density value of some specified functions at each point, and then normalising the densities over the realm to generate mass values. Both a digitised prior mixing mass function and digitised information transfer function are generated and used, via Bayes' Theorem, to compute posterior mass functions. Approximating posterior densities using continuous conjugate theory are evaluated, and the two sets of results are compared.
Additional Notes Reference number for report: UCDMS 2003/16

 
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Created: Wed, 18 Mar 2009, 11:15:21 EST by Maryanne Watson on behalf of School of Public Health