## TOPIC

Bayesian statistics is a type of statistical
analysis developed from the work of Thomas Bayes (1701-1761) and Pierre Simon
Marquis de Laplace (1749-1827). Although neglected for some time, Bayesian
methods have become prominent in many scientific fields in recent decades.

At Los Alamos, Bayesian methods are being applied to bioassay,
radiochemistry, and internal dosimetry.

## Bayesian Software

- Bayesian
Software Package I: Simple Bayesian inference sample problem as
self-teaching tool LA-CC 03-003
View Abstract /Close
**Guthrie
Miller, Luiz Bertelli**

This software is meant to be a self-teaching tool that allows the user to
thoroughly investigate and understand a simplified Bayesian inference
situation.

The objective is to correctly infer the "color" of an atom from
measurement data. Atoms can be white or black. The digital measuring instrument
used reads 0 for a white atom and 1 for a black atom. A quality control test
run with only white atoms shows that a small fraction alpha incorrectly measure
1 (“false positive rate”). Similarly, a small fraction beta of black atoms
measure 0 (“false negative rate”). If these error rates are 0, the measurements
are perfect, and there is no need for statistical inference.

In Bayesian inference, another quantity is needed in order to compute the
probability that an atom is indeed black from the measurement data (“the
posterior probability”). This is the prior probability that an atom is black,
which might be thought of as the posterior probability resulting from previous
measurements. If there is no prior knowledge, the prior probability would be ½.
However, in many situations it is known beforehand that black atoms are rare in
the measured population. This prior information can completely change the
interpretation of the measurement data.

A full description of this problem is contained in the paper: “The application
of Bayesian techniques in the interpretation of bioassay data”, by Miller,
Little, and Guilmette, Radiat Prot Dosimetry, 105: 333-338 (2003). A copy of
the paper is contained in the file RPDpaper.pdf, which is included in the
download. This problem is also discussed in the review paper "Analyzing
Bioassay Data Using Bayesian Methods--A Primer", by Miller, Inkret, Schillaci,
Martz, and Little, Health Physics 78, 598-613 (2000)

- Bayesian
Software Package II: Internal dosimetry verification and validation test
cases and Los Alamos Bayesian internal dosimetry codes ID and UF LA-CC 04-004

View Abstract /Close
**Guthrie Miller**

This software package serves two purposes: 1) it provides, for general use, internal dosimetry verification and validation cases for PU-238 and 2) it provides the Los Alamos ID and UF codes for Bayesian analysis of bioassay data to determine intake scenarios and internal dose.
The verification and validation cases use numerically generated data assuming ICRP-60 as well as Los Alamos developed forward biokinetic models, and Poisson/Lognormal statistics.

The Los Alamos ID code is a straightforward rigorous application of Bayes theorem using Markov Chain Monte Carlo as described in the paper "Bayesian Internal Dosimetry Calculations Using Markov Chain Monte Carlo" by Miller, Martz, Little, and Guilmette . The UF code uses the unfolding method described in the paper "Internal Dosimetry Intake Estimation Using Bayesian Methods" by Miller, Inkret, and Martz. In the UF code, the approach is to repetitively determine single intakes from data. Only one-dimensional integrations and discrete summations are involved, which can be computed very rapidly. Both codes optionally calculate exact Poisson likelihood functions using the method described in the paper "Using Exact Poisson Likelihood Functions in Bayesian Interpretation of Counting Measurements", although for the UF code this is valid only for single-intake situations. The prior probability distributions employed are described in the paper "Bayesian Prior Probability Distributions for Internal Dosimetry" by Miller, Inkret, Little, Martz, and Schillaci. The UF code is a plausible fast algorithm rather than being rigorously derived from Bayes theorem, and it is generally much faster than the ID code. It's shortcomings are 1) use of Gaussian uncertainty propagation using linearization techniques, 2) as a consequence, exact likelihood functions cannot be properly used if there are mutiple intakes, 2) for multiple intakes, data correlations introduced by tail substractions are not taken into account, and 3) dates of non-incident-related intakes cannot be varied.

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