Bayesian analyses informative priors

Special uninformative distributions are often used in Bayesian analysis to represent prior parameter uncertainty, in cases of minimum prior information on the parameters. The idea is often to select a prior distribution such that the results of the analysis will be dominated by the data and minimally influenced by the prior. 

Elicitation of jndgment may be involved in the selection of a prior distribution for Bayesian analysis. However, particularly because of developments in Bayesian computing, Bayesian modeling may be useful in data-rich situations. In those situations the priors may contain little prior information and may be chosen in such a way that the results will be dominated by the data rather than by the prior. The results may be acceptable from a frequentist viewpoint, if not actually identical to some frequentist results. 

Referring to the situation in question 2, one might think that an informative prior would outweigh the effect of the increasing sample size. With respect to the Bayesian analysis of the linear regression, analyze the way in which the likelihood and an informative prior will compete for dominance in the posterior mean. 

Many scientists ignore the prior information, and for cases where data are fairly good, this can be perfectly acceptable. However, chemical data analysis is most useful where the answer is not so obvious, and the data are difficult to analyse. The Bayesian method allows prior information or measurements to be taken into account. It also allows continuing experimentation, improving a model all the time. 

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. 

Dehning the values of the priors and their informativeness is therefore an essential part of any Bayesian analysis. 

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. 

Gastonguay, M.R., Gibiansky, L., Gillespie, W.R., Khoo, K.-C., and the PPRU Network. Population pharmacokinetics in pediatric patients using Bayesian approaches with informative prior distributions based on adults. Presented at East Coast Population Analysis Group (ECPAG), 1999. 

The idea of Bayesian updating is similar to our thinking process but it provides also the basis for quantification. We have a perception of different people and matters based on our experience, i.e., data. When a new event happens (i.e., new data is obtained), it modifies our perception. In other words, our perception is not only determined by the latest piece of information but it also depends on the original perception. In Bayesian analysis, the original perception is regarded as the prior information and the new piece of information is utilized to update our perception or mathematical model. 

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... 

Three Weakly Informative Prior Sp>ecifications for Bayesian Meta-Analysis... 

Bayesian methods are very amenable to applying diverse types of information. An example provided during the workshop involved Monte Carlo predictions of pesticide disappearance from a water body based on laboratory-derived rate constants. Field data for a particular time after application was used to adjust or update the priors of the Monte Carlo simulation results for that day. The field data and laboratory data were included in the analysis to produce a posterior estimate of predicted concentrations through time. Bayesian methods also allow subjective weight of evidence and objective evidence to be combined in producing an informed statement of risk. 

As mentioned earlier, incorporating prior information does not in itself constitute a Bayesian approach. Priors have been used in non-Bayesian settings in population PK analysis and other analyses. Applications using the PRIOR subroutine in NONMEM have been described previously (3,16). In this setting the prior information can be viewed as a penalty on the likelihood function, and its implementation is similar in spirit to the maximum a posteriori (MAP) procedures used commonly... 

In the second approach, a Bayesian regression analysis (e.g., Straub and Der Kiureghian 2008 loannou and Rossetto 2013) is adopted in order to take into accotmt prior information regarding the model s parameters, 0, especially when the available number of observations is small. Prior information is obtained from existing fragility functions or independent post-earthquake data of similar groups of assets. In addition, this... 

Clearly, the choice of the prior is crucial, as it influences our analysis. It is this subjective nature of the Bayesian approach that is the cause of controversy, since in statistics we would like to think that different people who look at the data will come to the same conclusions. We hope that the data will be sufficiently informative that the likelihood function is sharply peaked around specific valnes of 0 and a i.e., that the inference problem is data-dominated. In this case, the estimates are rather insensitive to the choice of prior, as long as it is nonzero near the peak of the likelihood function. When this is not the case, the prior influences the results of the analysis. 

---