---
title: "Fairness Metrics"
preamble: |
  \usepackage{longtable}
  \usepackage{bbm}
bibliography: '`r system.file("references.bib", package="mlr3fairness")`'
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Fairness Metrics}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  
---

```{r, include = FALSE}
library(mlr3)
library(mlr3fairness)
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```
# Fairness Measures

Fairness measures (or metrics) allow us to assess and audit for possible biases in a trained model. 
There are several types of metrics that are widely used in order to assess a model's fairness. 
They can be coarsely classified into three groups:

- **Statistical Group Fairness Metrics**: Given a set of predictions from our model, we assess for differences in one or multiple metrics across groups given by a *protected attribute* [@fairmlbook; @hardt2016equality]. 

- **Individual Fairness**: Basically requires that similar people are treated similar independent of the protected attribute [@dwork2012].
  We will briefly introduce individual fairness in a dedicated section below.

- **Causal Fairness Notions**: An important realization in the context of Fairness is, that whether a process is fair is often subject to the underlying causal directed acyclic graph (DAG). Knowledge of the DAG allows for causally assessing reasons for (un-)fairness. Since DAG's are often hard to construct for a given dataset, `mlr3fairness` currently does not provide any causal fairness metrics [@kilbertus2017avoiding].

## Statistical Group Fairness Metrics

One way to assess the fairness of a model is to find a definition of fairness that is relevant to a problem at hand. We might for example define a model to be fair if the chance to be accepted for a job given you are qualified is independent of a protected attribute e.g. `gender`. This can e.g. be measured using the `true positive rate`(TPR): in `mlr3` this metric is called `"classif.tpr"`. In this case we measure discrepancies between groups by computing differences `(-)` but we could also compute quotients.
In practice, we often compute absolute differences.

$$
  \Delta_{TPR} = TPR_{Group 1} - TPR_{Group 2}
$$

We will use metrics like the one defined above throughout the remainder of this vignette.
Some predefined measures are readily available in `mlr3fairness`, but we will also showcase how custom measures can be constructed below.

In general, fairness measures have a `fairness.` prefix followed by the metric that is compared across groups. We will thus e.g. call the difference in accuracies across groups `fairness.acc`.
A full list can be found below.

```{r, echo = FALSE}
knitr::kable(mlr3fairness:::mlr_measures_fairness)
```

## Assessing for Bias: A first look

This vignette assumes that you are familiar with the basics of `mlr3` and it's core objects.
The [mlr3 book](https://mlr3book.mlr-org.com/) can be a great resource in case you want to learn more about mlr3's internals.

We first start by training a model for which we want to conduct an audit.
For this example, we use the `adult_train` dataset.
Keep in mind all the datasets from mlr3fairness package already set protected attribute via the `col_role` "pta", here the "sex" column. To speed things up, we only use the first 1000 rows.

```{r}
library(mlr3fairness)
library(mlr3learners)

t = tsk("adult_train")$filter(1:1000)
t$col_roles$pta
```

Our model is a random forest model fitted on the dataset:

```{r}
l = lrn("classif.ranger")
l$train(t)
```

We can now predict on a new dataset and use those predictions to assess for bias:

```{r}
test = tsk("adult_test")
prd = l$predict(test)
```

Using the `$score` method and a measure we can e.g. compute the absolute differences in true positive rates.

```{r}
prd$score(msr("fairness.tpr"), task = test)
```

The exact measure to choose is often data-set and situation dependent. The [Aequitas Fairness Tree](https://textbook.coleridgeinitiative.org/chap-bias.html) can be a great ressource to get started.

We can furthermore simply look at the per-group measures:

```{r}
meas = groupwise_metrics(msr("classif.tpr"), test)
prd$score(meas, task = test)
```

## Fairness Measures - An in-depth look

Before, we have used `msr("fairness.tpr")` to assess differences in false positive rates across groups. But what happens internally?

The `msr()` function is used to obtain a `Measure` from a dictionary of pre-defined `Measure`s. We can use `msr()` without any arguments in order to print a list of available measures.
In the following example, we will build a `Measure` that computes differences in False Positive Rates making use of the `"classif.fpr"` measure readily implemented in `mlr3`.

```{r}
# Binary Class false positive rates
msr("classif.fpr")
```

The core `Measure` in `mlr3fairness` is a `MeasureFairness`. It can be used to construct arbitrary measures that compute a difference between a specific metric across groups. We can therefore build a new metric as follows:

```{r}
m1 = MeasureFairness$new(base_measure = msr("classif.fpr"), operation = function(x) {abs(x[1] - x[2])})
m1
```
This measure does the following steps:
- Compute the metric supplied as `base_measure` in each group defined by the `"pta"` column.
- Compute `operation` (here `abs(x[1] - x[2])`) and return the result.

In some cases, we might also want to replace the operation with a different operation, e.g. `x[1] / x[2]` in order to compute a different perspective.

`mlr3fairness` comes with two built-in functions that can be used to compute fairness metrics
also across protected attributes that have more than two classes.

- `groupdiff_absdiff`: maximum absolute difference between all classes (the default for all metrics)
- `groupdiff_tau`: minimum quotient between all classes

**Note**: Depending on the `operation` we need to set a different `minimize` flag for the measure, so subsequent operations based on the measure automatically know if the measure is to be minimized or maximized e.g. during tuning.

We can also use those operations to construct a measure using `msr()`, since `MeasureFairness` (key: `msr("fairness")`) can be constructed from the dictionary with additional arguments.

