Development of Bayesian procedures for the calibration of film badge dosimeters

Calibration of film badges uses data (z) from a “calibration" experiment. These data consist of pairs z_i={x_i,y_i} where x_i are exposures fixed and chosen before the experiment and _i are the corresponding film densities. To describe the saturation of film density at higher exposure levels the following model was used relating exposure and film density: y=β₀+β₁xe^-β₂x+μ where β₀, β₁, and β₂ are parameters and μ ~ N(0,σ) is the normally distributed observational error. Using Bayes’ Theorem and a non-informative prior for β₁, β₂, and a the expression for the joint posterior density of these parameters is obtained, assuming negligible background density β₀. Integration of the likelihood of a new observation, multiplied by the joint posterior density of β₁, β₂, and σ yields the predictive density which serves as a new “likelihood” in the derivation of the calibrative density. This density describes the remaining uncertainty about a new unknown exposure X_n corresponding to a new observed film density Y_n. The advantage of the procedure described here stems from the fact that the uncertainty about X_n is described quantitatively by the calibrative density. This procedure is applied to film badge calibration data sets used at Oak Ridge National Laboratory in the 1960's and 1970's and the results are compared to other calibration analysis.