Go to:Table of Contents

Entropy of kind t

Arimoto (1971) [3] considered generalized $f-$entropies involving a real function $f$ with some conditions (see section 3.5.4). Being an example of these generalized $f-$entropies, Arimoto came up to a generalized entropy involving a real parameter, here we call it, entropy of kind t, given by

$$_tH(P)={(2^{t-1}-1)}^{-1}\bigg[{\bigg(\sum_{i=1}^n{p^{1/t}_i}\bigg)}^t -1\bigg], \ t\neq 1,\ t>0,$$ (3.5)

for all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$. In this case also, we can easily verify that $\lim_{t\to 1}{_tH(P)}=H(P).$

Arimoto's main motivation in considering generalized$f-$entropies was to prove some important results on decision theory connected with bayesian probability of error.