pyhgf.updates.posterior.continuous.posterior_update_precision_continuous_node#

pyhgf.updates.posterior.continuous.posterior_update_precision_continuous_node(attributes: Dict, edges: Tuple[AdjacencyLists, ...], node_idx: int) → float[source]#

Update the precision of a state node using the volatility prediction errors.

Precision update from value coupling.

The new precision of a state node \(b\) value coupled with other input and/or state nodes \(j\) at time \(k\) is given by:

For linear coupling (default)

\[\pi_b^{(k)} = \hat{\pi}_b^{(k)} + \sum_{j=1}^{N_{children}} \kappa_j^2 \hat{\pi}_j^{(k)}\]

Where \(\kappa_j\) is the volatility coupling strength between the child node and the state node and \(\delta_j^{(k)}\) is the value prediction error that was computed before hand by pyhgf.updates.prediction_errors.continuous.continuous_node_value_prediction_error().

For non-linear value coupling:

\[\pi_b^{(k)} = \hat{\pi}_b^{(k)} + \sum_{j=1}^{N_{children}} \hat{\pi}_j^{(k)} * (\kappa_j^2 * g'_{j,b}(\mu_b^(k-1))^2 - g''_{j,b}(\mu_b^(k-1))*\delta_j)\]

Precision update from volatility coupling.

The new precision of a state node \(b\) volatility coupled with other input and/or state nodes \(j\) at time \(k\) is given by:

\[\pi_b^{(k)} = \hat{\pi}_b^{(k)} + \sum_{j=1}^{N_{children}} \frac{1}{2} \left( \kappa_j \gamma_j^{(k)} \right) ^2 + \left( \kappa_j \gamma_j^{(k)} \right) ^2 \Delta_j^{(k)} - \frac{1}{2} \kappa_j^2 \gamma_j^{(k)} \Delta_j^{(k)}\]

where \(\kappa_j\) is the volatility coupling strength between the volatility parent and the volatility children \(j\) and \(\Delta_j^{(k)}\) is the volatility prediction error given by:

\[\Delta_j^{(k)} = \frac{\hat{\pi}_j^{(k)}}{\pi_j^{(k)}} + \hat{\pi}_j^{(k)} \left( \delta_j^{(k)} \right)^2 - 1\]

with \(\delta_j^{(k)}\) the value prediction error \(\delta_j^{(k)} = \mu_j^{k} - \hat{\mu}_j^{k}\).

\(\gamma_j^{(k)}\) is the effective precision of the prediction, given by:

\[\gamma_j^{(k)} = \Omega_j^{(k)} \hat{\pi}_j^{(k)}\]

that was computed in the prediction step.

Parameters:

attributes: The attributes of the probabilistic nodes.
edges: The edges of the probabilistic nodes as a tuple of pyhgf.typing.Indexes. The tuple has the same length as node number. For each node, the index list value and volatility parents and children.
node_idx: Pointer to the value parent node that will be updated.
time_step: The time elapsed between this observation and the previous one.

Returns:

posterior_precision: The new posterior precision.