Decentralized Processing: An Information Theoretic Perspective

Decentralized Processing: 

An Information Theoretic Perspective 

Shlomo Shamai, , EE. Dept. Technion, Israel 

sshlomo@ee.technion.ac.il 

Communications Sciences Institute 

Department of Electrical Engineering 

USC Viterbi School of Engineering 

February 2, 2007 

Joint work with Amichai Sanderovich, Yossef Steinberg and 

Michael Peleg, , EE, Technion. 

1

Overview of presentation 

Decentralized Processing: Overview 

 

 

Nomadic transmitter and remote destination. 

Simple access points (agents) that encode but do not decode. 

Information theoretic upper and lower bounds. 

The Gaussian scalar channel – capacity through EPI. 

Decentralized MIMO: Multiple antenna Tx: 

 

 

 

Two schemes: Simple and Wyner-Ziv 

compression. 

Multiplexing gain of the schemes. 

Upper bounds for block fading. 

Impact of common feedback – cooperation mode. 

 

 

Two nomadic schemes. 

Limited benefits of the feedback – due to the nomadic setting. 

2

Decentralized Processing: 

Nomadic transmitter 

Overview 

Agent 

A1 

Transmitter 

S 

X 

Channel 

P(Y1,Y2|X) 

Y1 

Agent 

C1 

Destination 

D 

A2 

Y2 

C2 

3

Scheme description 

Memoryless broadcast channel without 

n 

feedback: 

n n n 

P( Y1 ,…, YT | X ) = ∏ P( Y1, i, …, YT , i 

| X 

i 

) 

i= 

1 

Connected via T (here T=2) non-interfering and 

reliable links with bandwidths 

C1, …, 

CT 

bits/channel use. 

One shot transmission (one block), but extends 

to steady-state state operation. 

Nomadic transmitter can not adapt to the 

different access points (agents), where each 

may use different ways of forwarding. 

Nomadic transmitter the agents lack the 

codebook knowledge. 

4

Scheme description 

Agents: 

Agents observe a memoryless noisy source 

similar to the CEO problem. 

Each agent encodes n ≤ k channel outputs 

into an index 

{ 

nC 

( 

n 

V 1, , 2 t 

} 

t 

∈ … , Vt = φt Yt 

). 

Each agent sends the k / n indices to the 

destination. 

Final destination: 

Decodes the transmitted message from the 

k / n 

received indices . 

{ } 

V 

1, … , T 

5

Problem statement 

Nomadic transmitters: 

 

Agents are ignorant of code-book structure. 

Objective: maximize the total achievable 

rate of the scheme over 

 

 

 

Agents encoding functions. 

Single letter distribution used for codebook 

selection. 

Decoding function. 

6

Some Relevant Literature 

Distributed source coding are directly applicable: 

 

 

 

 

 

 

 

The CEO problem [Berger, Zhang and Viswanathan 1996]. 

Upper bound on the sum rate distortion function for the CEO 

problem [Chen, Zhang, Berger and Wicker 2004]. 

Distributed Wyner-Ziv 

source coding over tree structure [Draper 

and Wornell 2004]. 

Rate Region of Quadratic Gaussian CEO [Oohama 

2005]. 

Network Vector Quantization [Fleming, Zhao and Effros 2004]. 

Rate Region of the Quadratic Gaussian two terminals source 

coding [Wagner, Tavildar and Viswanath2006]. 

Backward channels, Distributed WZ [Servetto 

2006]. 

7

Relevant literature 

Relevant channel coding problems: 

 

 

 

 

 

Parallel relay scheme [Schein/Gallager 

2001]. 

Relay channel [Cover and El-Gamal 

1979]. 

Multihopping for relay networks [Kramer, 

Gastpar and Gupta 2004]. 

User cooperation [Sendonaris, Erkip and 

Aazhang 2003]. 

MIMO relay problems [Wang, Zhang,Høst 

st- 

Madsen 2005]. 

