The Sigma-Pi-linked variant of this algorithm is used for time series analysis. A simpler way of implementing this method for non-acyclic architectures simplifies the algorithm of Lapedes and Farber, and is proved to be convergent in a probabilistic sense to the average of a Liapunov function by Amari. The parameters of a Bayesian network with hidden variables can be estimated using this technique after a conditional probability transformation. A shortcoming of it is explained using model selection criteria that suggest it minimizes the Kullback-Leibler divergence, and is independent of the number of hidden states. Its derivation involves the chain rule for partial differentiation across an activation function like 
The Sigma-Pi-linked variant of this algorithm is used for time series analysis. A simpler way of implementing this method for non-acyclic architectures simplifies the algorithm of Lapedes and Farber, and is proved to be convergent in a probabilistic sense to the average of a Liapunov function by Amari. The parameters of a Bayesian network with hidden variables can be estimated using this technique after a conditional probability transformation. A shortcoming of it is explained using model selection criteria that suggest it minimizes the Kullback-Leibler divergence, and is independent of the number of hidden states. Its derivation involves the chain rule for partial differentiation across an activation function like the logistic function, as proposed by Rumelhart, Hinton, and Williams, who called it the delta rule. The errors generated at the output is used to update weights in hidden layers and input for a feedforward architecture. Subject to overfitting, FTP name this supervised learning algorithm used to update multilayer artificial neural networks, which works by ►adjusting the weights of earlier layers.
The Sigma-Pi-linked variant of this algorithm is used for time series analysis.
{backpropagation} (or {backprop})