Prior noninformative

A noninformative prior distribution could be formed by setting each X to 1. 

The normal model can take a variety of forms depending on the choice of noninformative or infonnative prior distributions and on whether the variance is assumed to be a constant or is given its own prior distribution. And of course, the data could represent a single variable or could be multidimensional. Rather than describing each of the possible combinations, I give only the univariate normal case with informative priors on both the mean and variance. In this case, the likelihood for data y given the values of the parameters that comprise 6, J. (the mean), and G (the variance) is given by the familiar exponential... 

It is worthwhile noting that Box and Draper (1965) arrived at the same determinant criterion following a Bayesian argument and assuming that is unknown and that the prior distribution of the parameters is noninformative. 

Confidence intervals nsing freqnentist and Bayesian approaches have been compared for the normal distribntion with mean p and standard deviation o (Aldenberg and Jaworska 2000). In particnlar, data on species sensitivity to a toxicant was fitted to a normal distribntion to form the species sensitivity distribution (SSD). Fraction affected (FA) and the hazardons concentration (HC), i.e., percentiles and their confidence intervals, were analyzed. Lower and npper confidence limits were developed from t statistics to form 90% 2-sided classical confidence intervals. Bayesian treatment of the uncertainty of p and a of a presupposed normal distribution followed the approach of Box and Tiao (1973, chapter 2, section 2.4). Noninformative prior distributions for the parameters p and o specify the initial state of knowledge. These were constant c and l/o, respectively. Bayes theorem transforms the prior into the posterior distribution by the multiplication of the classic likelihood fnnction of the data and the joint prior distribution of the parameters, in this case p and o (Fignre 5.4). 

Aldenberg and Jaworska (2000) demonstrate that frequentist statistics and the Bayesian approach with noninformative prior results in identical confidence intervals for the normal distribution. Generally speaking, this is more the exception than the rule. 

For small data sets, the choice of the prior remains important. Investigators A and B could then reach agreement more readily by using a noninform tive prior. Such priors are discussed in the following sections. Another type of prior, appropriate for discrimination among rival models, will appear in Chapters 6 and 7. 

The construction of a noninformative prior is a nontrivial task, requiring analysis of likelihood functions for prospective data. The construction is simplest when the likelihood l 6) for a single parameter is data-translated in some coordinate d>(0) then the noninformative prior density takes the form p (t)) = const, over the permitted range of special form of a more general one derived by Jeffreys (1961) (see Section 5.4) we illustrate it here by two examples. 

Since i 0 y,a) is data-translated in 0. the function

[Pg.84]

Figure 5.2 Noninformative prior (dashed line) for the normal mean 6, and likelihood curves 9 y,(r) after 10 observations with O = 40. Compare with Figure 5.1.

Any change in the data translates this log-likelihood function along the Incr axis a distance exactly equal to the change in Ins. Thus, ln (lncr s) is data-translated in the coordinate 4> a) = Incr, as illustrated by the curves in Figure 5.3. Consequently, the noninformative prior for this problem is... 

Jeffreys (1961) gave a formal method for constructing noninformative priors... 

Two classes of parameters are needed in models of observations location parameters 6i to describe expected response values and scale parameters dg to describe distributions of errors. Jeffreys treated 6i and dg separately in deriving his noninformative prior this was reasonable since the two types of parameters are unrelated a priori. Our development here will parallel that given by Box and Tiao (1973, 1992), which provides a fuller discussion. The key result of this section is Eq. (5.5-8). 

Of special interest are those mappings 0(0) for which the determinant X (0) is independent of 0, so that the volume enclosed by the contour of Eq. (5.5-6) for any positive C is independent of 0m- A uniform p(0) then provides a noninformative prior, because the expected posterior probability content within any C-contour is then independent of 0m- For constant X (0) one also finds... 

In this chapter, Bayesian and likelihood-based approaches have been described for parameter estimation from multiresponse data with unknown covariance matrix S. The Bayesian approaches permit objective estimates of 6 and E by use of the noninformative prior of Jeffreys (1961). Explicit estimation of unknown covariance elements is optional for problems of Types 1 and 2 but mandatory for Types 3 and 4. 

Many Bayesian analyses utilize so-called noninformative priors (see examples in the WinBUGS manual (17)). The principal belief underlying their wide utility is to retain objectivity in relation to the current analysis. Should prior evidence influence the analysis of the current experiment, then the objectivity of the current analysis may be questioned, due to the subjective nature of priors and methods for their elicitation. In a philosophical sense, it might also be argued that it is equally nonobjective to ignore all previous evidence, no matter how applicable or strong the evidence might be. 

The use of noninformative priors itself is not without its difficulties, by the simple virtue that truly noninformative proper priors do not really exist. However, with very low precision terms (e.g., 0.0001) and an assumption of lognormality appropriate for many PK/PD parameters, the priors can be considered very vague. 

In the section on noninformative priors, a precision of 0.0001 corresponds to a variance of 10,000 (SD = 100), and if it were assumed that the underlying parameter distribution were lognormal (which is common in PK/PD problems), then the 95% interval of the priors would be essentially 0 and -foo. a possible example of a biologically plausible but still low-information prior follows. In this example the values of the parameters are chosen arbtitrarily and any mean values can be used that suit the likely situation. Any choice of mean values will require slight adjustment of the precision matrix however, this is quite straightforward. For a typical orally administered drug with an assumed fraction absorbed of 1, and mean population parameters for clearance, volume, and absorption rate constant that are the natural log of 1 (L/h), 40 (L), and 1 (h ), respectively (just over a 24 hour half-life), then the prior could be... 

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. 

We investigate two sets of prior distributions one fully noninformative (denoted by Bayesl) and ano er that shrinks the random baseline effects toward their grand means (Bayes2). For Bayesl, the aud d are assumed... 

UI data interval plot of odds ratios between drugs and placebo for discontinuation due to AE under four models. The safest and least safe drugs compared to placebo are indicated with closed symbols and thick lines. RE Freq, frequentist random effects model RE Bayesl, Bayesian random effects model with noninformative prior RE Bayes2, Bayesian random effects model with shrinkage prior RE AB, Bayesian arm-based random effects model. 

For the prior specification, noninformative priors in the form of normal distributions with mean 0 and very large variances can be used for p,. For the piecewise hazards, we assume normal hazards on so that x(t1i) N(0, k) for... 

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... 

Galiatsatou (Fig 38.5) estimates median and 95% confidence intervals of retiu n level estimates for wave heights, when extreme value parameters are estimated with (a) maximum likelihood (ML), (b) Bayesian with noninformative prior distributions, and (c) L-moments (LM) estimation procedures. 

Pig. 38.6. Surge levels against return period for the Eld station based on ML-estimates (dashed line) and the posterior Bayesian distributions (solid line) for the raw data, using (a) noninformative prior distributions and (b) information from the neighboring station Son. ... 

Here, we consider a method for generating priors that is objective in the sense that as long as different analysts agree to use this approach, they will independently come to (nearly) the same statistical conclusions given only the predictors, response data, model, and likehhood function. The basic technique, discussed below, is to identify a data-translation, or symmetry, property that the likelihood function has, and choose the prior so that the posterior density retains the same property. Such a prior does not give undue emphasis to any particular region of parameter space and thus is said to be noninformative. As we later show, for the single-response likelihood (8.64), a noninformative prior is... 

---