Gamma distribution conjugate prior

Statistical data analysis of operation time till failure shows that operation time till failure T as random variable follows Weibull distribution (according to performed goodness of fit tests). The parameters k and p are assumed as independent random variables with prior probability density functions p x)—gamma pdf with mean value equals to prior (DPSIA) estimate of k and variance—10% of estimate value, / 2(j)— inverse gamma (as conjugate prior (Bernardo et al, 2003 Berthold et al, 2003)) pdf with mean value equals to prior (DPSIA) estimate of p and variance—10% of estimate value. Failure data tj, j =1,2,. .., 28. Thus, likelihood function is... 

A number of issues arise in using the available data to estimate (he rates of location-dependent fire occurrence. These include the possible reduction in the frequency of fires due to increased awareness. Apostolakis and Kazarians (1980) use the data of Table 5.2-1 and Bayesian analysis to obtain the results in Table 5.2-2 using conjugate priors (Section 2.6.2), Since the data of Table 5.2-1 are binomially distributed, a gamma prior is used, with a and P being the parameters of the gamma prior as presented inspection 2.6.3.2. For example, in the cable- spreading room fromTable 5.2-2, the values of a and p (0.182 and 0.96) yield a mean frequency of 0.21, while the posterior distribution a and p (2.182 and 302,26) yields a mean frequency of 0.0072. 

Prior distributions are often chosen to simplify the form of the posterior distribution. The posterior density is proportional to the product of the likelihood and the prior density and so, if the prior density is chosen to have the same form as the likelihood, simplification occurs. Such a choice is referred to as the use of a conjugate prior distribution see Lee (2004) for details. In the regression model (1), the likelihood for /3, a can be written in terms of the product of a normal density on /3 and an inverse gamma density on a. This form motivates the conjugate choice of a normal-inverse-gamma prior distribution on (3, a. Additional details on this prior distribution are given by Zellner (1987). 

However, the prior distribution for prediction-error variance is taken to be the conjugate prior and it is the inverse Gamma distribution in this case ... 

Let Xj denote the actual downtime associated with test j for a fixed a>. We assume that Xj N(0, cr ), when the parameters are known. Here 0 is a function of o), but cr is assumed independent of a). Hence the conditional probability density of X = (Wj,j = 1,2,..., k),p(x I jS, 0-2), where /3 = (/3q, /3i), can be determined. We would like to derive the posterior distribution for the parameters fi and the variance given observations X = x. This distribution expresses our updated belief about the parameters when new relevant data are available. To this end we first choose a suitable prior distribution for fi and a. We seek a conjugate prior which leads to the normal-inverse-gamma (NIG) distribution p(/3, o ), derived from the joint density of the inverse-gamma distributed and the normal distributed fi. The derived posterior distribution will then be of the same distribution class as the prior distribution. 

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. 

Conjugate prior for gamma(n, A) distribution is the gamma(r, v) distri-bution. The conjugate prior for A will have the same form as the likelihood. Thus its shape is given by... 

When the observation comes from the Poisson p) distribution, the conjugate prior distribution for fx is gamma r, v). The posterior is ganuna r, i/) where the constants are found by... 

The Bayes conjugate is the gamma prior distribution (equation 2.6-11). When equations 2.6-9 and... 

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... 

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