In [1]:
%matplotlib inline
from IPython.display import display, Markdown, Latex
import numpy as np
import sklearn
import matplotlib
import matplotlib.pylab as plt
import sklearn.linear_model
import seaborn
import scipy.special
seaborn.set(rc={"figure.figsize": (8, 6)}, font_scale=1.5)
Let's generate some data with a simple model. There's a binary sensitive attribute $A$, predictor1 $p_1$ uncorrelated with the attribute, predictor2 $p_2$ correlated with the attribute, random noise $\epsilon$, and a binary label $y$ that's correlated with both predictors.
where
$$p1 \sim \mathrm{Normal}(0,1)$$$$p2 \sim \mathrm{Normal}(0,1) \iff A=a$$$$p2 \sim \mathrm{Normal}(1,1) \iff A=b$$$$\epsilon \sim \mathrm{Normal}(0,1)$$
In [2]:
n = 10000 # Sample size.
# Create an attribute.
attribute = np.choose(np.random.rand(n) > 0.5, ['a', 'b'])
# Create an uncorrelated predictor.
predictor1 = np.random.randn(n)
# Create a predictor correlated with the attribute.
disparity_scale = 1.0
predictor2 = np.random.randn(n) + ((attribute == 'b') * disparity_scale)
# Generate random noise.
noise_scale = 1.0
noise = np.random.randn(n) * noise_scale
# Calculate the probability of the binary label.
scale = 1.0
p_outcome = scipy.special.expit(scale * (predictor1 + predictor2 + noise))
# Calculate the outcome.
y = p_outcome > np.random.rand(n)
In [3]:
seaborn.set(rc={"figure.figsize": (8, 6)}, font_scale=1.5)
c0 = seaborn.color_palette()[0]
c1 = seaborn.color_palette()[2]
plt.figure(figsize=(8,6))
plt.bar([0], [y[attribute == 'a'].mean()], fc=c0)
plt.bar([1], [y[attribute == 'b'].mean()], fc=c1)
plt.xticks([0.0,1.0], ['$A=a$', '$A=b$'])
plt.xlabel("Attribute Value")
plt.ylabel("p(label=True)")
plt.ylim(0,1);
plt.figure()
plt.scatter(predictor1[attribute == 'b'], p_outcome[attribute == 'b'], c=c1, alpha=0.25, label='A=b')
plt.scatter(predictor1[attribute == 'a'], p_outcome[attribute == 'a'], c=c0, alpha=0.25, label='A=a')
plt.xlabel('predictor1')
plt.ylabel('p(label=True)')
plt.legend(loc='best')
plt.figure()
plt.scatter(predictor2[attribute == 'b'], p_outcome[attribute == 'b'], c=c1, alpha=0.25, label='A=b')
plt.scatter(predictor2[attribute == 'a'], p_outcome[attribute == 'a'], c=c0, alpha=0.25, label='A=a')
plt.xlabel('predictor2')
plt.ylabel('p(label=True)')
plt.legend(loc='best');
Above you can see that the probability of (actually) being in the positive class is correlated with the attribute and with both of the predictors.
In [13]:
display(Markdown("Condition Positive Fraction for each attribute class: (a,b): {:.3f} {:.3f}".format(
y[attribute == 'a'].mean(), y[attribute == 'b'].mean())))
display(Markdown(
"Members of the $A=b$ group are {:.0f}% more likely to have the positive label than $a$ group.".format(
100.0 * (y[attribute == 'b'].mean()/y[attribute == 'a'].mean() - 1.0))))
In [5]:
# Put the predictors into the expected sklearn format.
X = np.vstack([predictor1, predictor2]).T
# Initialize our logistic regression classifier.
clf = sklearn.linear_model.LogisticRegression()
# Perform the fit.
clf.fit(X, y)
# Model fit parameters:
clf.intercept_, clf.coef_
Out[5]:
Now generate predictions from the model.
