Fuzzy Information and Bayesian Statistics

The well known Bayes’ theorem for continuous stochastic models X~ f (·|θ), θ ∈ Ө with continuous parameter space Ө, a-priori density π(·), and precise sample x 1,..., x n , with likelihood function $$l(\theta ;x_1 ,...,x_n ) = \prod\limits_{i = 1}^n {f\left( {x_i \left| \theta \right.} \right)} for all \theta \in \Theta ,$$ reads $$\pi \left( {\theta \left| {x_1 ,...,x_n } \right.} \right) \propto \pi \left( \theta \right) \cdot l\left( {\theta ;x_1 ,...,x_n } \right).$$