Posterior probability density function

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

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)... 

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 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 ... 

FIG U RE 5.4 Bayesian normal density spaghetti plot random sample of 100 normal probability density functions (pdfs) drawn from the posterior distribution of p and o, given 7 cadmium NOEC toxicity data (dots) from Aldenberg and Jaworska (2000). 

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). 

Under this hierarchical model, the joint posterior distribution of all coefficients and parameters can be expressed as the product of the probability density functions at... 

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... 

According to Eq. (5.127), the posterior probability is computed from the probabihty density function for the considered class, p(x j), the prior probabihty for that class P(j), and the probability density function over ah classes p x). A sample is then assigned to that class J, for which the largest posterior probability is found. 

The Bayesian model updating approach, applied to structural identification in (Beck Katafygiotis 1998), is used to update the probability density function (PDF) of each model parameter according to measured data. Let be a set of measured data, Ms the model class, that is, the system of differential equations (Eq. 1), and 0s the vector of structural parameters for the chosen model class. Bayes theorem states that the posterior PDF p(0s Ds,Ms) of 0s is proportional to the product of the likelihood function p(Ds 0s,Ms) and the prior PDF p(0s Ms) the proportionality constant c is given by the inverse of the so-called evidence p(Ds Ms). 

The parameters 0, . .., 0 are assumed as random variables with known prior probability density functions (pdfs)p xf 1= 1,. .., w, with mean values equal to its estimates (of DPSIA). With respect to available failure data the posterior... 

But for safety it is usually acceptable to demonstrate pessimistic predictions. We can then look for a worst case prior distribution that one could assume for the inference. We can show that such a worst case does exist, as follows. Consider a probability density function for the unknown pfd, Q, made of two probability masses a mass Pp in Q = 0 and a mass (1 — Pp) in Q = q. Now, if we assume gjv to be close to either end of the interval [0,1], reliability predictions after observing failure-free demands will be very high. Indeed, in the two limiting cases, predicted reliability will be 1 if gjv = 1, one test is enough to show that P Q = <1n) = 0 and thus Q = 0 with certainty and if = 0, Q = 0 with certainty to start with.In between these extreme values, successM tests will reduce P Q = qw) and increase P Q = 0), but still leave a non-zero probabihty of future failure. Thus, posterior reliability as a function of q is highest at the two ends of the interval, and must have a minimum somewhere in between. ... 

We outline here how this difficulty can be overcome in principle. Regarding for instance case 1 above, we consider that, if a past product successfully underwent stringent scrutiny and a long operational life, we cannot declare it fault-free with certainty, hut it has a posterior distribution of pfd where most of the probability mass is either in 0 of close to it. The Lemma in appendix A shows that for worst-case reliability prediction, such a distribution can be substituted with a single probability mass in a point qs close to 0. Similar experience for multiple similar products will also give confidence, say a probability Pg, that the same property applies to the current product. So, for the current product, we can use a pessimistic probability density function of pfd similar to equation 4 ... 

If the prior probability density function of the pfd of a system is a mixture of probability density functions fqi, then substituting any subset of these component distributions with a set of single-point probability masses, one for each of the fqi thus substituted, will yield a pessimistic prediction of posterior reliability after observing failure-free demands. 

Let us suppose that the posterior pdf p Xf. I at time A — 1 is available and let us assume that Ihe noise sequences and are white, with known probability density functions and mutually independent. Moreover, let us assume thatdie initial state vector hasaknown pdf p xq ) and is also independent of noise sequences. The prediction stage evolves using the system model to obtain the prediction density of the state attime k viathe Chapman-Kolmogorov equation ... 

FIGURE 6.1 Bayes combination of a prior distribution and a likelihood function to obtain a posterior distribution for 0. The vertical axis (not shown) is probability density. 

An alternate argument for minimizing S 6) is to maximize the function i( 6,a Y) given in Eq. (6.1-10). This maximum likelihood approach, advocated by Fisher (1925). gives the same point estimate 6 as does the posterior density function in Eq. (6.1-13). The posterior density function is essential, however, for calculating posterior probabilities for regions of 6 and for rival models, as we do in later sections of this chapter. 

The posterior density function is the key to Bayesian parameter estimation, both for single-response and multiresponse data. Its mode gives point estimates of the parameters, and its spread can be used to calculate intervals of given probability content. These intervals indicate how well the parameters have been estimated they should always be reported. 

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. 

Sometimes, only one of the parameters is of interest to us. We don t want to estimate the other parameters and call them "nuisance" parameters. All we want to do is make sure the nuisance parameters don t interfere with our inference on the parameter of interest. Because using the Baye.sian approach the joint posterior density is a probability density, and using the likelihood approach the joint likelihood function is not a probability density, the two approaches have different ways of dealing with the nuisance parameters. This is true even if we use independent flat priors. so that the posterior density and likelihood function have the same shape. 

We have shown that both the likelihood and Bayesian approach arise from surfaces defined on the inference universe, the observation density surface and the joint probability density respectively. The sampling surface is a probability density only in the observation dimensions, while the joint probability density is a probability density in the parameter dimensions as well (when proper priors are used). Cutting these two surfaces with a vertical hyperplane that goes through the observed value of the data yields the likelihood function and the posterior density that are used for likelihood inference and Bayesian inference, respectively. 

In likelihood inference, the likelihood function is not considered a probability density, while in Bayesian inference the posterior always is. The main differences between these two approaches stem from this interpretation difference certain ideas arise naturally when dealing with a probability density. There is no reason to use the... 

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