In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". In the table, the values w, x, y and z give the relative weights of each corresponding condition and case.The figures denote the cells of the table involved in each metric, the probability being the fraction of each figure that is shaded. P(A|B) = Bayesian inference derives the posterior probability as a consequence of two antecedents, a prior probability and a "likelihood function" derived from a statistical model for the observed data.Ian Hacking noted that traditional "Dutch book" arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books.Hacking wrote "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. So the personalist requires the dynamic assumption to be Bayesian.For a given pair of values, μ and Σ, in the sample, we can generate a simulated dataset.The size of the simulated dataset is arbitrary, but should be large enough to generate a smooth distribution of P(A|B) and P(B|A).The distribution of belief over the model space may then be thought of as a distribution of belief over the parameter space.
Only this way is the entire posterior distribution of the parameter(s) used.Bayesian inference computes the posterior probability according to Bayes' theorem: – the posterior probability of a hypothesis is proportional to its prior probability (its inherent likeliness) and the newly acquired likelihood (its compatibility with the new observed evidence). Bayesian updating is widely used and computationally convenient.However, it is not the only updating rule that might be considered rational.I'm going to give you some background on what I'm trying to do here first.I'm doing sports analysis trying to find the best quarterback of the 2015 NFL season using passer rating and quarterback rating, two different measures of how the quarterback performs during a game.