This estimator is consistent because as we take more samples, the error becomes zero. However, why is it biased?

dvernet

I believe that it's biased because we're taking samples at fixed points over the image rather than at points that are randomly distributed. Even though it seems intuitively correct to average the samples as we do here, it is not true that $E[ I - \hat{I_{n}}] = 0$ precisely because we fix the points. If we chose them uniformly at random, the EV would equal 0.

PandaX

@dvernet Thanks, the explanation is good!

kmcrane

Right. In fact, if we picked random samples within each box, then we would basically be implementing the stratified sampling strategy described later in the slides. On this slide, however, we are picking the same fixed samples every time, which makes the estimator biased.

ak-47

Is this accurate?
"If your estimator has no randomness, it is either completely correct 100% of the time, or it is biased."

kmcrane

@ak-47: Hmm... for a finite number of samples I would say it's completely wrong 100% of the time. :-)

This estimator is consistent because as we take more samples, the error becomes zero. However, why is it biased?

I believe that it's biased because we're taking samples at fixed points over the image rather than at points that are randomly distributed. Even though it seems intuitively correct to average the samples as we do here, it is not true that $E[ I - \hat{I_{n}}] = 0$ precisely because we fix the points. If we chose them uniformly at random, the EV would equal 0.

@dvernet Thanks, the explanation is good!

Right. In fact, if we picked random samples

within each box, then we would basically be implementing the stratified sampling strategy described later in the slides. On this slide, however, we are picking the same fixed samples every time, which makes the estimator biased.Is this accurate? "If your estimator has no randomness, it is either completely correct 100% of the time, or it is biased."

@ak-47: Hmm... for a finite number of samples I would say it's completely wrong 100% of the time. :-)