8

Achievable rate 

Agents compress their received signal: 

( U Y ) 

t 

, ∈T 

where Ut 

depends only on Yt 

, i.e. the 

following are Markov chains: 

{ } 

Transmit U , 1 

… , UT 

to the destination, 

save rate by using the correlations 

between them (Wyner( 

Wyner-Ziv 

coding). 

t 

ε 

( , , 

) 

U −Y − X Y 

… … 

U 

… … 

t t 1, , t− 1, t+ 1, , T 1, , t− 1, t+ 

1, , T 

9



R < max I( X ; U1, …, U ) 

T 

T 

P( X ) P( Y1 

,…, Y | X ) ∏P( U | Y ) 

T t t 

t= 

1 

Where 

∑ 

t∈S 

C > I( U ; Y | U ) 

t S S S C 

for all 

S 

⊆ 

{ 1, …, 

T} 

10

The Gaussian channel 

For a Gaussian channel with: 

The capacity is 

2 2 

Y = X + N , E X = P , E N = P 

t t X t N t 

1 ⎛ 1− 

2 

R ≤ max min ∑[ C − r ] + log 1+ 

P ∑ 

0≤rt ≤Ct S⊆{ 1, …, 

T} ⎜ 

C 

⎝ 

t t 2 X 

t∈S 2 

t∈S PN 

Proof: through entropy power inequality and the 

contra-polymatroid 

region for the achievable 

rates [Tse 

et. al. 2004, Oohama 2005]. 

r t can be interpreted as the bandwidth used for 

noise quantization. 

−2r 

t 

t 

⎞ 

⎟ 

⎠ 

11


For symmetric agents, analytic solution 

gives: 

R 

1 ⎛ ⎛ 

= log ⎜ 1 + 2 SNR 

1 − 

2 ⎜ ⎜ 

2 

⎝ ⎝ 

Asymptotics: 

2 4C 

SNR + 2 (1 + 2 SNR) 

− SNR 

2 4C 

⎧SNR → ∞ : R = 2C 

⎪ 

⎨ 1 

⎪C → ∞ : R = log2 

1 + 2SNR 

⎩ 

2 

( ) 

⎞⎞ 

⎟ 

⎟⎟ 

⎠⎠ 

12


13

Decentralized MIMO: Scheme 

description 

Nomadic multi antenna transmitter 

Agent 

A1 

Transmitter 

S 

X1 

X2 

Y1 

Agent 

C1 

Destination 

D 

Y2 

A2 

C2 

14

Decentralized MIMO: Scheme 

description 

MIMO channel: 

t Tx antennas and agents with single antenna. 

Rayleigh fading channel: 

where 

Y ∈ C 

( ) 

h ∼ C N 0, I , i = 1, …, r, 

i 

Y = HX + N 

[ r ,1] , 

t 

r 

E ⎡ 

⎣ X X ⎤ 

⎦ ≤ P, 

[ t ,1] * 

X ∈ C , 

( ) 

H fully known to both agents and remote destination, 

can be fast fading (Shannon capacity), 

and block fading channel (average rate). 

Connected to the remote destination via orthogonal 

reliable links with bandwidths bits/channel use. 

C , , 1 

… Cr 

[ , , , ] 

T 

H = h1 h2 

… h r 

N ∼ CN 0, I 

r 

15

Relevant literature – Distributed 

MIMO 

Relevant channel coding problems: 

 

 

 

 

 

 

Gaussian MIMO channels [Telatar 

1999]. 

MIMO broadcast [H. Weingarten, Y. Steinberg, and S. 

Shamai 2004]. 

Capacity scaling laws in MIMO relay networks 

[Bolcskei, Nabar, Oyman and Paulraj 2006]. 

Capacity-achieving achieving input covariance for single-user 

MIMO [Tulino,, Lozano and Verdu 2006]. 

Quadratic gaussian two-terminal terminal source-coding 

coding 

[Wagner, Tavildar and Viswanath 2005]. 

Gaussian Many-Help 

Help-One Problem [Tavildar, 

Viswanath and Wagner 2007]. 

16

Achievable rate – simple 

compression 

Each agent needs to know only its own fading h i 

. 

Destination knows H. 

Agents compress their received signal ( i is the 

agent index): n 

n 

Y i 

to the codewords U i 

, 

nC 

Codebook size is 2 . 

i 

is defined by: 

U = Y + d , d ∼ N 0, P 

U i 

( ) 

i i i i D i 

P 

D 

i 

= 

h 

i 

2 

P / t + 1 

C 

2 −1 

17

Achievable rate – simple 

compression 

{ } 

Transmit U , 1 

… , Ur 

to the destination. 

Simple compression: no use of dependencies of 

{ U , 1 

… , U 

r} 

. 

{ } 

Then uses U , 1 

… , U 

r 

to decode X. 

