A Fair Post-processing Method Based on the MADD Metric for Predictive Student Models

Abstract

Predictive student models are increasingly used in learning environments. However, due to the rising social impact of their usage, it is now all the more important for these models to be both sufficiently accurate and fair in their predictions. To evaluate algorithmic fairness, a new metric has been developed in education, namely the Model Absolute Density Distance (MADD). This metric enables us to measure how different a predictive model behaves regarding two groups of students, in order to quantify its algorithmic unfairness. In this paper, we thus develop a post-processing method based on this metric, that aims at improving the fairness while preserving the accuracy of relevant predictive models’ results. We experiment with our approach on the task of predicting student success in an online course, using both simulated and real-world educational data, and obtain successful results. Our source code and data are in open access at https://github.com/melinaverger/MADD.

Appendices

Proofs of 3.2

We set C to be a random variable with probability density function $\mathcal {D}$, representing the predicted probability value of the output of the model $\mathcal {C}$, and S to be a random variable subject to Bernoulli distribution, representing the value of the sensitive parameter a. Thus, $\mathcal {D}^{G_0}$ and $\mathcal {D}^{G_1}$ are the probability density functions of the conditional distributions $C | S=0$ and $C | S=1$, respectively. According to the law of total probability, we have:

$$ \begin{alignedat}{2} &\quad &\mathbb {P}\left( C \le t \right) &= \mathbb {P}\left( C \le t \mid S=0 \right) \ \mathbb {P}\left( S=0 \right) + \mathbb {P}\left( C \le t \mid S=1 \right) \ \mathbb {P}\left( S=1 \right) \\ &\Longleftrightarrow & F(t) &= F^{G_0}(t) \ \mathbb {P}\left( S=0 \right) + F^{G_1}(t) \ \mathbb {P}\left( S=1 \right) \\ &\Longleftrightarrow & \mathcal {D}(t) &= \mathcal {D}^{G_0}(t) \ \mathbb {P}\left( S=0 \right) + \mathcal {D}^{G_1}(t) \ \mathbb {P}\left( S=1 \right) \end{alignedat} $$

Where $F, F^{G_0}, F^{G_1}$ are the cumulative distribution functions (CDFs) of $\mathcal {D}$, $\mathcal {D}^{G_0}$, $\mathcal {D}^{G_1}$, respectively. Since $\mathbb {P}\left( S=0 \right) + \mathbb {P}\left( S=1 \right) = 1$, f is a linear combination of $\mathcal {D}^{G_0}$ and $\mathcal {D}^{G_1}$, and $\mathcal {D}$ lies between $\mathcal {D}^{G_0}$ and $\mathcal {D}^{G_1}$ in the function space (i.e., $\mathcal {D}^{G_0}, \mathcal {D}, \mathcal {D}^{G_0}$ are collinear.

This property is also true for estimators obtained from observed values. In fact, the definition of the sequence of the heights of the histogram is: for the m equal sub-intervals $\left[ \frac{k-1}{m}, \frac{k}{m} \right]$ for all $k \in \{1, \ldots , m\}$ on [0, 1],

$$\begin{aligned} &D_{G_0}^a &:= \left\{ d_{G_0, k} \mid \forall k \in \{ 1,\ldots , m \} \right\} , &\text { with } d_{G_0, k} &:= &\frac{N_{G_0, k}}{n_0} &:= &\frac{1}{n_0}\sum _{i \in G_0} \textbf{1}_{\hat{p}_i \in I_k} \\ &D_{G_1}^a &:= \left\{ d_{G_1, k} \mid \forall k \in \{ 1,\ldots , m \} \right\} , &\text { with } d_{G_1, k} &:= &\frac{N_{G_1, k}}{n_1} &:= &\frac{1}{n_1}\sum _{i \in G_1} \textbf{1}_{\hat{p}_i \in I_k} \\ &D_{G} &:= \left\{ d_{G, k} \mid \forall k \in \{ 1,\ldots , m \} \right\} , &\text { with } d_{G, k} &:= &\frac{N_{G_0, k} + N_{G_1, k}}{n_0 + n_1} &:= &\frac{1}{n_0 + n_1}\sum _{i \in G} \textbf{1}_{\hat{p}_i \in I_k} \end{aligned}$$

