Simulating from a constructed posterior distribution

A problem often faced by those using Bayesian inference is the difficulty of determining the normalizing integral for the posterior distribution. For all but the simplest likelihood functions this can be a complex integral equation. Although sophisticated commercial software products like Mathematica®, Mathcad® and Maple® are available to perform these equations for the analyst, many integrals remain intractable and have to be solved numerically. That means that the calculation has to be redone every time new data is acquired or a slightly different problem is encountered.

For the risk analyst using Monte Carlo techniques, the normalizing part of the Bayesian inference analysis can be bypassed if one is using simulation to obtain the posterior distribution. For a constructed posterior distribution, ModelRisk offers two functions that enable us to again bypass the normalizing step: the VoseDiscrete({x},{p}) distribution and the VoseRelative(min, max, {x},{p}):

The VoseDiscrete function defines a discrete distribution where the allowed values are given by the {x} array and the relative likelihood of each of these values is given by the {p} array. An example of how this is used is given with the Tigers problem.

The VoseRelative function defines a continuous distribution with a minimum = min, a maximum = max, and several x-values given by the array {x} each of which has a relative likelihood "density" given by the {p} array. An example of how this is used is given with the Turbine blade problem

The reason that these two functions are so useful is that the user is not required to ensure that for the Discrete distribution the probabilities in {p} sum to one and for the General distribution the area under the curve equals one. The functions normalize themselves automatically.

Simulating from a constructed posterior distribution

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