Add Brier Score metrics

[Vincent-Maladiere]

What does this PR implement?

This is the implementation of the Brier score metrics module:

• BrierScoreComputer
• BrierScoreSampler to be moved to the GBIncidence source code
• brier_score
• brier_score_incidence
• integrated_brier_score
• integrated_brier_score_incidence

This also implements the IPCWEstimator used by the BrierScoreComputer. We have a dependency on lifelines but not on scikit-survival.

Remarks

- This fixes a subtle bug in y_binary, making our previous implementation of brier_score() incorrect. It now matches scikit-survival results.
Edit: y_true_binary is 1 when we observed an event, and 0 otherwise, which is correct for our Gradient Boosting classification or regression target.
However, when computing the Brier Score metric, we actually need np.logical_not(y_true_binary). Indeed, when we observed an event, the survival probability drops to 0. Otherwise, it stays at 1. The Brier Score uses the survival probability, not the incidence probability.

• brier_score computes is the regular Brier Score like scikit-survival does, where the y_pred input is the survival probability. On the contrary, brier_score_incidence follows the Kretowska formula for competing events and expects y_pred to be the incidence probability for the kth cause of failure instead. Under the hood though, brier_score relies on brier_score_incidence, by simply inputting 1 - y_pred.

  This is needed to disambiguate our Brier Score implementation. We could also only expose brier_score_incidence without the regular brier_score, but I thought it was handy to have it, instead of telling the user to perform 1 - y_pred for binary events, all the time.

• When running

  brier_score(y_train, y_test, y_pred, times)

  times must be the times used for computing y_pred —of shape (n_samples, n_times)— otherwise, the Brier Score will be incorrect. This is tricky to check when y_pred is not a dataframe, so we only check for shapes.

[ogrisel]

First quick pass of feedback.

[ogrisel]

Here is the suggestion related to https://github.com/soda-inria/hazardous/pull/10/files#r1257862724.

[ogrisel]

Let's add some tests to check some mathematical properties:

• IPCWEstimator should constantly predict 1.0 when the training data has no censored values,
• check that assert_allclose(ipcw.predict([0]), [1.0]) when ipcw is fit on random data, even when the shorter time in the training set is censored.
• check that ipcw.predict([t + eps]) is strictly larger than 1.0 when ipcw is fit on random
  censored data and t is the smallest censored duration in the training set.
• check that IPCWEstimator predicts monotonically increasing values on a np.linspace(0, t_max, 100) when fit on random censored event data.
• check that when a training set has deterministic censoring beyond a given t value (100% of the largest training events are censored), predicting with large t values is clipped to 1 / ipcw.min_censoring_survival_prob for different values of min_censoring_survival_prob.

[ogrisel]

More feedback.

[ogrisel]

Ok @Vincent-Maladiere I think this is ready for final pass of review on your end.

[ogrisel]

I rendered the HTML of the doc locally but I realize that we do not have a CI job to do it for the PR. Only for publishing the result in the merge of the PR.

I think the easiest way to achieve this would be to configure a circle CI job for this project. Let's do that later in a separate PR.