# Binary log likelihood when there is a misclassification issue

$t$ is a binary target variable equal to $0$ or $1$, and below are the binary distribution and the log-likelihood error function.

$p(t|x,w)=y(x,w)^t(1-y(x,w))^{1-t}$

$E(W)=-\displaystyle\sum_{n=1}^N (t_n \ln y_n + (1-t_n) \ln (1-y_n))$

But how should I write down the error function if there is a misclassification problem on a training data set? The exact wording is:

There is a probability $\epsilon$ that the class label on a training data point has been incorrectly set.

It's said that the error function above is obtained when $\varepsilon$ = 0.

Would appreciate any help or hint.

For each sample $i$, let $m_i=T$ if there is a misclassification and $m_i=F$ otherwise. Then:
$$p(t|x,w)=p(t|x_i,w,m_i=F)P(m_i=F)+p(t|x,w,m_i=T)P(m_i=T).$$
Then the case $m_i=T$ flips $y_i$ to $1-y_i$. Can you finish it from here?