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 $\mu$, height-at-mean parameter $h$. The pdf is $C(x;\mu ,h)={\pi }^{-1}(\frac{h}{(x-\mu {)}^2+h^2})$.

Limited to positive deviation

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