In [6]:
p = clf.predict_proba(X)[:,1]
yhat = p > 0.5
In [7]:
plt.figure(figsize=(8,6))
plt.bar([0, 2], [y[attribute == 'a'].mean(), y[attribute == 'b'].mean()],
fc=c0, label='Actual Labels')
plt.bar([1, 3], [yhat[attribute == 'a'].mean(), yhat[attribute == 'b'].mean()],
fc=c1, label='Predicted Labels')
plt.xticks([0.0, 1.0, 2.0, 3.0], ['$A=a$ \n actual', '$A=a$ \n pred', '$A=b$ \n actual', '$A=b$ \n pred'])
plt.ylabel("Positive Label Fraction")
plt.ylim(0,1)
plt.legend(loc='best');
In [8]:
display(Markdown("Condition Positive fraction for each attribute class: (a,b): {:.0f}% {:.0f}%".format(
100.0 * y[attribute == 'a'].mean(), 100.0 * y[attribute == 'b'].mean())))
display(Markdown("Predicted Positive fraction for each attribute class: (a,b): {:.0f}% {:.0f}%".format(
100.0 * yhat[attribute == 'a'].mean(), 100.0 * yhat[attribute == 'b'].mean())))
display(Markdown("""
So the initial {:.0f}% disparity in the _actual_ labels is amplified by the model.
**Members of the $A=b$ group are {:.0f}% more likely to have the positive _predicted_ label than $b$ group.**
The model has amplified the initial disparity by a factor of {:.2f}.
""".format(
100.0 * (y[attribute == 'b'].mean()/y[attribute == 'a'].mean() - 1.0),
100.0 * (yhat[attribute == 'b'].mean()/yhat[attribute == 'a'].mean() - 1.0),
(yhat[attribute == 'b'].mean()/yhat[attribute == 'a'].mean()) /
(y[attribute == 'b'].mean()/y[attribute == 'a'].mean())
)))
In [9]:
fpr_all, tpr_all, t_all = sklearn.metrics.roc_curve(y, p)
fpr_falseclass, tpr_falseclass, t_falseclass = sklearn.metrics.roc_curve(y[attribute == 'a'], p[attribute == 'a'])
fpr_trueclass, tpr_trueclass, t_trueclass = sklearn.metrics.roc_curve(y[attribute == 'b'], p[attribute == 'b'])
plt.plot(fpr_falseclass, tpr_falseclass, label='attribute=a', alpha=0.5, lw=3)
plt.plot(fpr_all, tpr_all, label='all', alpha=0.5, lw=3)
plt.plot(fpr_trueclass, tpr_trueclass, label='attribute=b', alpha=0.5, lw=3)
plt.legend(loc='best')
plt.xlabel("FPR")
plt.ylabel("TRP");
In [10]:
plt.plot(t_falseclass, tpr_falseclass, label='attribute=a', lw=3)
plt.plot(t_all, tpr_all, label='all', lw=3)
plt.plot(t_trueclass, tpr_trueclass, label='attribute=b', lw=3)
plt.legend(loc='best')
plt.xlabel("threshold")
plt.ylabel("TPR");
So it looks like the model will actually perform better in terms of TPR (aka recall) for group $A=b$.
Now let's check the false positive rate $P(\hat{y}=0 \vert y=1)$ vs the score threshold:
In [11]:
plt.plot(t_falseclass, fpr_falseclass, label='attribute=a', lw=3)
plt.plot(t_all, fpr_all, label='all', lw=3)
plt.plot(t_trueclass, fpr_trueclass, label='attribute=b', lw=3)
plt.legend(loc='best')
plt.xlabel("threshold")
plt.ylabel("FPR");
We find that the false positive rate is much higher at all thresholds for the $A=b$ group. If the negative class is preferred (e.g. in a model predicting fraud, spam, defaulting on a loan, etc.), that means we're much more likely to falsely classify an actually-good member of group $b$ as bad, compared to a actually-good member of group $a$.
In [12]:
def fpr(y_true, y_pred):
fp = (np.logical_not(y_true) & y_pred).sum()
cn = np.logical_not(y_true).sum()
return 100.0 * fp * 1.0 / cn
display(Markdown("""
At a threshold of model score $= 0.5$, the false positive rate is {:.0f}% overall, **{:.0f}% for group $a$,
and {:.0f}% for group $b$**.
""".format(
fpr(y, yhat), fpr(y[attribute=='a'], yhat[attribute=='a']), fpr(y[attribute=='b'], yhat[attribute=='b']))))
We've demonstrated that a model trained on correct labels, and with no directly access to a particular attribute, can be biased against members a group who deserve the preferred label, merely because that group has a higher incidence of the non-prefered label in the training data. Furthermore, this bias can be hidden, because it's only revealed by comparing false positive rates at a fixed threshold, not other performance metrics.