Part V

BAYESIAN INFERENCE AND FUZZY INFORMATION

In standard Bayesian statistics all unknown quantities are described by stochastic quantities and their probability distributions. Consequently, so are the parameters θ of stochastic models X ~ f (·|θ); θ ∈ Θ in the continuous case, and X ~ p(·|θ);θ ∈ Θ in the discrete case. The a priori knowledge concerning θ is expressed by a probability distribution π(·) on the parameter space Θ, called a priori distribution.

If the parameter θ is continuous, the a priori distribution has a density function, called a priori density.

For discrete parameter space Θ = {θ1,…,θm} the a priori distribution is a discrete probability distribution with point probabilities π(θj),j = 1(1)m.

If the observed stochastic quantity X ~ p(·|θ); θ ∈ {θ1,…,θm} is also discrete, i.e. the observation space M_{X} of X is countable with

and observations 1,…,n of X are given [sample data D = (1,…,n)], then the conditional distribution Pr(|D) of the parameter is obtained via Bayes’ ...