Exponential distribution conjugate prior

For exponential distribution, the prior distribution of the failure rate X is usually characterized by its conjugate distribution, Gamma distribution (Zhang Jiang 2000, Wang 1993) whose probability density function is. 

Another popular choice is the class of conjugate prior distributions. A prior distribution is said to be conjugate to a class of likelihood functions p D 0, C) if the resulting posterior distributions p 0 T>, C) are in the same family as the prior distribution [214]. For example, let s say the likelihood function has the form of exponential distribution (of x) ... 

The Bayesian model assumes CV events are exponentially distributed and uses a conjugate gamma prior for each treatment group. For randomization group G (G = 1 for confrol or G = 2 for freafmenf), we use the prior... 

In this chapter we go over the cases where the posterior distribution can be found easily, without having to do any numerical integration. In these cases, the observations come from a distribution that is a member of the exponential family of distributions, and a conjugate prior is used. The methods developed in this chapter will be used in later chapters as steps to help us draw samples from the posterior for more complicated models having many parameters. 

Where the observation distribution comes from an exponential family, and the prior comes frirni the family that is conjugate to the observation distribution. 

Thus when the observations are from a one-dimensional exponential family, and the prior is from the conjugate family, the posterior is easily found without any need for integration. All that is needed is the simple updating formulas for the constants. We will now look at some common distributions that are members of the one-dimensional exponential family of densities. 

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