"Support for Bayesian Statistics" Guthrie Miller et al.
"Support for Bayesian Statistics" by Guthrie Miller, Harry F. Martz , Mario E. Schillaci, CHP, Don Berry, William Inkret, CHP, and Thomas Little, published in the Health Physics Society Newsletter, pp. 28-29, March, 1998
We appreciate Gary Kramer's reporting of the Bayesian statistics workshop held in conjunction with the 43rd Conference on Bioassay, Analytical, and Environ-mental Radiochemistry. The spirit of the workshop was certainly captured by the immortal words of Allen Brodsky's classic ballad, especially composed for the concluding panel discussion in Charleston, "Oh the Bayesians and Classicists should be friends . . ." To supplement Gary's general review in the January 1998 Health Physics Newsletter, we believe it will be helpful to the readership to provide some more detailed comments.
Bayesian statistics is increasingly used in many scientific and medical applications. For example, the U.S. Nuclear Regulatory Commission has stated that "it is generally believed by most probability risk assessment (PRA) analysts that, for PRA application to complex systems, such as nuclear power plants, the advantages of Bayesian methods outweigh the disadvantages." Thus, Bayesian methods are widely used in PRA. Its natural simplicity is scientifically appealing. Instead of calculating the probabilities of observing a measured value above some limit, assuming no activity is present, the Bayesian framework directly calculates the probability that activity is present given the measured value. However, to do this requires some probabilistic information regarding the presence of activity (the prior probability distribution).
More information on application of Bayesian methods to health physics is available on the Web at http://www.pnl.gov/bayesian.
Gary states, "One difficulty that was never satisfactorily resolved was the inability of Bayesian statistics to deal with negative numbers (sometimes a background count is higher than a sample count)." During the first two days of the workshop, there was considerable discussion of the Bayesian treatment of a counting measurement where a significant background is present, using the fact that the distribution of counts is Poisson, rather than assuming normal distributions of gross and background counts. The Bayesian method elegantly handles this problem and gives the probability of true counts t as proportional to the Poisson likelihood function (nN/n!)e-n, where N is the measured number of gross counts and n is the average number of gross counts, given by t plus the average background counts b. This Bayesian result uses a prior probability distribution oft that is uniform for all positive values of t, which would be appropriate in the absence of other information about the distribution of t, and assumes that the background count average b is well known. This gives the following expression for the probability distribution of t, given that we have measured N gross counts,
C(b +t)N e-(b+t),
where C is a normalization factor that is determined by the requirement that the probability of all different values of t sum up to 1. Note that the Bayesian method, unlike the classical method, gives us exactly what we are interested in, namely the probability distribution of true counts given the measurement result. The standard technique of just subtracting background counts and assuming a normal distribution of uncertainties, which is valid when the number of counts is sufficiently large, also is elegantly handled using Bayesian statistics. Subtracting average background counts from measured gross counts may yield negative results, but we wouldn't want to report a negative dose or a negative environmental contamination level. The prior probability distribution certainly excludes negative true results. Doses and contamination levels must be positive! Isn't the real question of interest here, "What is the mathematical basis for handling negative net results in the classical, currently accepted framework?"
Gary goes on to mention the dual problem of false positives and false negatives. This is a valid question for both the Bayesian and non-Bayesian to explore. One strength of the Bayesian framework is the ability to model the measurement process based on all pertinent information, including historical experience and experience from objectively similar situations. This model allows the Bayesian to estimate the expected number of false positives and false negatives for a given situation or population.
As was pointed out at the workshop, in a typical radiation protection situation where true positives are rare, the Bayesian decision level will be higher than the classical decision level, resulting in more false negatives and fewer false positives. Using the classical decision level is equivalent to assuming that false negatives are approximately 250 times more costly than false positives. The cost of false negatives and false positives is a legitimate subject for debate. We believe that in the occupational monitoring situation, false positive errors are more costly than false negative errors.
For example, one of the most serious plutonium incidents in the history of Los Alamos occurred because a continuous air monitor that had repeatedly given false alarms was ignored in a situation where, in fact, significant amounts of 239Pu were airborne. Also, the magnitude of the doses missed in the increased number of false negatives obtained with the Bayesian method is likely to be small. A truly large dose will be caught by either method.
The Bayesian framework provides more informative statements regarding a result, beyond the currently accepted "negative" or "positive." A Bayesian analysis would state "negative," meaning there is at least a 95 percent probability that the dose is less than XX, where XX is given (for example, for tritium internal dosimetry, XX is a fraction of a mrem, for plutonium it is much greater). A generalized question we have frequently encountered in litigation cases, from both plaintiff and defendant counsel, is what is the probability that the dose is less than some value of interest. Doesn't the public often ask at hearings and in newspaper editorials, "What is the chance that the measured result will affect me?" It would seem that a scientific society concerned with educating the public and judicial system regarding facts about radiation measurement would be very excited about a mathematical approach that could provide a technically sound answer to such questions. Bayesian methods do exactly that.
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Revised: March 6, 1998