The achievable rate is: 

⎛ ⎛ 1 

⎞ 

⎜ ⎜ 

P 1 

⎟ 

⎜ 

D 

+ 

⎜ 1 

⎟ 

P 

RSC = EH log 

2 

det ⎜ Ir 

+ ⎜ ⋱ ⎟ HH 

⎜ t ⎜ 

⎟ 

⎜ ⎜ 

1 ⎟ 

⎜ ⎜ 

PD 

+ 1⎟ 

⎝ ⎝ 

r ⎠ 

* 

⎞ 

⎟ 

⎟ 

⎟ 

⎟ 

⎟ 

⎟ 

⎠ 

18

Achievable rate WZ compression 

Use the dependencies between receptions to increase 

compression efficiency. 

Requires the knowledge of H in all agents. 

Define 

Only now: 

U = Y + d 

i i i 

d 

i 

∼ 

, 

1 

N 0, , 1 2 

i 

P + 1 

( ) 

−ri 

( H ) 

PD 

= − 

D 

i 

 

( ) 

r H 

i is bandwidth wasted on noise compression (this is 

due to nomadic assumption: no decoding, so only source 

coding possible [Oohama2005]). 

19

Achievable rate WZ compression – 

The achievable rate: 

cont. 

( 

r 

{ ( )} 

) 

i= 

1 

RWZ = E 

⎡ 

H 

max min RS ri H , H 

S 

, C 

⎢⎣ 

0≤ri 

( H ) ≤C 

S⊆{ 1, …, 

r} 

( 

r 

{ ( )} 

, , ) ≜ ⎡ − ( ) 

i= 

1 

∑ ⎣ 

R r H H C C r H 

S i S i 

C 

i∈S 

⎛ 

P ⎛ ⎧⎪ 

1 ⎫⎪ 

+ log2 

det ⎜ I + diag 

S ⎨ ⎬ 

⎜ t ⎜ ⎪ PD 

+ 1 

i 

⎝ ⎝ ⎩ ⎪⎭ 

⎤⎦ 

⎤ 

⎥⎦ 

i∈S 

⎞ 

H 

S 

H 

⎟ 

⎠ 

* 

S 

⎞ 

⎟ 

⎟ 

⎠ 

Notice that i 

is the bandwidth left for signal 

transmission. 

( ) 

H ≜ H i, j , i ∈ S, j = 1, …, 

t 

S 

( ) 

C − r H 

20

Achievable rate WZ compression – 

cont. 

( ) 

{ } 1 

The optimization over ri 

H in WZ 

i = 

compression needs to be performed for any 

The optimization is concave, and thus can be 

performed efficiently. 

As t → ∞, the channel matrix hardens, and 

no need for H in the agents, since it is 

required only for the WZ binning resolution. 

r 

H 

21

Multiplexing Gain 

Multiplexing gain = 

( ) 

( P) 

R P 

lim 

P→∞ 

log2 

For simple compression, when t ≥ r and 

C = 

the compression noise di 

in the definition of 

is with power 

2 2 

hi 

P / t + 1 hi 

P / t + 1 

PD 

= = 

i C 

2 −1 C= 

log( P) 

P −1 

Upper bounded by: 

P 

* 

2 2 

hi 

P / t + 1 hi 

P / t + 1 

= max 

= max 

i 

C 

2 −1 = log( ) i P −1 

D C P 

log 2 

Ui 

( P) 

22


Thus full multiplexing Gain is attained (inequality( 

since we use the largest compression noise, there are 

others smaller) 

⎛ r 

P 1 ⎞ ⎛ 

* 

Pλi 

/ t ⎞ 

RSC ≥ EH log2 det Ir 

+ HH = ∑ Eλ 

log 

2 

1+ 

⎜ i 

t 1 P ⎟ ⎜ * i 1 

1 P ⎟ 

* 

⎝ + 

D ⎠ = ⎝ + 

D ⎠ 

( ) { } 

* 

un − ordered eignvalues HH = λi 

RSC 

( P) 

lim ≥ r 

P→∞ 

log 

2 ( P) 

When t < r, only subset of t agents can be 

used. 

23


Wyner-Ziv 

compression is always better than the 

simple compression, and thus, trivially: 

2 

( P) 

( P) 

RWZ 

lim m, 

P→∞ 

log 

≥ m ≜ min { r, 

t} 

24

Nomadic joint UB 

For upper bounds on average rate with 

block fading (not valid for fast fading): 

Assume that the agents can fully 

cooperate. 

Equivalent to m parallel channels, with 

signal to noise ratios: 

{ λ P t i } 

results in almost simple MIMO. 

