Computation of Posterior Mode for Location Parameter and Conditional Expectation for Scale Parameter of Normal Sequence

Abstract

In accordance with Bayesian perspective analysis of Normal Sequence, instead of using usual type of priors rather than Non-informative, Normal-gamma priors, we, here introduce a combination of novel priors such as Double- Exponential and Half-Normal distributions as a sparsity promoting priors respectively, for the location and scale parameters are considered assuming independence, to carryout the Inferential procedure. Suppose that the prior for the parameters behaves similar to that of the density with heavy tailed nature rather Double Exponential prior suitably describes the practical situation for location parameter than usual type of other priors. In this case analytical derivations of posterior distributions for the parameters are too tedious, almost all impossible and it will lead to a complicated numerical integration process. To overcome this complication we conceive an ideology of developing a new type of algorithmic procedure for finding posterior mode, mean and variance of the parameters. It is the first newly developed algorithmic procedure and before that no one was tried to find this type of algorithmic procedure. In particular, surmounts the difficulty of hard numerical integration of location parameter over infinite range, we developed this algorithmic procedure and it was achieved through C- language in which posterior mode and mean obtained as Bayes estimates of location parameter. Conditional Bayes estimates of scale parameter are also obtained. To illustrate the methodology of the novel procedure, simulation study carried out and shown that their better performance through computing the Bayesian point estimates and mean square error of the Bayes estimates of posterior mode, mean and variance of both the parameters. Among all the three measures, posterior mode performs well. It gives very closeestimates for the original parameter values and it has the less mean square error (MSE) compared with MSE of posterior mean and variance. Our newly developed algorithmic procedure is performed well and the results are shown in the tables.