Distribution of misclassified testing samples Suppose we train some classifier $f$ on samples from an unknown distribution $\mathbb{P}$ via empirical risk minimization or some other technique. What is known about the distribution of misclassified samples? Presumably it is not also $\mathbb{P}$.
 A: I will first classify the setting that I think, you are talking about. I hope, I am right. We have a classifier $f$ that classifies according to some true classification function $F$. Hence, $f$ is constructed such that $F \approx f$ in terms of empirical risk. We now sample $x \sim \mathbb{P}$ and you classify $x$, hence $F(x) \neq f(x)$. 
Question: What is the conditional distribution of $x$ given that $F(x) \neq f(x)$? In particular, what is $\mathbb{P}(x \in \cdot | F(x) \neq f(x))$
That is a question that we can try to approach with Bayesian statistics. In that case, we need to know $\mathbb{P}$ though, which is the prior measure here. This I hope is not a too strong assumption. $\mathbb{P}$ has probability density function $\pi_0$. I propose the following approach:
What is a suitable likelihood in this setting? We can use the error measure $\mathrm{Error}(F,f,x)$ to construct it. The likelihood of $F = f$ given a certain $x$ is then $$\exp\left(-\mathrm{Error}(F,f,x)\right)$$
But we need the likelihood of $F \neq f$ at a certain $x$. We can model it as follows Let some $M \in \mathbb{R}$ exist that is the maximum of $\mathrm{Error}(\cdot,\cdot,\cdot)$. Then we consider 
$$\exp\left(-M+\mathrm{Error}(F,f,x)\right)$$ as the wanted likelihood. 
Bayes formula tells us that the probability density function $\pi_p$ of the posterior measure $\mathbb{P}(x \in \cdot | F(x) \neq f(x))$ is given by:
\begin{align}
\pi_p(x) = C \cdot f(x) \exp\left(-M+\mathrm{Error}(F,f,x)\right) \propto \pi_0(x) \exp\left(\mathrm{Error}(F,f,x)\right),
\end{align}
since $C, M$ are independent of $x$. 
You can now use some Markov chain Monte Carlo method to approximate $\mathbb{P}(x \in \cdot | F(x) \neq f(x))$. In case you do, I would be very interested in the results. Please let me know. :)
