Inverse Gamma distribution

Table 5. The mean and various quantiles of Inverse Gamma distributions with X = 1 and v = 1,..., 10. For other values of X, multiply the table entries by X to obtain the appropriate quantile...

Gaussian Prior for the Coefficients and Inverse Gamma Distribution for the Prediction-error Parameter... 

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 ... 

It can be easily shown that the conditional PDF p(h V, C, n) is Gaussian with mean b (n) and covariance matrix a /NA, which depend on n. Furthermore, the conditional PDF p(a T), C, n) follows the inverse Gamma distribution with shape parameter a and scale parameter p given by ... 

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. 

Based on the smck-pipe database we assign the parameters of the prior distribution, the normal-inverse-gamma (NIG) distribution p(j8, a ), which is derived from the joint density of the inverse-gamma distributed and the normal distributed p. For p we then need to specify the means m and variances (we consider Pq and P independent) and for the two parameters ai and 2 of the inverse-gamma distribution. The following values were used for jar A ... 

Inversion of this transform gives the gamma distribution... 

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). 

When a system has long-time correlation, for which we expect fractional power scaling of excess heat, our assumption of the Boltzmann equilibrium distribution may always not be valid. Actually some power distributions such as the Tsallis distribution [14] have been reported at the edge of chaos [15]. A superstatistical equilibrium distribution is written as a superposition of Boltzmann distributions with different temperatures. Beck and Cohen [13] considered many types of distributions for the inverse of temperature. For example, they chose Gaussian, uniform, gamma, log-normal, and others. In particular, the Tsallis distribution is realized for gamma distribution. We will show that excess heat can be written as a superposition of correlation functions... 

Before we close this section, we will comment on the area of the hysteresis loop. When the decay constant of a correlation functions depends on the inverse of temperature, we can expect various behavior for the area of the hysteresis loop. In the case of the Tsallis distribution, the inverse of temperature is distributed as a gamma distribution. If the decay constant is proportional to the temperature, the area of the histeresis loop decays as a modified Bessel function for the large period of external transformation. On the other hand, if the decay constant is proportional to the inverse of temperature, we can expect the fractional power scaling. 

Alternatively, the inverse variance of residual uncertainty can be assumed to arise from a gamma distribution. 

A third method to simulate random variables is convolution, where the desired random variates are expressed as a sum of other random variables that can easily be simulated. For example, the Erlang distribution is a special case of the Gamma distribution when the shape parameter is an integer. In this case, an Erlang random variate with shape parameter can be generated as the sum of j3 exponential random variates each with mean a. A last method to simulate random variables is decomposition (sometimes called composition), where a distribution that can be sampled from is composed or decomposed by adding or subtracting random draws into a distribution that cannot be simulated. Few distributions are simulated in this manner, however. These last two methods are often used when the first two methods cannot be used, such as if the inverse transformation does not exist. 

Fig. 1-7. A definition sketch for the proposed upscaling scheme depicting (a) gamma distribution of central pore lengths with % = 2, and hypothetical six bins (note the inverse relationships between 3 and 1.) (b) three different filling stages in the population of unit cells (represented by l. l.fl) defined at three chemical potential values to g (dry-wet) and (e) the rcNtdllng hypothetical characteristic curve.

The Inverse Problem Determination of the Pore Volume Distribution The analysis presented so far can be used to solve the inverse problem, that is if we know the amount adsorbed versus pressure, the equation (3.9-27) can be used to determine the constants for the pore volume distribution provided that we know the shape of the distribution a-priori. We shall handle this inverse problem by assuming that a mesopore volume distribution can be described by the double Gamma distribution as given in eq.(3.9-22). With this form of distribution, the amount adsorbed can be calculated from eq.(3.9-27), and the result is ... 

Is determined by the CDF F (W ) and the Inverse cumulative standard normal distribution In order to facilitate the evaluation of p j, empirical formulae have been developed recently. In case the probability distribution of the RV s (Y or S ) are determined by a Gamma distribution in eq. (7), p v/iWj can be obtained with sufficient accuracy in terms of py/ty/j and the parameters and vy/j ... 

The inverse problem is to obtain h k) empirically from measured C t). Giay analyzed Tiytten et al. s HDS data and foimd that h k) followed a gamma distribution with Y = 2. Others have also addressed the inverse problem." ... 

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... 

CSIRO Minerals has developed a particle size analyzer (UltraPS) based on ultrasonic attenuation and velocity spectrometry for particle size determination [269]. A gamma-ray transmission gauge corrects for variations in the density of the slurry. UltraPS is applicable to the measurement of particles in the size range 0.1 to 1000 pm in highly concentrated slurries without dilution. The method involves making measurements of the transit time (and hence velocity) and amplitude (attenuation) of pulsed multiple frequency ultrasonic waves that have passed through a concentrated slurry. From the measured ultrasonic velocity and attenuation particle size can be inferred either by using mathematical inversion techniques to provide a full size distribution or by correlation of the data with particle size cut points determined by laboratory analyses to provide a calibration equation. 

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