After having provided a number of training examples for , usually will still make some errors, particularily if the training environment is noisy. How can we model the reliability of 's predictions?
We introduce an additional `confidence module' (not necessarily a neural network) whose input at time is the real vector and whose output at time is the real vector , where the real vector is the internal state of . At time there is a target output for the confidence module. should provide information about how reliable 's prediction can be expected to be   .
In what follows, is the th component of a vector , denotes the expectation operator, denotes the dimensionality of vector , denotes the absolute value of scalar , denotes the conditional probability of given , and denotes the conditional expectation of given . For simplicity, we will concentrate on the case of for all . This means that 's and 's current outputs are based only on the current input. There is a variety of simple ways of representing reliability in :
1. Modelling probabilities of global prediction failures. Let be one-dimensional. Let . can be estimated by , where is the number of those times with and where is the number of those times with .
2. Modelling probabilities of local prediction failures. Let be -dimensional. Let for all appropriate . can be estimated by , where is the number of those times with and where is the number of those times with .
Variations of method 1 and method 2 would not measure the probabilities of exact matches between predictions and reality but the probability of `near-matches' within a certain (e.g. euclidian) tolerance.
3. Modelling global expected error. Let be one-dimensional. Let
4. Modelling local expected error. Let be -dimensional. Let