17
Fuzzy predictive distributions
Let X ~ Pθ;θ ∈ Θ be a stochastic model with a priori distribution π(·). Then in standard Bayesian inference the predictive distribution of X, given observed data D, is denoted by X|D.
There are three practically important situations for predictions:
- discrete X and discrete Θ;
- discrete X and continuous Θ;
- continuous X and continuous Θ.
For the standard situation predictive distributions conditional on observed samples D = (x1,…,xn) are available.
In the case of fuzzy a priori distributions and fuzzy samples (x1*…, xn*) of X the concept of predictive distributions has to be generalized.
17.1 Discrete case
For discrete stochastic quantities X ~ p(·|θ), θ ∈ Θ = {θ1,…,θm}, and sample x1,…,xn of X, the a posteriori probabilities π (θj|x1,…,xn) = 
are obtained as explained at the beginning of Part V.
The predictive distribution of X|x1,…,xn is given by its predictive probabilities 
for all x ∈ Mx where Mx is the observation space of X, i.e. the set of all possible values of X.
In the case of fuzzy a posteriori distribution π*(θj|x1,…,xn) the so-called fuzzy predictive distribution of X|x1,…,xn is given by its fuzzy probabilities
The fuzzy a posteriori probabilities ...
