Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities

Details Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities

2011-07-29

arxiv.org/abs/1107.6027v2

Computer Science

Information Theory

Submitted to IEEE Transactions on Information Theory

Scientific paper

In many signal detection and classification problems, we have knowledge of the distribution under each hypothesis, but not the prior probabilities. This paper is aimed at providing theory to quantify the performance of detection via estimating prior probabilities from either labeled or unlabeled training data. The error or {\em risk} is considered as a function of the prior probabilities. We show that the risk function is locally Lipschitz in the vicinity of the true prior probabilities, and the error of detectors based on estimated prior probabilities depends on the behavior of the risk function in this locality. In general, we show that the error of detectors based on the Maximum Likelihood Estimate (MLE) of the prior probabilities converges to the Bayes error at a rate of $n^{-1/2}$, where $n$ is the number of training data. If the behavior of the risk function is more favorable, then detectors based on the MLE have errors converging to the corresponding Bayes errors at optimal rates of the form $n^{-(1+\alpha)/2}$, where $\alpha>0$ is a parameter governing the behavior of the risk function with a typical value $\alpha = 1$. The limit $\alpha \rightarrow \infty$ corresponds to a situation where the risk function is flat near the true probabilities, and thus insensitive to small errors in the MLE; in this case the error of the detector based on the MLE converges to the Bayes error exponentially fast with $n$. We show the bounds are achievable no matter given labeled or unlabeled training data and are minimax-optimal in labeled case.

Affiliated with

Also associated with

No associations

Rating

Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-319579