# Bayesian decision theory, the maximum local mass method, and color constancy

### Bayesian analysis of the problem a b = 1.

Assuming uniform prior probabilities over the graphed region, (a)
shows the posterior probability for gaussian observation noise of
variance $0.18$. The noise broadens the geometric solution into a
hyperbola--shaped ridge of maximum probability. (b) Note the
different thickness of the ridge; some parts have more local
probability mass than others, even though the entire ridge has a
constant maximum height.

### Three loss functions, and the expected loss

Left column: Three loss functions. Plots show penalty for
guessing parameter values offset from the actual value, taken to be
the plot center. (a) Minus delta function loss, assumed in MAP
estimation. Only {\em precisely} the correct answer matters. (b)
Squared error loss (a parabola), used in MMSE estimation. Very wrong
guesses can carry inordinate influence. (c) Minus local mass loss
function. Nearly correct answers are rewarded while all others carry
nearly equal penalty. Right column: Corresponding expected loss, or
Bayes risk, for the y = a b problem. Note: loss increases
vertically, to show extrema. (d) Expected loss for MAP estimator is
minus the posterior probability. There is no unique point of minimum
loss. (e) The minimum mean squared error estimate, (1.3, 1.3)
(arrow) does not lie along the ridge of solutions to a b = 1. (f)
The minus local mass loss favors the point (1.0, 1.0) (arrow), where
the ridge of high probability is widest. There is the most probability
mass in that local neighborhood.

### References

D. H. Brainard and W. T. Freeman,
Bayesian Color Constancy
, Journal of
the Optical Society of America, A, 14(7), pp. 1393-1411, July, 1997.
pdf file.

- W. T. Freeman and D. H. Brainard,
Bayesian decision theory, the
maximum local mass estimate, and color constancy, Fifth
Intl. Conference on Computer Vision, IEEE Computer Society, Cambridge,
MA, U.S.A, June, 1995, pp. 210 - 217.
TR94-23.

William T. Freeman
(freeman@merl.com)