pyhgf.updates.posterior.continuous.posterior_update_mean_continuous_node#

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

Update the mean of a state node using the value prediction errors.

Mean update from value coupling.

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

For linear value coupling:

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

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 beforehand. If the child node is an input node, this value was computed by pyhgf.updates.prediction_errors.inputs.continuous.continuous_input_value_prediction_error(). If the child node is a state node, this value was computed by pyhgf.updates.prediction_errors.nodes.continuous.continuous_node_value_prediction_error().

For non-linear value coupling:

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

Mean update from volatility coupling.

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

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

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 is 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:

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

with \(\Omega_j^{(k)}\) the predicted volatility computed in the prediction step pyhgf.updates.prediction.predict_precision().

If the child node is an input node, the volatility prediction error \(\Delta_j^{(k)}\) was computed by pyhgf.updates.prediction_errors.inputs.continuous.continuous_input_volatility_prediction_error(). If the child node is a state node, this value was computed by pyhgf.updates.prediction_errors.nodes.continuous.continuous_node_volatility_prediction_error().

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 the node number. For each node, the index lists the value and volatility parents and children.
node_idx: Pointer to the value parent node that will be updated.
node_precision: The precision of the node. Depending on the kind of volatility update, this value can be the expected precision (ehgf), or the posterior from the update (standard).

Returns:

posterior_mean: The new posterior mean.