```{r}
m2 = msr("fairness", operation = groupdiff_absdiff, base_measure = msr("classif.tpr"))
```

This allows us to construct pretty flexible metrics e.g. for regression settings:

```{r}
m2 = msr("fairness", operation = groupdiff_absdiff, base_measure = msr("regr.mse"))
```

### Non-binary protected groups

While fairness measures are widely defined or used with binary protected attributes, we can easily extend
fairness measures such that they work with non-binary valued protected attributes.

In order to do this, we have to supply an `operation` that reduces the desired metric measured in each subgroup
to a single value. Two examples for such operations are `groupdiff_absdiff` and `groupdiff_tau` but custom functions can also be applied. 
Note, that **mlr3** `Measure`s are designed for a scalar output
and `operation` therefore always has to result in a single scalar value.

## Composite Fairness Measures

Some fairness measures also require a combination of multiple Fairness Metrics.
In the following example we show how to compute the mean of two fairness metrics,
here false negative and true negative rates and create a new `Measure` that computes the mean (see `aggfun`) of those metrics:

```{r}
ms = list(msr("fairness.fnr"), msr("fairness.tnr"))
ms

m = MeasureFairnessComposite$new(measures = ms, aggfun = mean)
```


## How to compare the fairness performance between different learners using Benchmarks

In this example, we create a `BenchmarkInstance`. Then by using `aggregate()` function they could access the fairness measures easily. 
The following example demonstrates the process to evaluate the fairness metrics on Benchmark Results:

```{r, warning = FALSE}
design = benchmark_grid(
  tasks = tsks("adult_train"),
  learners = lrns(c("classif.ranger", "classif.rpart"),
    predict_type = "prob", predict_sets = c("train", "test")),
  resamplings = rsmps("cv", folds = 3)
)

bmr = benchmark(design)

# Operations have been set to `groupwise_quotient()`
measures = list( msr("fairness.tpr"), msr("fairness.npv"), msr("fairness.acc"), msr("classif.acc") )

tab = bmr$aggregate(measures)
tab
```

## Metrics aggregation - groupdiff_* 

For `MeasureFairness`, **mlr3** computes the `base_measure` in each group specified by the `pta` column. 
If we now want to return those measures, we need to aggregate this to a single metric - e.g. using one of the 
`groupdiff_*` functions available with mlr3. See `?groupdiff_tau` for a list. 
Note, that the `operation` below also accepts custom aggregation function, see the example below.


```{r}
msr("fairness.acc", operation = groupdiff_diff)
```

We can also report other metrics, e.g. the error in a specific group: 

```{r}
t = tsk("adult_train")$filter(1:1000)
mm = msr("fairness.acc", operation = function(x) {x["Female"]})
l = lrn("classif.rpart")
prds = l$train(t)$predict(t)
prds$score(mm, t)
```

## Individual Fairness

Individual fairness notions were first proposed by [@dwork2012].
The core idea comes from the principle of *treating similar cases similarly and different cases differently*.
In contrast to statistical group fairness notions, this notion allows assessing *fairness* at an individual level and 
would therefore allow determining whether an individual is treated fairly. 
A more in-depth treatment of individual fairness notions is given by [@binns2020apparent].

In order to translate this from an abstract concept into practice, we need to define two distance metrics:
- A distance metric $d(x_i, x_j)$ that measures how *similar* two cases $x_i$ and $x_j$ are
- A distance metric between treatments, here the predictions of our model $f$: $\phi(f({x}_i), f({x}_j))$.

Intuitively, we would now want, that if $d(x_i, x_j)$ is small, the difference in predictions  $\phi(f({x}_i), f({x}_j))$ should also be small.
This essentially requires Lipschitz continuity of $f$ with respect to $d$. 
Given a Lipschitz constant $L > 0$, we can write this as:

 $$ \phi(f({x}_i), f({x}_j)) \leq L \cdot d(x_i, x_j).$$

Currently, `mlr3fairness` does not support individual fairness metrics, but we aim to introduce such metrics in the future.


## Using metrics for non-mlr3 predictions

We can similarly employ mlr3 metrics on predictions stemming from different models.
To do so, we create a `data.table` containing the different components.

```{r}
# Get adult data as a data.table
train = tsk("adult_train")$data()
mod = rpart::rpart(target ~ ., train)

# Predict on test data
test = tsk("adult_test")$data()
yhat = predict(mod, test, type = "vector")

# Convert to a factor with the same levels
yhat = as.factor(yhat)
levels(yhat) = levels(test$target)

compute_metrics(
  data = test, 
  target = "target",
  prediction = yhat,
  protected_attribute = "sex",
  metrics = msr("fairness.acc")
)
```
