Predicting Probabilities

A common assumption about classifiers goes like this: when a model reports that it is very certain about a prediction, that prediction is probably right, and the more certain the model sounds, the more trustworthy it is. This sounds reasonable, but it quietly mixes two separate questions. The first question is what the model says about its own confidence. The second is how often the model is actually correct when it says that. This article follows the gap between those two questions and the property that closes it, which is called calibration.

Read the requirement in words: among all the predictions where the model reports confidence p, the fraction that turn out correct should be p. The claim is about a group of predictions that share the same stated confidence. A single prediction made at 70% confidence can be right or wrong, and calibration says nothing about that one outcome. It describes what happens across many predictions made at that confidence level.

Classifiers that expose uncertainty estimates give a probability for each class. In a binary problem with two classes, those two probabilities add up to 1: $P(\text{class A}) + P(\text{class B}) = 1$. Because they must sum to 1, exactly one class is above 50% and that class is the predicted one. The other class is necessarily below 50%.

The source also describes how this plays out across a whole dataset. In its example the classifier is relatively certain for most points, so most points sit comfortably on one side of 50%. That overall picture tells you how the model is behaving, but it does not by itself tell you whether the certainty is deserved. Whether the reported uncertainty reflects real uncertainty in the data depends on the model and its parameters.

Everything so far concerned binary classification. The decision_function and predict_proba methods also work in the multiclass setting, where there are more than two classes. The source illustrates this on the Iris dataset, a three-class problem. Both methods provide uncertainty estimates in binary and multiclass settings, but they report those estimates in different forms. The source does not say what those forms are beyond this, so the reading below focuses on the probabilities themselves.

In the multiclass case, the predicted label is the class with the highest probability. The 50% test used in the binary case does not carry over directly, because with three classes the top probability may fall below 50% while still being the largest. In the table above, class B at 0.61 is the predicted class even though the next class is at 0.24. Identifying the predicted class is a matter of comparing values, not of checking a single threshold.

The source ties the reliability of a model's uncertainty to its complexity, and it makes two claims that point in opposite directions. First, a model that is more overfitted tends to make more certain predictions, even if those predictions might be wrong. Second, a model with less complexity usually has more uncertainty in its predictions. Both claims matter when you read a model's reported confidence.

This matters for calibration because an overfitted model can look confident for the wrong reason. Its high certainty reflects how closely it has fit the training data, not necessarily how often it will be right on new inputs. So certainty alone cannot be read as calibration. A calibrated model's stated confidence has to line up with how often it is correct, and an overfitted model can report high confidence while that alignment fails.

The belief that high confidence means a good model separates from calibration at one specific step: when the stated confidence is treated as a verified claim without checking it against observed outcomes. The first question asks what the model says about its confidence. Calibration asks the second question, namely whether the model is correct about as often as it says across predictions made at that confidence. Only the second question establishes that the stated confidence means anything.