(1) The posterior probability distribution
of vehicle pose is estimated using the control information and the motion model.
Based on the Bayesian method, the posterior probability distributions
of the three parameters of Weibull model for data2 were obtained.
The joint posterior probability distribution
for the post-retrofit regression parameter vector E and the variance [[sigma.sub.2] is defined in Equation (7), with y equal to the post-retrofit gas use and X equal to [X.sub.post] as defined in Equation (9).
After the posterior probability distribution
is estimated, the probability of coexistence can be calculated as the volume under the posterior probability density curve where [[Delta].sub.1] [less than] 0 and [[Delta].sub.2], [less than] 0, since this is the parameter space where the condition for coexistence is met.
where [P.sub.d](t) is the posterior probability distribution
obtained by conditioning on the document d, and (1 + [[Lambda].sub.d]) is the factor by which the prior probability is to be modified to obtain the posterior probability.
Calculate the posterior probability distribution
p([S.sup.h] | D); that is,
Figure 9 shows comparison results of the three posterior probability distributions
(i.e., the basic probability, attached probability, and combined probability).
In a Bayesian analysis, prior distributions for the model parameters are combined with a likelihood function for the data to give posterior probability distributions
for the parameters, from which inference is based (Gelman et al., 1995; Ellison, 1996; Wade, 2000).
It should be noted that the axes limits are chosen to display the regions of the highest probability in the marginal posterior probability distributions
for the parameters and, in certain cases, these regions do not contain the true parameter values (with the result that some of the panels in Figure 2 do not contain either the solid square or solid vertical line representing the true parameter values).
These graphs also serve a diagnostic function as the suitability of the convergence (behavior of the curves) may be assessed, The sample monitoring tool offers several other very useful diagnostics including trace, history, autocorrelation and kernel density curves, as well as the actual statistical outputs of the parameters and quantiles (Credible Intervals in Bayesian parlance) around them These outputs constitute the estimates of the posterior probability distributions
, which may be further utilized in statistical inference.
At 21% ambient oxygen concentration in the pore space, the mean values (average values of the posterior probability distributions
) of the initial oxygen flux, the oxygen diffusion coefficient through the oxidation product layer, and the surface reaction rate constant were 1.3 x [10.sup.-7] mol/[m.sup.2].s, 2.9092 x [10.sup.-13] [m.sup.2]/s, 3.2278 x [10.sup.-7] [mol.sup.0.45]/[m.sup.0.35].[s.sup.-1], respectively.