pyhgf.math.gaussian_predictive_distribution#
- pyhgf.math.gaussian_predictive_distribution(x: float, xi: Array | ndarray | bool_ | number | bool | int | float | complex, nu: float) float[source]#
Density of the Gaussian-predictive distribution.
This distribution is parametrized by hyperparameters from the exponential family as:
\[\begin{cases} \mathcal{NP}(x, \xi, \nu) := \sqrt{\frac{1}{\pi(\nu+1)(\xi_{x^2}-\xi_{x}^2)}} \frac{\Gamma(\frac{\nu+2}{2})}{\Gamma(\frac{\nu+1}{2})} \left( 1+\frac{(x-\xi_{x})^2}{(\nu+1)(\xi_{x^2}-\xi_x^2)} \right) ^{-\frac{\nu+2}{2}} \end{cases}\]- Parameters:
- x
The point at which the density is evaluated.
- xi
Hyperparameter updated by the sufficient statistics of the observed variables.
- nu
Hyperparameter over the number of valid observation (pseudo-counts).
- Returns:
- y
The probability density evaluated at x.
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