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,

versus

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).

Pre-processing

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

or

It is what we observe on our datasets.

In-processing

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.