Tag Archives: calibration

Exposé au séminaire de statistique (StatQAM)

Tomorrow, Ewen Gallic will present some recent work at the StatQAM statistical seminar, on calibration, with Agathe Fernandes Machado, François Hu, and Emmanuel Flachaire. It will substantially be based on our recent paper From Uncertainty to Precision: Enhancing Binary Classifier Performance through Calibration

The assessment of binary classifier performance traditionally centers on discriminative ability using metrics, such as accuracy. However, these metrics often disregard the model’s inherent uncertainty, especially when dealing with sensitive decision-making domains, such as finance or healthcare. Given that model-predicted scores are commonly seen as event probabilities, calibration is crucial for accurate interpretation. In our study, we analyze the sensitivity of various calibration measures to score distortions and introduce a refined metric, the Local Calibration Score. Comparing recalibration methods, we advocate for local regressions, emphasizing their dual role as effective recalibration tools and facilitators of smoother visualizations. We apply these findings in a real-world scenario using Random Forest classifier and regressor to predict credit default while simultaneously measuring calibration during performance optimization.

To illustrate, consider predictions about the gender of the person on the picture, including probabilities (confidence), obtained from https://www.picpurify.com/demo-face-gender-age.html, with fake pictures, from https://www.nytimes.com/interactive/2020/11/21/science/artificial-intelligence-fake-people-faces.html.

From Uncertainty to Precision: Enhancing Binary Classifier Performance through Calibration

Our paper From Uncertainty to Precision: Enhancing Binary Classifier Performance through Calibration, written with Agathe Fernandes Machadoa, Emmanuel Flachaire, Ewen Gallic and François Hu is now online on ArXiv,

The assessment of binary classifier performance traditionally centers on discriminative ability using metrics, such as accuracy. However, these metrics often disregard the model’s inherent uncertainty, especially when dealing with sensitive decision-making domains, such as finance or healthcare. Given that model-predicted scores are commonly seen as event probabilities, calibration is crucial for accurate interpretation. In our study, we analyze the sensitivity of various calibration measures to score distortions and introduce a refined metric, the Local Calibration Score. Comparing recalibration methods, we advocate for local regressions, emphasizing their dual role as effective recalibration tools and facilitators of smoother visualizations. We apply these findings in a real-world scenario using Random Forest classifier and regressor to predict credit default while simultaneously measuring calibration during performance optimization.

Fairness and discrimination, PhD Course, #5 Models and Data

For the fifth course, we will discuss machine learning and standard techniques used to get predictive models, and to assess accuracy of those models.

GLM (possibly constrained)

Classically, we use a penalized version of least squares (but this can be adapted to GLMs, when penalizing the negative log-likelihood).  Because of Karush–Kuhn–Tucker conditions, having a constraint on the parameter is equivalent to the following penalized problem, when the constraint is on the \ell_2 norm of \boldsymbol{\beta},

We can also consider the \ell_1 norm of \boldsymbol{\beta},

Those two approaches can be see as a trade-off between accuracy (here the empirical risk on the left) and complexity of the model (on the right). And we can also consider a mixture of the two norms,

As we will see, it will also be possible to consider some penality related to fairness and discriminiation measures (in-processing).

Classifier and ROC Curves

We will also recall metrics used in the context of classification, such as the ROC curve

Each point of the curve can be related to two areas related to the distributions of the scores (in the two groups), for the same threshold – namely the false positive rate and true positive rate

Based on the ROC curve, we can define the AUC, the area under the ROC curve,

But for classifiers, the important challenge is to have calibrated scores, meaning that we want the score to be interpreted as the true underlying probability.

Calibration

Well-calibration is defined as follows

or (with different notations)

It is a well know properties in several applications.

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-07.png

The plot on the right is the calibration plot,

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-10.png

We can easily get that plot

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-09.png

This concept is related to the question “do probabilities returned by some model represent reals probabilities ?” For instance, below, we have pictures generated as some sort of geodesic between two pictures, with a woman on the top left, and a man in the bottom right, published in the New York Times. And below, “probabilities” given by  https://www.picpurify.com/demo-face-gender-age.html.

We could agree that it is rather strange that probabilities (to have a man) do not increase continuously, but on top, with extremely high confidence, the model predicts that the picture is the one of a woman, and below, also with extremely high confidence, that the person is a man…

Data, observations vs. experiments

Then, after concept and notations related to models, we will talk about data. More specifically, the distinction between observations and experimentations.

Another popular classification is the one discussed by Judea Pearl.

So we will talk about association, correlation, causal inference, and counterfactuals.

“Correlated variables” or proxies

One important issue, is that with massive data, one can easily get a (good) proxy of almost any sensitive variable.

The concept is related to comonotonicity, or perfect correlation.

But this is clearly too strong, so we will discuss depedence measures, too.

Independence properties

Recall that independence is defined as follows

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-11.png

and we can consider a weaker form, based on null-covariance

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-12.png

or null-correlation

(sidenote, this correlation measure is bounded, and those bounds are related to Hardy-Littlewood inequality and optimal transport)

An interesting measure is the maximal correlation

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-13.png

or we can consider a weaker version, without consider all possible transformation, but only a subset

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-14.png

Another important concept is the one of conditional independence

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-16.png

(the later will be used in the context of causal graphs).

Causality

Before talking about causality, recall that what non-independence mean…

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-17.png

We can then construct causal graphs, or “directed acyclic graphs”

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-20.png

where nodes are the variables used in the model, and the outcome (usually that the end of the causal graph). Then we define paths

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-18.png

and the concept of d-separation

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-19.png

This concept is related to the statistical property of conditional independence

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-21.png

More precisely, we have the following Markov property on causal graphs

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-22.png

For example, for such a graphical model,

the joint distribution is \mathbb{P}[x_1,x_2,x_3,x_4]=\mathbb{P}[x_1]\times \mathbb{P}[x_2|x_1]\times \mathbb{P}[x_3|x_2]\times \mathbb{P}[x_4|x_3]and for the graphical model below

we have\mathbb{P}[x_1,x_2,x_3,x_4]=\mathbb{P}[x_1]\times \mathbb{P}[x_2]\times \mathbb{P}[x_3|x_1,x_2]\times \mathbb{P}[x_4|x_3]Those graphs can be related to structural models (with idiosyncratic noise denoted U), since

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-23.png

Potential outome

Another important concept is the concept of counterfactuals, and potential outome. In an ideal world, we would have observed the outome in both cases, with and without the treatement

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-24.png

but in real life, it’s only one of them,

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-25.png

And the goal will be, somehow, to estimate what the non-observed outcome would be. And then, classical quantites we wish to estimate are the average treatement effect, and the conditional version, based on some covariates.

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-26.png

This concept will be related to counterfactual fairness actually, when the “treatement” will be the sensitive attribute.

Twin network representation of the counterfactual

Finally, we will consider a so-called “twin network representation”. Consider a DAG, associated with some simple structural model

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-27.png

Based on a structural model, we can get values of idiosyncratic noise component

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-28.png

Then, we use those values on the twin representation, when the treatement is not 0, but 1. Counterfactuals are created by using the same noises

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-29.png

The difference between the two outcomes is the treatement effect, or the disparate treatement

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-30.png

or more generally, we write

https://freakonometrics.hypotheses.org/files/2024/01/cours-slides-fairness-31.png

This is an idea used in Plecko & Meinshausen, 2019, in the context of fairness, but we will discuss this more, later on…