/sci/ - Science & Math » Thread #11662136

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you should be able to solve this.

Anonymous Tue 12 May 04:05:22 2020 No.11662136 View Reply Original Report

Quoted By: >>11662143 >>11664138

Write $n$ iid draws of two RVs $(x,y)$ distributed over a finite set as $(x^n, y^n)$ . Fix $R>0$ . What is the min of $n^{-1}H(x^n|f(y^n))$ over $f$ with $H(f(y^n))\leq nR$ , taking $n\to \infty$ , where $H(\cdot|\cdot)$ denotes (conditional) Shannon entropy?

Anonymous

Anonymous Tue 12 May 2020 04:07:17 No.11662143 Report

Quoted By: >>11662164 >>11662512

>>11662136
50/50.
It either happens or it doesn’t.

Anonymous

Anonymous Tue 12 May 2020 04:08:59 No.11662147 Report

Quoted By: >>11662164

How to get /sci/ to do your homework:
step1 - write down the question
step2 - add something like "you should be able to solve this" or "If you can't solve this, you don't belong here"

Anonymous

Anonymous Tue 12 May 2020 04:13:49 No.11662164 Report

Quoted By:

>>11662147
if this is homework for any class you've taken then i would love to see those course notes

>>11662143
it's a nonrandom liminf, the answer will be some function of the joint distribution.

Anonymous

Anonymous Tue 12 May 2020 05:43:26 No.11662512 Report

Quoted By:

>>11662143

based

Anonymous

Anonymous Tue 12 May 2020 16:34:00 No.11664091 Report

Quoted By:

bump idiots

Anonymous

Anonymous Tue 12 May 2020 16:44:24 No.11664122 Report

Quoted By: >>11664460

by the way i believe this procedure for building $f$ will produce the right answer. Don't know what the construction causes the limit to calculate out to, and (critically) I don't have a converse that says you can't build significantly better $f$ but it's my best guess so far.

Form a pre-codebook $\mathcal{C}^\ast=\{c_\lambda\}_\lambda$ by randomly uniformly picking typical sequences for $y^n$ . Choose enough that $\{T(x^n|y^n=c_\lambda)\}_\lambda$ is with high probability a cover for most of $T(x^n)$ . Pick a random mapping $\phi:\mathcal{C}^\ast \to \{1,2,\dots, 2^{nR}\}$ . Then choose $f$ to map $\bar{y}^n$ to the $m$ that minimizes $\min_{c\in \phi^{-1}(m)} \left| T(x^n|y^n=c)\cap T(x^n|y^n=\bar{y}^n) \right|$

Anonymous

Anonymous Tue 12 May 2020 16:49:03 No.11664138 Report

Quoted By: >>11664143

>>11662136
Are we supposed to find the optimal function f?

Anonymous

Anonymous Tue 12 May 2020 16:50:19 No.11664143 Report

Quoted By:

>>11664138
all that matters in the end is the limit. finding a negligibly-suboptimal f is just a potential means for computing the limit.

Anonymous

Anonymous Tue 12 May 2020 18:25:45 No.11664460 Report

Quoted By:

>>11664122
Nah this is probably wrong

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