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Bayesian regularization backpropagation


net.trainFcn = 'trainbr'
[net,tr] = train(net,...)


trainbr is a network training function that updates the weight and bias values according to Levenberg-Marquardt optimization. It minimizes a combination of squared errors and weights, and then determines the correct combination so as to produce a network that generalizes well. The process is called Bayesian regularization.

net.trainFcn = 'trainbr' sets the network trainFcn property.

[net,tr] = train(net,...) trains the network with trainbr.

Training occurs according to trainbr training parameters, shown here with their default values:


Maximum number of epochs to train


Performance goal


Marquardt adjustment parameter


Decrease factor for mu


Increase factor for mu


Maximum value for mu


Maximum validation failures


Minimum performance gradient


Epochs between displays (NaN for no displays)


Generate command-line output


Show training GUI


Maximum time to train in seconds

Validation stops are disabled by default (max_fail = inf) so that training can continue until an optimal combination of errors and weights is found. However, some weight/bias minimization can still be achieved with shorter training times if validation is enabled by setting max_fail to 6 or some other strictly positive value.

Network Use

You can create a standard network that uses trainbr with feedforwardnet or cascadeforwardnet. To prepare a custom network to be trained with trainbr,

  1. Set NET.trainFcn to 'trainbr'. This sets NET.trainParam to trainbr’s default parameters.

  2. Set NET.trainParam properties to desired values.

In either case, calling train with the resulting network trains the network with trainbr. See feedforwardnet and cascadeforwardnet for examples.


Here is a problem consisting of inputs p and targets t to be solved with a network. It involves fitting a noisy sine wave.

p = [-1:.05:1];
t = sin(2*pi*p)+0.1*randn(size(p));

A feed-forward network is created with a hidden layer of 2 neurons.

net = feedforwardnet(2,'trainbr');

Here the network is trained and tested.

net = train(net,p,t);
a = net(p)


This function uses the Jacobian for calculations, which assumes that performance is a mean or sum of squared errors. Therefore networks trained with this function must use either the mse or sse performance function.


trainbr can train any network as long as its weight, net input, and transfer functions have derivative functions.

Bayesian regularization minimizes a linear combination of squared errors and weights. It also modifies the linear combination so that at the end of training the resulting network has good generalization qualities. See MacKay (Neural Computation, Vol. 4, No. 3, 1992, pp. 415 to 447) and Foresee and Hagan (Proceedings of the International Joint Conference on Neural Networks, June, 1997) for more detailed discussions of Bayesian regularization.

This Bayesian regularization takes place within the Levenberg-Marquardt algorithm. Backpropagation is used to calculate the Jacobian jX of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to Levenberg-Marquardt,

jj = jX * jX
je = jX * E
dX = -(jj+I*mu) \ je

where E is all errors and I is the identity matrix.

The adaptive value mu is increased by mu_inc until the change shown above results in a reduced performance value. The change is then made to the network, and mu is decreased by mu_dec.

Training stops when any of these conditions occurs:

  • The maximum number of epochs (repetitions) is reached.

  • The maximum amount of time is exceeded.

  • Performance is minimized to the goal.

  • The performance gradient falls below min_grad.

  • mu exceeds mu_max.


[1] MacKay, David J. C. "Bayesian interpolation." Neural computation. Vol. 4, No. 3, 1992, pp. 415–447.

[2] Foresee, F. Dan, and Martin T. Hagan. "Gauss-Newton approximation to Bayesian learning." Proceedings of the International Joint Conference on Neural Networks, June, 1997.

Introduced before R2006a