Question

Category A A1. Prove that the entropy function Hin satisfies the grouping property H.(P1, ...,Pm) = Hm-1 (P1 + P2, P3, ...,Pm) + (P1 + P2) H2(- Pi + P2 Pi + P2

Answer 1

Here we assume that $H_m$ is defined as:

ed 607:07 -= (und 6.61d) “H

Let us start from the RHS as:

= Hn-1(P1 + P2, P3, ...., Pm) + (p1 + p2) H (_P1, P2 P1 + P2 p1 + P2

Expanding the above by the definition of $H_m$:

= Hm-1 (pı+P2, P3, ..., Pm)+(p1+p2) - Pl_log (_P1_) - P2-log_P2 = im-1 (p1+P2, P3, ..... Pm) Pl+P P1 + P2 P1 + P2 P1 + P2 P 1 + P2)

Cancellation of
P1 + pz
from numerator and denominator:

1 P2 = Hm-1(pi + p2, P3, ...., Pm) + -pi log P1 - P1 + P2 -p2 log P1 + P2)

= Hm-1(P1+P2, P3, ...., Pm)+1-P1 log p1 + pı log (P1 + P2) - p2log P2 + pa log (p1 + P2)

= Hm-1 (P1+P2, P3, ....,Pm)+[-P1 log P1 - p2log P2 + (p1 + p2) log (P1 + P2)

= [-(p1 + p2) log (P1 + P2) - P3 logP3, ...., Pmlog Pm + [-pı log pi - pa log p2 + (p1 + p2) log (p1 + p2)]

Cancellation of
(p1 +p2) log (p1 + P2)
from the two bracketed expressions and collecting rest of the terms:

= 1-pilog P1 - p2 log P2 - P3 log P3, ...., Pmlog Pm

= Hm (P1. ....pm)

Hence proved!!!!!!!!

hope this helps!

*********** PLEASE THUMBS UP!!!!!!!!! ***********