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03 Mode-deviation penalizers

For the important exponential decay for squared deviation from mean, see exponential distribution family chapter.

Exponential decay and bilateral-exponential decay distributions are described elsewhere.

Inverse squared decay

Aka Cauchy distribution. Parameters: mean/ mode $\mean$, height-at-mean parameter $h$. The pdf is $C(x; \mean, h) = \pi^{-1} (\frac{h}{(x - \mean)^{2} + h^{2}})$.

Limited to positive deviation

Aka half-Cauchy distribution. Parameters: mean/ mode $m$, height-at-mean parameter $h$. Range of the random variable is $[m, \infty]$. The pdf is $C(x; \mean, h) = 2\pi^{-1} (\frac{h}{(x - m)^{2} + h^{2}})$.