Imagine we have a binary classification model: predicting if a person $x$ is
F (female) or
M (male). However, the target variable that we observe (say, $y$) is noisy. So there is some other variable that we don't observe with the true value (say, $g$)
We don't know $g$, but we know the conditional distribution $P(g \mid y)$:
- $P(g = F \mid y = F) = 0.76$, $P(g = M \mid y = F) = 0.24$
- $P(g = F \mid y = M) = 0.20$, $P(g = M \mid y = M) = 0.80$
Additionally, we know the marginal distribution of $y$:
- $P(y = F) = 1/3$, $P(y = M) = 2/3$
Now we train a model $P(y \mid x)$ on these $y$ (say, logistic regression).
How can we correct the output of the model to account for the noise in the data? That is, how do we get $P(g \mid x)$ from $P(y \mid x)$ using $P(y)$ and $P(g \mid y)$?