Use of the Bayesian statistics and the product of probabilities in the ionizing radiation field

Abstract

Abstract Nowadays, the probability of the intersection (PI) of two or more stochastic events or processes is calculated as the product of probabilities (PPs). The Bayes’ theorem (BT) is widely used in the ionizing radiation field. We will show the PI is not only obtained as the PPs; but the minimum of their probabilities; and demonstrate that terms P(B|A) and P(A|B) in the BT are not new probabilistic metrics, but the own respective probabilities of B and A events. Mathematical derivations based on strong probabilistic foundations, and with their respective illustrations were our methodology. There are demonstrations of: 1) The two ways for determining the PI; and 2) Incoherencies of the BT. The tumor control probability (TCP) and normal tissue non-complication probability (NTCP0) of the radiation oncology treatments to patients with more than one target, calculated respectively as the product of TCP, and NTCP0 of each treatment, are excellent-practical examples in the determination of the PI using PPs. Given previously explained conditions of the BT terms; the use of this theorem should be re-considered. The current determination of the PI using the PPs is not valid for stochastic variables belonging to a stochastic event or process.