13.9 Bias

Another approach to variance reduction is to introduce bias into the computation: sometimes knowingly computing an estimate that doesn’t actually have an expected value equal to the desired quantity can nonetheless lead to lower variance. An estimator is unbiased if its expected value is equal to the correct answer. If not, the difference

beta equals upper E left-bracket upper F right-bracket minus integral f left-parenthesis x right-parenthesis normal d x

is the amount of bias.

Kalos and Whitlock (1986, pp. 36–37) gave the following example of how bias can sometimes be desirable. Consider the problem of computing an estimate of the mean value of a uniform distribution upper X Subscript i Baseline tilde p over the interval from 0 to 1. One could use the estimator

StartFraction 1 Over upper N EndFraction sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper X Subscript i Baseline comma

or one could use the biased estimator

one-half max left-parenthesis upper X 1 comma upper X 2 comma ellipsis comma upper X Subscript upper N Baseline right-parenthesis period

The first estimator is in fact unbiased but has variance with order upper O left-parenthesis upper N Superscript negative 1 Baseline right-parenthesis . The second estimator’s expected value is

0.5 StartFraction upper N Over upper N plus 1 EndFraction not-equals 0.5 comma

so it is biased, although its variance is upper O left-parenthesis upper N Superscript negative 2 Baseline right-parenthesis , which is much better.

The pixel reconstruction method described in Section 7.8 can also be seen as a biased estimator. Considering pixel reconstruction as a Monte Carlo estimation problem, we’d like to compute an estimate of

upper I left-parenthesis x comma y right-parenthesis equals double-integral f left-parenthesis x minus x Superscript prime Baseline comma y minus y Superscript prime Baseline right-parenthesis upper L left-parenthesis x prime comma y Superscript prime Baseline right-parenthesis normal d x prime normal d y Superscript prime Baseline comma

where upper I left-parenthesis x comma y right-parenthesis is a final pixel value, f left-parenthesis x comma y right-parenthesis is the pixel filter function (which we assume here to be normalized to integrate to 1), and upper L left-parenthesis x comma y right-parenthesis is the image radiance function.

Assuming we have chosen image plane samples uniformly, all samples have the same probability density, which we will denote by p Subscript c . Thus, the unbiased Monte Carlo estimator of this equation is

upper I left-parenthesis x comma y right-parenthesis almost-equals StartFraction 1 Over upper N p Subscript c Baseline EndFraction sigma-summation Underscript i equals 1 Overscript upper N Endscripts f left-parenthesis x minus x Subscript i Baseline comma y minus y Subscript i Baseline right-parenthesis upper L left-parenthesis x Subscript i Baseline comma y Subscript i Baseline right-parenthesis period

This gives a different result from that of the pixel filtering equation we used previously, Equation (7.12), which was

upper I left-parenthesis x comma y right-parenthesis equals StartFraction sigma-summation Underscript i Endscripts f left-parenthesis x minus x Subscript i Baseline comma y minus y Subscript i Baseline right-parenthesis upper L left-parenthesis x Subscript i Baseline comma y Subscript i Baseline right-parenthesis Over sigma-summation Underscript i Endscripts f left-parenthesis x minus x Subscript i Baseline comma y minus y Subscript i Baseline right-parenthesis EndFraction period

Yet, the biased estimator is preferable in practice because it gives a result with less variance. For example, if all radiance values upper L left-parenthesis x Subscript i Baseline comma y Subscript i Baseline right-parenthesis have a value of 1, the biased estimator will always reconstruct an image where all pixel values are exactly 1—clearly a desirable property. However, the unbiased estimator will reconstruct pixel values that are not all 1, since the sum

sigma-summation Underscript i Endscripts f left-parenthesis x minus x Subscript i Baseline comma y minus y Subscript i Baseline right-parenthesis

will generally not be equal to p Subscript c and thus will have a different value due to variation in the filter function depending on the particular left-parenthesis x Subscript i Baseline comma y Subscript i Baseline right-parenthesis sample positions used for the pixel. Thus, the variance due to this effect leads to an undesirable result in the final image. Even for more complex images, the variance that would be introduced by the unbiased estimator is a more objectionable artifact than the bias from Equation (7.12).