pyhgf.math.gaussian_predictive_distribution

pyhgf.math.gaussian_predictive_distribution(x, xi, nu)

Density of the Gaussian-predictive distribution.

This distribution is parametrized by hyperparameters from the exponential family as:

\[\begin{cases} \mathcal{NP}(x, \xi, \nu) := \sqrt{\frac{1}{\pi(\nu+1)(\xi_{x^2}-\xi_{x}^2)}} \frac{\Gamma(\frac{\nu+2}{2})}{\Gamma(\frac{\nu+1}{2})} \left( 1+\frac{(x-\xi_{x})^2}{(\nu+1)(\xi_{x^2}-\xi_x^2)} \right) ^{-\frac{\nu+2}{2}} \end{cases}\]

See [1] for more details.

Parameters:

• x (float) – The point at which the density is evaluated.

• xi (Array | ndarray | bool | number | bool | int | float | complex) – Hyperparameter updated by the sufficient statistics of the observed variables.

• nu (float) – Hyperparameter over the number of valid observation (pseudo-counts).

Returns:

The probability density evaluated at x.

Return type:

y

References

[1]

Mathys, C., & Weber, L. (2020). Hierarchical Gaussian filtering of sufficient statistic time series for active inference. In Communications in Computer and Information Science. Active Inference (pp. 52–58). doi:10.1007/978-3-030-64919-7_7