# probability calibration and Brier score

Assume that I have a binary classification problem. The outcome from classification I am mostly interested in is the well-calibrated probabilities.

The first way to check this is the calibration plot (or reliability curve).

The question: is it fair enough to judge about calibration based on Brier score?

Assume that we have "enough" data. Would the classifier with smaller Brier score provide a rather better reliability curve?

My concern comes from the fact that the probability from a classifier are the conditional probabilities. Therefore, I do not see the intuition of applying the Brier score to conditional probabilities.

The short answer is that it only makes sense to calculate the Brier score for the conditional probabilities, $$\hat y = P(y=1|X)$$, where $$y$$ is the outcome, $$\hat y$$ is your prediction, and $$X$$ are your predictors. In other words, $$\hat y$$ is the probability that $$y=1$$, conditional on this particular value of the predictors, $$X$$.

The Brier score in this case is just

$$\frac{1}{N}\sum_i^N (\hat y_i - y_i)^2$$

What other kinds of probability could there be? The only other option here is the marginal probability, $$P(y=1)$$. We can estimate this by simply counting the proportion of times $$y=1$$ in the data. Clearly, it doesn't make sense to use this value when calculating the Brier score!

Would the classifier with smaller Brier score provide a rather better reliability curve?

Yes. If your classifier predicts $$\hat y = 1$$ in all cases where $$y = 1$$ and $$\hat y = 0$$ where $$y = 0$$, it has a Brier score of $$0$$. If it does the opposite, it has a score of $$\pm1^2 = 1$$. In most cases, such perfect predictions won't be possible, but a good classifier can still be well calibrated, for instance by predicting $$\hat y = 0.5$$ in cases where $$y = 1$$ half the time and $$y=0$$ the rest. A classifier that does this will have the lowest possible Brier score on your data.

• Why doesn’t it make sense to calculate Brier score for a model that always guesses the prior probability?
– Dave
Sep 7 '20 at 13:04
• I mean, you could do it, and it makes sense mathematically, but it doesn't tell you anything you want to know. My main point is that @ABK was possibly a bit confused about the difference between conditional and marginal probability.
– Eoin
Sep 7 '20 at 13:48
• To elaborate, if you decided to plug marginal $P(y)$ into the Brier score calculations instead of $P(y|X)$, you would get a score close to 0 if the data is very imbalanced (most zeros, or mostly ones), and a score of $\pm 0.5^2 = 0.25$ if it's 50% zeros and 50% ones.
– Eoin
Sep 7 '20 at 15:02