Bayesian point estimation

Christensen, R. Huffman, M. D. 1985. Bayesian point estimation using the predictive distribution. The American Statistician, 39, 4, Part 1 319-321. 

Bayesian point estimate of parameter 0, is an expected value calculated using posterior pdf... 

Table 9.2 Comparing Bayesian point estimates calculated from sample with MLE...

For simplicity and in order to avoid potential misrepresentation of the experimental equilibrium surface, we recommend the use of 2-D interpolation. Extrapolation of the experimental data should generally be avoided. It should be kept in mind that, if prediction of complete miscibility is demanded from the EoS at conditions where no data points are available, a strong prior is imposed on the parameter estimation from a Bayesian point of view. 

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. 

A point estimate of-2 LL (also termed the deviance (denoted D(9))) at its maximum is suggested to make the model lit appear better than it should in reality, and in a Bayesian sense averaging over the deviance values (for all values of the posterior distribution of the parameters) would provide a more appropriate choice and so the BIC, now BIC, can be written... 

Forecasted mean values of indicators calculated with Bayesian method point estimates of random... 

The purpose of this paper is to present the quantification of the risk rates, that is the probability per unit of time (hour) for each and every of the 63 hazards. The paper is organised as follows Section 2 outlines the modelling of the arrival of occupational accidents as a Poisson random process and briefly describes the procedure followed for identifying the number of accidents and the exposure of the Dutch working population to the occupational hazards during a given period of time. With these two sets of data point estimates of the risk rates are obtained. Section 3 presents the assessment of the uncertainties associated with this estimation following a Bayesian approach. Finally Section 4 discusses the obtained results. 

The first type of inference is where a single statistic is calculated from the sample data and is used to estimate the unknown parameter. From the Bayesian perspective, point estimation is choosing a value to summarize the posterior distribution. The most important summary number of a distribution is its location. The posterior mean and the posterior median are good measures of location and hence would be good Bayesian estimators of the parameter. Generally we will use the posterior mean as our Bayesian estimator since it minimizes the posterior mean squared error... 

Zobrist J, Reichert P (2006) Bayesian estimation of export coefficients from diffuse and point sources in Swiss watersheds. J Hydrol 329 207-223... 

Bayesian probability theory157 can also be applied to the problem of NMR parameter estimation this approach incorporates prior knowledge of the NMR parameters and is particularly useful at short aquisition times158 and when the FID contains few data points.159 Bayesian analysis gives more precise estimates of the NMR parameters than do methods based on the discrete Fourier transform (DFT).160 The amplitudes can be estimated independently of the phase, frequency and decay constants of the resonances.161 For the usual method of quadrature detection, it is appropriate to apply this technique to the two quadrature signals in the time domain.162-164... 

A few programs are now available that allow the efficient simultaneous data analysis from a population of subjects. This approach has the significant advantage that the number of data points per subject can be small. However, using data from many subjects, it is possible to complete the analyses and obtain both between- and within-subject variance information. These programs include NONMEM and WinNON-MIX for parametric (model dependent) analyses and NPEM when non-parametric (model independent) analyses are required. This approach nicely complements the Bayesian approach. Once the population values for the pharmacokinetic parameters are obtained, it is possible to use the Bayesian estimation approach to obtain estimates of the individual patient s pharmacokinetics and optimize their drug therapy. 

Inclusion of the posthoc option instructs NONMEM to obtain the Bayesian post hoc ETA estimates when the first-order method is used. These effects and other relevant parameters can be output into a table using the table record. Thereafter, the distribution of the effects can be characterized, including skewness if present. Both the mixture model and the nonmixture models need to be reestimated with the first-order method, as one cannot compare the mofs in a meaningful way between models differing only in estimation method. The mof has dropped 676 points between the nonmixture model (see r5.txt) and the mixture model (r4.txt). Furthermore, the mixture model run has now concluded with a successful covariance step. A choice has to made whether to make two plots (one for each subpopulation) or one (after all, the etas all share the same distribution). The latter approach is shown in Figure 28.2. Similar plots can be generated for each subpopulation. 

N. Flournoy, A clinical experiment in bone marrow transplantation Estimating a percentage point of quantal response curve, in Case Studies in Bayesian Statistics, C. Gastonis, J. S. Hodges, R. E. Kass, and N. D. Singpurwalla (Eds.). Springer, New York, 1993, pp. 324-336. 

Unfortunately there is nothing in statistical theory that allows determination of the correct error estimate taking into account model uncertainty. Bayesian model averaging has been advocated as one such solution, but this methodology is not without criticism and is not an adequate solution at this point in time. At this point the only advice that can be given is to be aware of model uncertainty and recognize it as a limitation of iterative model development. 

The Bayesian, of course, will also wish to adjust naive treatment estimates. This is because the posterior estimate at any point must reflect not only what the data say about the model ( the likelihood ) but prior belief. This sort of adjustment is sometimes referred to as shrinkage. The posterior estimate shrinks the naive estimate towards the prior estimate. The more data available the less the shrinkage that takes place but, in the Bayesian scheme, there is no direct dependence on the stopping rule. As discussed in section 19.2.1, there is an indirect dependence, since the stopping rule itself will depend on the prior distribution. 

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