Estimation of Probabilities of Three Kinds of Petrologic Hypotheses with Bayes Theorem

Abstract

Physical-chemical explanations of the causes of variations in rock suites are evaluated by comparing predicted to measured compositions. Consistent data turn an explanation into a viable hypothesis. Predicted and measured values seldom are equal, creating problems of defining consistency and quantifying confidence in the hypthesis. Bayes theorem leads to methods for testing alternative hypotheses. Information available prior to data collection provides estimates of prior probabilities for competing hypotheses. After consideration of new data, Bayes theorem updates the probabilities for the hypotheses being correct, returning posterior probabilities. Bayes factors, B, are a means of expressing Bayes theorem if there are two hypotheses, H ₀ and H ₁. For fixed values of the prior probabilities, B > 1 implies an increased posterior probability for H ₀ over its prior probability, whereas B < 1 implies an increased posterior probability for H ₁ over its prior probability. Three common problems are: (1) comparing variances in sets of data with known analytical uncertainties, (2) comparing mean values of two datasets with known analytical uncertainties, and (3) determining whether a data point falls on a predicted trend. The probability is better than 0.9934 that lava flows of the 1968 eruption of Kilauea Volcano, Hawaii, are from a single magma batch. The probability is 0.99 that lava flows from two outcrops near Mount Edziza, British Columbia, are from different magma batches, suggesting that the two outcrops can be the same age only by an unlikely coincidence. Bayes factors for hypotheses relating lava flows from Volcano Mountain, Yukon Territory, by crystal fractionation support the hypothesis for one flow but the factor for another flow is so small it practically guarantees the fractionation hypothesis is wrong. Probabilities for petrologic hypotheses cannot become large with a single line of evidence; several data points or datasets are required for high probabilities.