Bayesian Inference

Bayesian inference, which forms a normative and rational method for belief updating, is applied for force coefficient determination here. Bayesian inference models are used to update a user's belief about an uncertain variable when new information becomes available (e.g., an experimental result). Bayes' rule is given by

where {A j &} is the prior distribution about an uncertain event, A, at a state of information,&; {B j A,&} is the likelihood of obtaining an experimental result B given that event A has occurred; {B j &} is the probability of obtaining experimental result B (without knowing that A has occurred); and {A j B,&} is the posterior belief about event A after observing the result, B.According to Bayes’ rule, the product of the prior and likelihood functions is used to determine the posterior belief. This is the process of learning, i.e., updating the prior belief given the new data B to obtain the posterior belief. Note that {B j&} acts as a normalizing constant when updating probability density functions

For the case of updating the four force coefficients in Eqs. (8)–(11) using experimental force data, Bayes’ rule is written as

where is the posterior distribution of the force coefficients given measured values2 of the mean forces in the x and y directions,

coefficients, and

and

is the prior distributions of the force

is the likelihood of obtaining the