And because for all $k \in \{1,\ldots , m\}$, we have:

$$\begin{aligned} d_{G, k} &= \frac{N_{G_0, k} + N_{G_1, k}}{n_0 + n_1} = \frac{n_0 \ d_{G_0,k} + n_1 \ d_{G_1,k}}{n_0 + n_1} \\ &= \frac{n_0}{n_0 + n_1} d_{G_0,k} + \frac{n_1}{n_0 + n_1} d_{G_1,k} \end{aligned}$$

Also, $f^{(G_0)}, f, f^{(G_1)}$ are based on $D_{G_0}^a, D_{G}, D_{G_1}^a$, respectively:

$$\begin{aligned} f^{(G_0)}(x) &:= \sum _{k=i}^m d_{G_0, k} \textbf{1}_{x \in I_k} \\ f^{(G_1)}(x) &:= \sum _{k=i}^m d_{G_1, k} \textbf{1}_{x \in I_k} \\ f(x) &:= \sum _{k=i}^m d_{G, k} \textbf{1}_{x \in I_k} \end{aligned}$$

Therefore, $f^{(G)}(x) = \frac{n_0}{n_0 + n_1} f^{(G_0)}(x) + \frac{n_1}{n_0 + n_1} f^{(G_1)}(x)$, so $f^{(G_0)}, f, f^{(G_1)}$ are also collinear (see Fig. 3c). This is not a coincidence; in fact, as histogram estimators, when $(n_0, n_1) \rightarrow +\infty$, $\left( f^{(G_0)}, f, f^{(G_1)} \right) \rightarrow \left( \mathcal {D}_{G_0}^a, \mathcal {D}_{G}, \mathcal {D}_{G_1}^a \right)$.

Proof of 3.3

According to Inverse transform sampling [2], we have the following two theorems:

Theorem 1

Let $\mathcal {A}$ be a distribution and $F_{\mathcal {A}}$ be the cumulative distribution function of that distribution. If X obeys the distribution $\mathcal {A}$ i.e. $X \sim \mathcal {A}$, then $F_{\mathcal {A}}(X) \sim \mathcal {U}_{[0, 1]}$, where $\mathcal {U}_{[0,1]}$ is a uniform distribution over [0, 1].

Theorem 2

Let $U \sim \mathcal {U}_{[0,1]}$ and $F^{-1}_{\mathcal {A}}$ be the generalised inverse function of $F_{\mathcal {A}}$, then $F^{-1}_{\mathcal {A}}(U) \sim \mathcal {A}$.

Take $G_0$ as an example. By definition, the newly generated prediction $\overline{p}_{i}^{(\lambda )}$ is $\overline{\operatorname {CDF}}_{(G_0)}^{-1(\lambda )}(\operatorname {CDF}_{(G_0)}(\hat{p}_i))$, and by Theorem 1, we have $\operatorname {CDF}_{(G_0)}(\hat{p}_i) \sim \mathcal {U}_{[0,1]}$, so $\overline{\operatorname {CDF}}_{(G_0)}^{-1(\lambda )}(\operatorname {CDF}_{(G_0)}(\hat{p}_i))$ obeys the newly generated distribution according to Theorem 2. Furthermore, since the CDF is monotone increasing and the inverse function does not change the monotonicity, $\overline{\operatorname {CDF}}_{(G_0)}^{-1(\lambda )}$ is also monotone increasing, which means that $\forall i,j, \hat{p}_i \ge \hat{p}_j \Longrightarrow \overline{p}_{i}^{(\lambda )} \ge \overline{p}_{i}^{(\lambda )}$. The conclusion on $G_1$ follows in the same way.