In [SSSK2005] there was no H. We copy that 

result for a fixed H: 

⎛ 

R ( H ) ≤ max log2 

⎜1 + Pλi 

/ t 

Bi 

2 −1 

⎞ 

Bi 

2 + Pλ 

/ t 

∑ 

⎟ 

∑Bi 

≤Ctotal 

⎝ i ⎠ 

Bi 

≥0 

25

Nomadic separated UB 

Another upper bound is based on: 

r 

1 

( 

n 

) 

1 

( 

n 

R ≤ I X ; V{ } 

| H ≤ ∑ I X ; V ) 

1, , 

i 

| H 

… r 

n n i = 1 

th 

Vi 

is the message sent form the i agent. 

Every term in the sum can be treated as a single- 

agent model, number of antennas at the source is 

not important. Invoking results of [SSSK2005] for 

single antenna, for every term of the above sum 

(loose: not accounting for the single source 

setting): 

r ⎛ 

Ci 

2 2 −1 

⎞ 

R ( H ) ≤ ∑log2 1 + P / t hi 

C 

2 

⎜ 

i 

i= 

1 

2 + P / t h ⎟ 

⎝ 

i ⎠ 

26

Performance of 4 by 4 system, with C=2 

8 

7 

R [bits/sec] 

6 

5 

4 

3 

Nomadic Joint upper bound 

Cut-set 1 

Nomadic seperated upper bound 

Cut-set 2 

Simple compression 

Wyner-Ziv 

2 

4 6 8 10 12 14 16 

P [dB] 

27

Performance of t=4 by r=8 system, with C=1 

8 

7 

R [bits/sec] 

6 

5 

4 

3 

Nomadic Joint upper bound 

Cut-set 1 

Nomadic seperated upper bound 

Cut-set 2 

Simple compression 

Wyner-Ziv 

2 4 6 8 10 12 14 16 

P [dB] 

28

Asymptotics – simple compression 

Consider the case of: 

r, 

t → ∞ 

r Ctotal 

while τ = and C = , C fixed, ind. . of r. 

t 

r 

Here we take min/max on 

P Di 

to get upper/lower 

bounds: 

2 

h a. s. 

identical asymptotically: 

∀i 

: i 

→1 

t 

simple compression: 

⎛ P 1 ⎞ 

⎛ 

* 

Pλ 

/ t ⎞ 

EH log2 det Ir + HH → mEH log ⎜ 

2 

1+ 

⎟ 

⎜ C / r 

t 1+ P ⎟ ⎜ 1+ P + 1 2 −1 

⎟ 

Ctotal 

P 

r 

So lim RSC 

(no dependence on τ = ). 

r→∞ 

1 P 

Notice that 

total 

( ) ( ) 

⎝ D* 

⎠ ⎝ ⎠ 

= + 

total 

⎛ 

log det 

⎝ 

* 

EH 

2 ⎜ Ir 

+ HH ⎟ → rP 

t P

Feedback link: 

t 

= 1, r = 2. 

Y1 

Agent 

A1 

C1 

Transmitter 

S 

X 

Channel 

P(Y1,Y2|X) 

Cf 

Destination 

D 

Y2 

Agent 

A2 

C2 

30

Relevant literature – Feedback - 

Cooperation 

Feedback to one agent, in asymmetric case 

[Schein-Gallager 

2001]. 

Coding for interactive communication [Schulman 

1996]. 

Cooperative relay broadcast channels [Liang- 

Veeravalli 2005]. 

Cooperative receivers [Dabora-Servetto 

2005\6]. 

Network-Coding in Interference Networks 

[Smith-Vishwanath 

2006]. 

31

Feedback link - setting 

The final destination has finite capacity 

feedback – common to the two agents. 

Cooperation is limited to two phases. 

The final destination does not reveal 

codebook to agents. 

Feedback link is used to improve 

compression quality. 

Network coding can be used. 

C f 

32


First phase: the agents (1,2) transmit M1M 

and M2, M 

respectively. 

The final destination receives these 

messages and forwards MfM 

through the 

feedback link, involving network coding. 

Second phase: the first agent sends M’1 M 

and the second M’2, M , through the loss less 

links. 

33


We have that: 

1 log 2 ( M' M t t ) ≤ C , t 1, 2 

t 

= 

n 

1 log 2 ( M f ) ≤ C 

n 

f 

where 

t t t 

( 

n 

Y ) 

( Y 

n 

) 

M = φ , t = 1, 2 

M' = φ ' ,M 

t t t f 

( ) 

M = φ M ,M 

f 1 2 

34

Feedback link – achievable rate #1 

First phase: each agent uses WZ to 

n 

n 

compresses Yt 

into U 

t. 

Then sends the corresponding bin by Mt. M 

n 

The destination uses 1 2 to decode 

U , 

and then sends Mf = φ ( M 

1, M2 

). 

