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In some machine learning models there is a predict_proba function for the overall class probability based on all of the features in the model. Is there a way to find the individual probability contribution of each feature? For example, if there is a 85% chance of raining based on three features: cloud coverage, humidity, and temperature. Is there a way to determine that dense cloud coverage by itself means there is a 60% chance of rain, high humidity means there is a 70% chance of rain, and high temperature means there is a 10% chance of rain, but these three features combined produce an 85% chance of rain? Looking for a general direction or materials to read. Thanks in advance.

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You can look at what is known as the "marginal effect" of variables. This doesn't quite give you what you want, because all marginal effects do not sum to the probability, but it is a way to say "how much does the probability change on average when this variable changes". This is a technique that can apply to models more generally, though you typically see them used with GLMs like logistic regression because people want to do inference on the quantity.

Working with your example, let's say one of our variables is a binary indicator for "Is Cloudy Outside". To compute the marginal effect of cloudiness, simple compute the average difference in predicted probabilities when all observations have the variable "Is Cloudy = 1" and when all observations have "Is Cloudy=0".

Mathematically, Let $f$ be a model which produces estimated probabilities $\hat{p}$ from a given set of data $\{ (\mathbf{x}_1, w_1, y_1), \dots, (\mathbf{x}_n, y_n) \}$. Here, $w$ is the variable for which we want to compute the marginal effect, $\mathbf{x}$ are other variables in the model, and $y$ is the label.

The marginal effect would then be

$$ \dfrac{1}{n} \sum f(\mathbf{x}_i, w_i=1) - f(\mathbf{x}_i, w_i=0) \>.$$

This is easy to do in code. Here is an example in R. Let's first make a problem

set.seed(0)
N <- 10000
x <- rbinom(N, ,m 0.5)
w <- rbinom(N, ,m 0.5)
y <- rbinom(N, 1, 0.25 + 0.05*x + 0.075*w)

From the code, we already know the marginal effect of $w$, its 0.075. So on average, changing $w$ from 0 to 1 will yield a 7.5% increase in probability. Let's fit a model to these data using fit <- glm(y~x+w, family = binomial()) and compute the marginal effect

p_w_1 <- predict(fit, newdata = list(x=x, w=rep(1,N)), type = 'response')
p_w_0 <- predict(fit, newdata = list(x=x, w=rep(0,N)), type = 'response')
effect = mean(p_w_1 - p_w_0)
effect
>[1] 0.06910256

Our estimated marginal effect of the variable $w$ is ~7%. This approach lines up with what libraries would give us

library(marginaleffects)
library(tidyverse)
marginaleffects(fit) %>% 
  summary()
 Term  Effect Std. Error z value   Pr(>|z|)   2.5 %  97.5 %
1    x 0.04832   0.009159   5.276 1.3218e-07 0.03037 0.06627
2    w 0.06895   0.009130   7.552 4.2956e-14 0.05105 0.08684

Model type:  glm 
Prediction type:  response

You could apply these approaches to any model you like to get an estimated marginal effect.

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  • $\begingroup$ Thank you! This is very helpful. Can this be used for continuous independent variables as well? Can it be applied to any model that has a predict_proba function, or only logistic regression? $\endgroup$
    – jw32022
    Commented Jan 3, 2023 at 18:43
  • $\begingroup$ @jw32022 This can be used for continuous independent variables as well, yes. This can be used with any model that produces a probability. You should probably make sure the probability estimates are calibrated, else the marginal effects may not make sense to interpret. $\endgroup$ Commented Jan 3, 2023 at 20:36
  • $\begingroup$ Thank you! Calibrated - is that like platt scaling? $\endgroup$
    – jw32022
    Commented Jan 3, 2023 at 21:08
  • $\begingroup$ Plat scaling is a mode of re-calibrating estimators. Calibration is related, I suggest you search for questions on that topic here. $\endgroup$ Commented Jan 3, 2023 at 21:15

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