Tag Archives: adversarial

Fairness and discrimination, PhD Course, #9 Mitigation, Pre-processing and In-processing

Finally, after defining (and quantifying) “group fairness“ and “individual fairness“, we can now start to discuss the idea of mitigating a possible discrimination. Here, we will see, how based on some data that were initially collected, and a model (a pricing model), it is possible to remove the discrimination in our pricing model.

Biases everywhere

As mentioned previously, insurance princing is based on the use of different datasets, at least one from “claims” and one from “underwriting”. And obviously, there might be biases in those data, conscious, or intended, or not.

Somehow, idea of tackling the problem from the end, as proposed, may not be the right one, and it might be better to tackle it from the beginning, through the biases in the data. The outcome of models will be less discriminatory if we could get rid of sexist or racist bias in underwriting, or even in the assessment of claims costs. Unfortunately, I cannot discuss that here since I do not have data that could be used to assess selection biases related to sensitive attributes.

On mitigation…

From a philosophical perspective, asking for mitigation might lead to some paradoxes.  I mention here two statements, by two judges in the U.S., that have very opposite perspective on the same problem,


I will not talk much about those philosophical aspects (discussed a bit more in the textbook), we will not discuss how we can achieve fairness if required.

Interestingly, we have a nice property, on the price to pay to achieve fairness (price in terms of risk)

More precisely, we have the following result,

Interestingly, we have not only a lower bound, we can actually reach that bound (we will discuss that point next week).


The first approach is related to the idea of “distorting” inputs, to get legitimate explanatory variables that are uncorrelated with sensitive ones.

If that makes sense in the context of linear models, it is not working well in the general case.

But one should not be (too) suprised: as mentioned in a previous post, on independence (and correlation), there is no statistical guarantees to keep it with nonlinear transforms of variables


It is what we observe on our datasets.


An alternative is to use a penalized approach, where fairness is added as the constraint in the optimization procedure. For example Zafar et al. (2017), considered the following approach, with a constraint based on the covariance between the outcome and the sensitive attribute. We can adapt it for non-linear models.

We can look at the evolution of \widehat{\boldsybol{\beta}} as a function of c.

We can also visualize the evolution of predictions,

(including prediction for unwaware model, blind to the sensitive attribute)

On this slide, we can see that we have a tradeoff between accuracy and fairness.

It is also possible to visualize the distance between the distributions of scores in the two groups. We can see that c\to0 gives here strong fairness actuarlly, since Wasserstein distance tends to 0.

An alternative that we can find in the litterature (that I include in the in-processing section) is based on adversarial learning,

Formally, it is

which is related to minimax theorems

Standard references about adversarial learning and fairness are the following

Next week, we will discuss post-processing approaches.

A Fair Pricing Model via Adversarial Learning

We recently uploaded a revised version of our joint paper A Fair Pricing Model via Adversarial Learning, on ArXiv.

At the core of insurance business lies classification between risky and non-risky insureds, actuarial fairness meaning that risky insureds should contribute more and pay a higher premium than non-risky or less-risky ones. Actuaries, therefore, use econometric or machine learning techniques to classify, but the distinction between a fair actuarial classification and `discrimination’ is subtle. For this reason, there is a growing interest about fairness and discrimination in the actuarial community Lindholm et al. (2022). Presumably, non-sensitive characteristics can serve as substitutes or proxies for protected attributes. For example, the color and model of a car, combined with the driver’s occupation, may lead to an undesirable gender bias in the prediction of car insurance prices. Fairness in insurance pricing is a relatively new and much-requested topic, especially in light of new laws and regulations and past issues encountered in practice (Embrechts and Wüthrich, 2022; Frees and Huang, 2021; Gao and Wüthrich, 2018). Consequently, companies/regulators are looking for new methodologies to ensure a sufficient level of fairness while maintaining an adequate accuracy of predictive models. This paper discusses the importance of adapting the traditional fairness algorithms to specific real-life applications and, in particular, to insurance pricing. We claim that mitigating undesired biases with a generic fair algorithm can be counterproductive insurance. We will show that traditional Fair-ML as adversarial methods are not currently adequate for insurance pricing. Therefore, for these purposes, we have developed a more suitable and effective framework to satisfy a fairness objective while maintaining a sufficient level of predictor accuracy. Inspired by the recent approaches, Blier et al. (2021) and Wuthrich et al. (2021), that have shown the value of autoencoders in pricing, we will show that (2) it can be generalized to multiple pricing factors (geographic, car type), (3) it is more adapted for a fairness context (since it allows to debias the set of pricing components): We extend this main idea to a general framework in which a single whole pricing model is trained by generating the geographic and car pricing components needed to predict the pure premium while mitigating the unwanted bias according to the desired metric. 

There are more examples in this revised version, including the case of a non-binary target, and a non-binary senssitive attribute, such as a spatial one