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
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 over the interval from 0 to 1. One could use the estimator
or one could use the biased estimator
The first estimator is in fact unbiased but has variance with order . The second estimator’s expected value is
so it is biased, although its variance is , 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
where is a final pixel value, is the pixel filter function (which we assume here to be normalized to integrate to 1), and 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 . Thus, the unbiased Monte Carlo estimator of this equation is
This gives a different result from that of the pixel filtering equation we used previously, Equation (7.12), which was
Yet, the biased estimator is preferable in practice because it gives a result with less variance. For example, if all radiance values 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
will generally not be equal to and thus will have a different value due to variation in the filter function depending on the particular 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).