Marginal likelihood

A.3.6 Maximum Penalized Marginal Likelihood (MPML) Approach... 

Equation (28.5) gives the average of the marginal likelihoods over the r subpopulations. Thus, it can be seen how the mixture probability (Pik(d)) can vary between subpopulations (as k varies) and between individuals (as i varies) due, for example. 

This original FO-approximation algorithm to provide population estimates has proven surprisingly adequate for many pharmacokinetic-pharmacodynamic problems and was also important in the past when computer processor speed was slower than it is today. However, more recent algorithms are developed around the marginal likelihood. The joint probability distribution for Y and t can be written as... 

Nuisance parameters are generally eliminated by computing the marginal likelihood. In the two-dimensional case, with random variables X and Y, the marginal likelihood can be obtained by integrating out the nuisance parameter from the joint likelihood. For example,... 

Marginal likelihoods can easily be extended to more than two-dimensions. Suppose that L(X), L(Y), and L(Z) are the likelihood functions for random variables X, Y, and Z, respectively. Then the joint likelihood is... 

If V = ZtGZ + R, then after a lot of algebraic manipulation (see Davidian and Giltinan (1995) for details), the marginal likelihood can be expressed as... 

Because the individual likelihoods were normally distributed and s was independent of U, the resulting marginal likelihood was solvable and itself normally distributed. For nonlinear mixed effect models or when is not independent of U, the evaluation of the marginal likelihood cannot be computed analytically and must be evaluated numerically. It is how the individual software packages evaluate the marginal likelihood function that differentiates them. 

Chib, S. (1995). Marginal likelihood fiom the Gibbs output. Journal of the American Statistical Association, 90 1313-1321. 

Chickering, D. H. and Heckerman, D. (1997) Efficient approximations for the marginal likelihood of Bayesian networks with hidden variables. Machine Learning, 29 181-212. 

For a drift Brownian motion, the differences of two consecutive performance measurements are independently, normally distributed with mean and variance proportional to the time lag between two measurements. Utilizing this property, Doksum Normand (1995) discussed the maximum likelihood estimation of the parameters in degradation mean function and variance. Therefore, the solution to model parameter estimation is obtained by maximizing the marginal likelihood function, which in turn is obtained by integrating out the random coefficient from the joint condi-... 

In this expression, p(H) is referred to as the prior probability of the hypothesis H. It is used to express any information we may have about the probability that the hypothesis H is true before we consider the new data D. p(D H) is the likelihood of the data given that the hypothesis H is true. It describes our view of how the data arise from whatever H says about the state of nature, including uncertainties in measurement and any physical theory we might have that relates the data to the hypothesis. p(D) is the marginal distribution of the data D, and because it is a constant with respect to the parameters it is frequently considered only as a normalization factor in Eq. (2), so that p(H D) x p(D H)p(H) up to a proportionality constant. If we have a set of hypotheses that are exclusive and exliaus-tive, i.e., one and only one must be true, then... 

After the likelihood, the perceived consequences of the precursors have to be established in order to determine the perceived risk. To determine these consequences, expert opinions and past lessons are needed. From accident reports and several multidisciplinary experts in the company, the precursors are listed according to their perceived consequences in terms of critical, marginal, and negligible. The consequences are established by deriving a scenario, perceived as highly likely, and determining the consequences if this scenario would occur. 

Alteration in Allende chondmle C6 is concentrated at the margins where 5 Mg values are low (Fig. 17). Evaluation of the likelihood that low 8 Mg values could be the result of aqueous alteration will require studies of Mg isotope fractionation during low-T alteration of terrestrial mafic rocks. An alternative explanation is that low 8 Mg and alteration resulted from condensation (collision frequency of gaseous Mg is greater for the lighter isotopes). 

Figure 2.8 Dose-response curves for pharmacological effect and toxic effect, illustrating the EDS0 and TD50. The proximity of the curves for efficacy and toxicity indicates the margin of safety for the compound and the likelihood of toxicity occurring in certain individuals after doses necessary for the desired effect.

Limited Information Maximum Likelihood Estimation). Consider a bivariate distribution for x and y that is a function of two parameters, a and fi The joint density is j x,y a,p). We consider maximum likelihood estimation of the two parameters. The full information maximum likelihood estimator is the now familiar maximum likelihood estimator of the two parameters. Now, suppose that we can factor the joint distribution as done in Exercise 3, but in this case, we have, fix,y a, ft) — f(y x.a.f )f(x a). That is, the conditional density for y is a function of both parameters, but the marginal distribution for x involves only... 

A further area in which sequential extraction continues to be applied successfully is in assessment of the likelihood of mobilisation of metal contaminants from sediment-derived soil. When dredged sediment is used to reclaim land from the coastal margins or applied to arable soil to improve fertility, there is concern that potentially toxic elements accumulated under reducing conditions may be released on exposure to an oxygen-rich environment. Sequential extraction can be used to characterise the sediment prior to application, or to monitor changes in metal availability in the soil with time (e.g. Singh et al, 1998). 

In an analysis of data from the National Institute of Mental Health Collaborative Depression Study in 643 patients with affective disorders who were followed up after fluoxetine was approved by the FDA in December 1987 for the treatment of depression, nearly 30% (n = 185) took fluoxetine at some point (18). There was an increased rate of suicide attempts before fluoxetine treatment in those who subsequently took fluoxetine. Relative to no treatment, fluoxetine and other antidepressants were associated with non-significant reductions in the likelihood of suicide attempts or completions. Severity of psychopathology was strongly associated with increased risk, and each suicide attempt after admission to the study was associated with a marginally significant increase in the risk of suicidal behavior. The authors concluded that the results did not support the speculation that fluoxetine increases the risk of suicide. 

The margin of exposure (MOE) is dehned as a benchmark dose divided by the dose from exposure. As the latter becomes smaller, the MOE becomes larger and the likelihood of any adverse health effect as a result of the exposure becomes smaller, or even zero. The larger the MOE, then the more conhdence there is that no adverse health effect will be observed as a result of the exposure. 

An estimated lower bound on the margin of exposure obtained by combining default constants provides no indication of the relative likelihood or frequency of that MOE or any other MOE greater than the exaggerated lower bound. On the other hand, the characterization of the MOE obtained by using probability distributions and probabilistic techniques provides a quantitative assessment of the relative likelihood of each of the different possible values for the MOE. 

Risk The likelihood that an individual will develop a specified adverse health effect. Risk can be characterized in quantitative terms, such as the probabihty of the adverse health effect or the margin of exposure which is the ratio of the dose with a specified probability of the adverse health effect and an individual s dose from exposure (Sielken, Ch. 8). 

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