# AUC variance for estimators with randomness (e.g. Random Forest)

I know that we can calculate the standard error for the AUC for all estimators, assuming that the conditional density is fixed. What I'd like to do, however, is additionally account for the randomness in certain estimators such as random Forest. I know that we can do a nonparametric bootstrap, where we resample and refit the estimator each time. However, I'm looking for a less computationally expensive approach.

Thanks!

I would just model the fact that each tree in a binary random forest votes for class 0 or class 1. This means that for $T$ trees, you have $T$ Bernoulli trials for each sample, and each sample $i$ has a different probability $p_i$ of being classified as 1 by a tree. These Bernoulli trials are IID because trees in a random forest are IID. A tree votes for class 1 some number of times $0\le k_i \le T$.
Adopting a Bayesian framework, we can leverage the beta-binomial model. The posterior distribution over $p_i$ for each sample is
$$p_i \sim \text{Beta}(\alpha+k_i, \beta + T-k_i)$$ where $(\alpha,\beta)$ is a prior belief about $p_i$. I wouldn't worry over choosing $(\alpha,\beta)$ too much; a uniform prior of $\alpha=\beta=1$ implies two "psuedo-RF-trees" voted for class 0 and class 1 once for each sample.
The rest is just monte carlo: for each $i$, sample from the posterior distribution and compute the ROC curve. Repeat this $M$ times, and you have a distribution of $M$ ROC curves drawn from this model.