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Transforming an "Uninformative" Distribution

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This Demonstration shows the effect of transforming a uniformly distributed variable. Naively we may choose to use the uniform distribution to represent a state of no information, perhaps as an uninformative prior for Bayesian inference. However, if
X
is uniformly distributed, even simple transformations of
X
may not be—in this example the phenomenon is illustrated with powers of
X
. To give a concrete example, suppose
X
represents the radius of a circle. Then if
X
is uniformly distributed (we might say we were completely "uninformed" about
X
) then we know something about the area of the circle (the distribution of
X
squared is not uniform)! This suggests that the concept of an "uninformative" distribution is not as simple or clearly defined as it first appears.

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