Stochastic models
The reasonable expectation from the stochastic model $M$
$$ M(x_1, x_2, x_3, ... , x_n)$$
is return of 1D distribution, which depends on features $x_1, x_2, x_3, ... , x_n$. It is not multivariate distribution by its nature. All features
are always given, they can't be variables (they are provided) and result or target is random, but fit some distribution.
Many applications claim the capability of building stochastic models, but in reality they provide fake result. After finding that deterministic
modeling is not possible and there is always a random error, they offer some very approximate and frequently not accurate estimation of either
variance or confidence interval, assuming normal distribution of targets. And the logic sounds usually as follows: "we don't know anything, but
we can see random errors, so we assume they are normal and try to estimate confidence interval at least for entire dataset or, if possible,
find how it depends on features". That is kind of trivial task and not what we try to solve on this site.
The goal is to provide the code which supports recognition of input dependent multimodality, i.e. the probability
densities with multiple peaks like it is shown below
The validation of modeled probability density is called calibration. We provide calibrating tests estimating the accuracy of probabilistic modeling.
Types of uncertainty
1. Aleatoric
Example: an income of working for paycheck individuals depending on demographic factors, such as education, profession, age, sex, marital status and so on.
2. Epistemic
Example: the area of triangle given by coordinates of the vertices.
Probabilistic classification
The probabilistic classification model simply provides probability for each class, while deterministic classification returns the class label.

