Posterior probability density

FIGURE 5.5 Bayesian posterior normal probability density function values for SSD for cadmium and its Bayesian confidence limits 5th, 50th, and 95th percentiles (black) and Bayesian posterior probability density of the HC5 (gray). 

FIGURE 5.7 Bayesian posterior probability density of the fraction affected at median log (HC5) for cadmium. 

Reilly (1970) gave an improved criterion that the next event be designed to maximize the expectation. R, of information gain rather than the upper bound D. His expression for R with a known is included in GREG-PLUS and extended to unknown a by including a posterior probability density p((j s,t e) based on a variance estimate with Oe error degrees of freedom. The extended R function thus obtained is an expectation over the posterior distributions of y and cr. 

Let 0 denote the vector of parameters for the current model. A point estimate, 6, with locally maximum posterior probability density in the parameter space, is obtained by minimizing a statistical objective function... 

The Bayes distribution or the posterior probability density function Fpos(0 fc), which serves the evaluation of Bis, is ... 

Although very popular and useful in many situations, the minimization of the least-squares norm is a non-Bayesian estimator. A Bayesian estimator [28] is basically concerned with the analysis of the posterior probability density, which is the conditional probability of the parameters given the measurements, while the likelihood is the conditional probability of the measurements given the parameters. If we assume the parameters and the measurement errors to be independent Gaussian random variables, with known means and covariance matrices, and that the measurement errors are additive, a closed form expression can be derived for the posterior probability density. In this case, the estimator that maximizes the posterior probability... 

Use V to denote the measured data of a system and consider it as the vector f in Equation (2.18). Then, the updated/posterior probability density function (PDF) of the parameters 0 is ... 

In Eq. (9.43) f(X) is the prior probability density function. It reflects the— subjective—assessment of component behaviour which the analyst had before the lifetime observations were carried out. L(EA.) is the likelihood function. It is the conditional probability describing the observed failures under the condition that f(5t) applies to the component under analysis. Eor failure rates L(E/X) is usually represented by a Poisson distribution of Eq. (9.30) and for unavaUabUities by the binomial distribution of Eq. (9.35). The denominator in Eq. (9.43) serves for normalizing so that the result lies in the domain of probabilities [0, 1] f(X/E) finally is the new probability density function, which is called posterior probability density function. It represents a synthesis of the notion of component failure behaviour before the observation and the observation itself. Thus it is the mathematical expression of a learning process. 

In addition to recursive filters, other model-based estimation-theoretic approaches have been developed. For example, in the Wiener filter described above, one can use random field models (see Section in) to estimate the power spectra needed. Alternatively, one can use MRF models to characterize the degraded images and develop deterministic or stochastic estimation techniques that maximize the posterior probability density function. 

BM was applied to calculate posterior probability density functions of the parameters of Weibull distribution (formula (3)), Bayesian point esti-... 

When a new measurement Zk+i is available, the noise parameters can be updated. Using the Bayes theorem, the posterior probability density function (PDF) of the noise parameter vector given the measurement data set Dk+i is given by (Yuen 2010a)... 

The maximum posterior probability density of 0 conditional on D for model class Cj is given by... 

In a Bayesian context, the information about the set of modal parameters 0 that can be inferred from the FFT data Fk is encapsulated in the posterior probability density function (PDF) of... 

The posterior probability density of the unknown parameters 0 is calculated as shown in Eq. 1 ... 

The posterior probability density function p(0 I x) follows directly from Eq. 1. 

Gaussianity is made for the state and noise com-ptMients. The general particle filter, however, does not make any prior assumption on the state distri-butiOTi. Instead, the posterior probability density function (PDF), p(Xjtlyi jt), is approximated via a set of random samples, also known as support points x k, i = I,. .., N, with associated weights This means that the probability density function at time k can be approximated as follows ... 

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