Modelling uncertainty

Calibration

The problem of getting accurate estimates of the uncertainty of the prediction(s) of a classifier or regressor.

For example, if a binary classifier gives scores of 0.9 and 0.1 for classes A and B that does not necessarily mean it has a 90% chance of being correct. If the actual probability of being correct (class A) is far from 90% we say that the classifier is poorly calibrated. On the other hand, if the model if it really does have a close to 90% chance of being correct we can say the classifier is well calibrated.

Binary classification

  1. Train the classifier \hat{y} = f(x) in the normal way
  2. Construct a dataset with, for each row in the original dataset, the predicted score and the actual label.
  3. Fit an isotonic regression \bar{y} = g(\hat{y}) to this data, trying to predict the label given the score. \bar{y} can be used as a well-calibrated estimate of the true probability.

Multi-class classification

Reduce the problem to n one-vs-all binary classification tasks and use the method in the preceding section for each of them. Normalise the resulting distribution to ensure it sums to 1.

Regression

TODO

Measuring uncertainty

This section describes methods for estimating the uncertainty of a classifier. Note that additional methods may be necessary to ensure that this estimate is well-calibrated.

Classification

The uncertainty for a predicted probability distribution over a set of classes can be measured by calculating its entropy.

Regression

Unlike in classification we do not normally output a probability distribution when making predictions for a regression problem. The solution is to make the model output additional scalars, describing a probability distribution.

This could be:

  • The Gaussian distribution. This only requires two parameters but may be over-simplifying if there aren’t strong theoretical reasons to believe the distribution ought to be Gaussian or at least unimodal.
  • A categorical distribution. This option allows a great degree of flexibility but requires a relatively large number of parameters. It also makes learning harder since the model has to learn for itself that the 14th category is numerically close to the 13th and 15th (Salimans et al., 2017).
  • A mixture model. If the number of mixtures is chosen well this can represent a good middle ground between descriptiveness and efficiency.

Here is an example in full, using the normal distribution:

The network outputs two numbers describing the Normal distribution N(\mu,\sigma^2). \mu is the predicted value and \sigma^2 describes the level of uncertainty.

  • The mean \mu, outputted by a fully-connected layer with a linear activation.
  • The variance \sigma^2, outputted by a fully-connected layer with a softplus activation. Using the softplus ensures the variance is always positive without having zero gradients when the input is below zero, as with the ReLU.

The loss function is the negative log likelihood of the observation under the predicted distribution:

L(y,\mu,\sigma) = - \frac{1}{2}\log(\sigma^2) - \frac{1}{2n \sigma^2}\sum_{i=1}^n (y - \mu)^2