Marginal densities

Next, we calculate the derived first-order densities (sometimes called marginal densities)... 

Mandelstam, S., 356,371,377,381,664 Mandelstam s postulate, 376 Many particle state, 540 Margenau distribution, 49 Margenau, Henry, 49,391 Marginal densities of probability density functions, 138... 

An important concept is the marginal density function which will be better explained with the joint bivariate distribution of the two random variables X and Y and its density fXY(x, y). The marginal density function fxM(x) is the density function for X calculated upon integration of Y over its whole range of variation. If X and Y are defined over SR2, we get... 

Figure 4.8 A bivariate probability density function. The slice parallel to the y axis represents the marginal density fxM(x).

Gelfand and Smith, 1990] Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. J. Am. Statist. Assoc., 85 398-409. 

Tierney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81, 82-86. 

Transport resistance as quantified by 8((2) or (4)) is dependent on the structure of the fracture network and prevailing boundary conditions for flow. The key issue for PA applications is how to infer the distribution of P based on site characterization data. The full description requires the joint density fi t,P), however, for more strongly tracers the marginal density PP) is sufficient for most practical purposes. 

For PA applications, we need to infer P statistics based on site characterization data. More specifically, we need to infer the marginal density fip). Several approaches for estimating yf ) are possible, however, at this time there is no general consensus on which one is most preferable. In view of the heterogeneity and complexity of fractured porous media, and uncertainties involved, it is most likely that several complementary approaches for estimating fip) from field data need to be used, rather than a single approach. 

Gelfand AE, Smith AFM (1990) Sampling-based approaches to calculating marginal densities. 

Assuming the maximal number of evaluations of the proposed GGS (31 points) for applying traditional GGS, one performs 145500 evaluations of the involved FCDs (about 16% more evaluations than the proposed GGS). Table 1 shows some statistical metrics from the resulting marginal densities based on the traditional and proposed GGS according to the abovementioned input parameters. One can see the better accuracy provided by the proposed GGS even under a reduced number of evaluated points of the involved FCDs. 

Figure 4. Marginal densities for the BN extracted from Brewer et al. (1996) according to the traditional and proposed GGS.

It is readily seen that the transform of the marginal density function is given by ... 

Bayesian statistics has a single way of dealing with nuisance parameters. Because the joint posterior is a probability density in all dimensions, we can find the marginal densities by integration. Inference about the parameter of interest 0i is based on the marginal posterior g 0i data), which is found by integrating the nuisance parameter 2 out of the joint posterior, a process referred to as marginalization ... 

For a discussion of alternative marginalization methods, consult Chen et al. (2000). calc.g.1Dmarginal.m uses the kernel method to construct 1-D marginal densities concurrently for multiple parameters, and is used by... 

X.pred, y, fun.yhat, theta.O, sigma.O, and MCOPTS take the same definitions as in Bayes.MCMC.pred.SR. j.plot.l D contains the numbers of the parameters whose marginal densities are desired. val lo and val.hi are vectors that set the lower and upper limits of the histogram for each parameter. N.bins is die number of bins in each histogram (we use... 

We next use the MCMC approach to study again the hypothesis that the reaction A + B C is elementary, given the data in Table 8.1. We compute the 95% HPD for the 2-D maiginal posterior density p(92,di y). If this HPD region contains the point ( 2 = 1, 3 = 1), we cannot support the conclusion that the hypothesis is false (i.e., the reaction is not elementary). We compute this marginal density using MCMC simulation 1 ... 

This MCMC routine is used in turn Ity other routines that compute marginal posterior densities and generate HPD regions. 1-D marginal densities and their corresponding HPD... 

The following code computes the marginal density for the rate constant from the composite data and the corresponding 95% HPD region ... 

As expected, the confidence interval using both data sets is tighter than those computed using either one individually. The 1-D marginal posterior density is shown in Figure 8.10. Compare the breadth of this marginal density to that obtained from the data of Table 8.3 alone, Fignre 8.9. 

For single-response data, Bayes.MCMC.pred.SR.m computes the expectation of a vector g 6,a). This routine is used in turn by Bayes MCMC.1DmarginaLSR.m and Bayes.MCMC 2Dmarginai SR.m to compute 1-D and 2-D marginal densities. The outputs of these routines are used to compute 1-D and 2-D HPD regions by Bayes.1D HPD SR.m and Bayes 2D HPD SR.m respectively. 

The most probable parameter vector is computed from multiresponse data using si m anneaLMR.m. The resulting marginal posterior density on 9 is used to compute expectations of g 9) by Bayes MCMC pred.MR.m. Marginal densities and HPD regions are computed by similar routines to those above, with MR substituted for 3R. 

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