BAYESIAN ESTIMATION WITH INFORMATIVE PRIORS

Two examples with informative priors are given here. The first deals with well-defined prior knowledge and a discrete parameter space the second illustrates subjective priors and a continuous parameter space. 

Example 5.2. Estimation with Known Prior Probabilities 

Four coins are provided for a game of chance. Three of the coins are fair, giving y(Heads) = p(Tails) = 1/2 for a fair toss, but the fourth coin has been altered so that both faces show Heads, giving p(Heads) = 1 and y(Tails) = 0. 

A coin is randomly chosen from the four and is tossed six times, giving the following data  

Let 6 = 0 denote the choice of a fair coin and 6=1 the choice of the altered one. Then the information in the first paragraph gives the prior probabilities 

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The Bayesian estimator must converge to the maximum likelihood estimator as the sample size grows. The posterior mean will generally be a mixture of the prior and the maximizer of the likelihood function. We do note, however, that the likelihood will only dominate an informative prior asymptotically - the Bayesian estimator in this case will ultimately be a mixture of a prior with a finite precision and a likelihood based estimator whose variance converges to zero (thus, whose precision grows infinitely). Thus, the domination will not be complete in a finite sample. 

Sensitivity analysis is about asking how sensitive your model is to perturbations of assumptions in the underlying variables and structure. Models developed under any platform should be subject to some form of sensitivity analysis. Those constructed under a Bayesian framework may be subject to further sensitivity analysis associated with assumptions that may be made in the specihcation of the prior information. In general, therefore, a sensitivity analysis will involve some form of perturbation of the priors. There are generally scenarios where this may be important. First, the choice of a noninformative prior could lead to an improper posterior distribution that may be more informative than desired (see Gelman (18) for some discussion on this). Second, the use of informative priors for PK/PD analysis raises the issue of introduction of bias to the posterior parameter estimates for a specihed subject group that is, the prior information may not have been exchangeable with the current data. 

A practical challenge of Bayesian meta-analysis for rare AE data is that noninformative priors may lead to convergence failure due to very sparse data. Weakly informative priors may be used to solve this issue. In the example of the previous Bayesian meta-analysis with piecewise exponential survival models, the following priors for log hazard ratio (HR) (see Table 14.1) were considered. Prior 1 assumes a nonzero treatment effect with a mean log(HR) of 0.7 and a standard deviation of 2. This roughly translates to that the 95% confidence interval (Cl) of HR is between 0.04 and 110, with an estimate of HR to be 2.0. Prior 2 assumes a 0 treatment effect, with a mean log(HR) of 0 and a standard deviation of 2. This roughly translates to the assumption that we are 95% sure that the HR for treatment effect is between 0.02 and 55, with an estimate of the mean hazard of 1.0. Prior 3 assumes a nonzero treatment effect that is more informative than that of Prior 1, with a mean log(HR) of 0.7 and a standard deviation of 0.7. This roughly translates to the assumption that we are 95% sure that the HR... 

In mathematical statistics theory it is well known that Bayesian method allows a combination of two kinds of information prior (for instance, generic statistic data, subjective option of experts) and measurements or observations (Bernardo et al, 2003 Berthold et al, 2003). Bayesian method allows updating estimates of all parameters in the model with a single new obtained observation, i.e. Bayesian method does not require to have new information on the values of all factors involved in the created model. 

Under certain conditions we may have some prior information about the parameter values. This information is often summarized by assuming that each parameter is distributed normally with a given mean and a small or large variance depending on how trustworthy our prior estimate is. The Bayesian objective function, SB(k), that should be minimized for algebraic equation models is... 

Bayesian analysis provides another alternative (also computationally intensive) to deal with very weak signals and avoid FT artifacts. This method, which uses probability theory to estimate the value of spectral parameters, permits prior information to be included in the analysis, such as the number of spectral lines when known or existence of regular spacings from line splittings due to spin coupling. Commercial software is available for Bayesian analysis, and the technique is useful in certain circumstances. 

Figure 3.13 Model parameter estimates as a function of the prior standard deviation for clearance. A 1-compartment model with absorption was fit to the data in Table 3.5 using a proportional error model and the SAAM II software system. Starting values were 5000 mL/h, 110 L, and 1.0 per hour for clearance (CL), volume of distribution (Vd), and absorption rate constant (ka), respectively. The Bayesian prior mean for clearance was fixed at 4500 mL/h while the standard deviation was systematically varied. The error bars represent the standard error of the parameter estimate. The open symbols are the parameter estimates when prior information is not included in the model.

Partial observation of a Bayesian network implies that there are some unknown variables such as hidden variables or variables with missing values. If the network structure is unknown, the topology of the graph is also partially or completely unknown. When the network structure also partially known through prior information such as literature and experimental information, the unknown parameters of the network can be estimated based on the conditional probabUity inference aforementioned. However, if there is no prior knowledge about the network stmeture, the inference on this kind of Bayesian network becomes difficult and often computationally challenging to estimate both the optimal stmeture and the parameters of the network (Murphy and Mian, 1999). 

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