  In situations like tax declarations or analyzes of household budgets we would
like to automatically evaluate credibility of exogenous variable (declared
income) based on some available (endogenous) variables - we want to build a
model and train it on provided data sample to predict (conditional) probability
distribution of exogenous variable based on values of endogenous variables.
Using Polish household budget survey data there will be discussed simple and
systematic adaptation of hierarchical correlation reconstruction (HCR)
technique for this purpose, which allows to combine interpretability of
statistics with modelling of complex densities like in machine learning. For
credibility evaluation we normalize marginal distribution of predicted variable
to $\rho\approx 1$ uniform distribution on $[0,1]$ using empirical distribution
function $(x=EDF(y)\in[0,1])$, then model density of its conditional
distribution $(\textrm{Pr}(x_0|x_1 x_2\ldots))$ as a linear combination of
orthonormal polynomials using coefficients modelled as linear combinations of
features of the remaining variables. These coefficients can be calculated
independently, have similar interpretation as cumulants, additionally allowing
to directly reconstruct probability distribution. Values corresponding to high
predicted density can be considered as credible, while low density suggests
disagreement with statistics of data sample, for example to mark for manual
verification a chosen percentage of data points evaluated as the least
credible.