Each agent then decodes the compressed 

n 

n 

vector U3 

− t, from Mf 

with the SI Yt 

. 

Network coding effect example: 

( ) φ ( ) 

M = φ M ⊕ M 

f 1 2 

M ,M ( 

n 

) 1 

U 

2 

35


Second phase: each agent uses WZ (with 

SI at agents & destination) to compresses 

n 

n 

into Z t 

, given U3 

−t 

. 

Then sends the corresponding bin: M’t. 

The destination finally decodes 

( 

n n 

Z ) 1 

, Z2 

from the received M’1, M 

M’2. 

Since agents have better SI than D, they 

can decode U n 

3 −t . 

n 

Y t 

36


Achievable rate: 

R < max I( X ; U1, U2, Z1, Z2) 

With the constraints: 

( 1; 1 

| 

2 ) ≤ 

( 2; 2 

| 

1) 

≤ 

( ) 

I U Y U 

I Y , Y ; U , U ≤ 2C 

1 2 1 2 

C 

I U Y U C 

f 

f 

( ; | , , ) ≤ − ( ; | ) 

I Z Y U U Z C I U Y U 

1 1 1 2 2 1 1 2 

( ) 2 

; 

2 

| 

1 

, 

2 

, 

1 

≤ − ( 2; 2 

| 

1) 

I Z Y U U Z C I U Y U 

f 

( , ; , | , ) ≤ 2 − ( , ; , ) 

I Z Z Y Y U U C I Y Y U U 

⎫ 

⎪ 

⎬ 

⎪ 

⎪⎭ 

Standard WZ 

1 2 1 2 1 2 1 2 1 2 

Conditional-WZ 

on remaining 

BW 

37


( ) 

n n 

For Gaussian setting, if side information 

U1 , U 

2 

is known at decoder (destination): no rate gain 

by sending it to the encoder (agents). 

WZ = Conditional Rate-Distortion. 

This mean that for Gaussian setting, the scheme 

can do the same as without feedback. 

For other setting (such as binary), the scheme 

can use the feedback for rate improvement. 

38


( ) 

n n 

The destination does not decode U at the first 

1 

, U 

2 

phase. 

For decoding at the agents, first phase requires: 

I ( U ) ( ) 

1 

, Y1 | Y2 ≤ I U1 , Y1 | U 

2 (since U1 −Y1 −Y2 

, U 

2 ). 

A1 and A2 have better SI than D. 

Second phase: each agent uses WZ to compress 

( 

n n 

Y ) 

t 

, U 

n 

3 − t into Z t 

. 

Then sends the corresponding bin: M’t. 

The destination finally jointly decodes 

( 

n n n n 

U ) from the received M1, M 

M’1, 

M2, 

M’2. 

1 

, Z1 , U 

2 

, Z2 

n n 

n n 

Better than decoding , and Z , Z separately. 

( U ) 1 

U ( ) 

2 

1 2 

39

Feedback link – achievable rate 

Achievable rate is: 

R < max I( X ; U , U , Z , Z ) 

But, with the constraints: 

( 1; 1 

| 

2 ) 

( ; | ) 

I U Y Y 

2 2 1 

≤ 

C 

I U Y Y ≤ C 

f 

f 

⎫ ⎪⎬ 

⎪⎭ 

1 2 1 2 

#2 

Standard WZ 

( ) 1 

; 

1 

| 

1 

, 

2 

, 

2 

max ( 1; 1 

| 

2 ), ( 2; 2 

| 

2, 

1) 

{ } 

I Z Y U U Z + I U Y Y I U Y Z U ≤ C 

( ) 2 

; 

2 

| 

1 

, 

2 

, 

1 

max ( 2; 2 

| 

1) , ( 1; 1 

| 

1, 

2 ) 

{ } 

I Z Y U U Z + I U Y Y I U Y Z U ≤ C 

I ( Z1, Z2; Y1 , Y2 | U1, U 

2 ) + I ( Y1 , Y2 ; U1, U 

2 ) ≤ 2C 

Remaining BW under 

Conditional-WZ 

decoding at both agents and D 

40


For Gaussian symmetric channel, the 

achievable rate with no feedback is with 

equality in the non-diagonal constraint: 

( ) 

I U , U ; Y , Y ≤ 2C 

1 2 1 2 

The achievable rate in the second scheme, is 

limited by the same constraint: 

( ) ( ) 

I Z , Z ; Y , Y | U , U + I Y , Y ; U , U ≤ 2C 

1 2 1 2 1 2 1 2 1 2 

Thus, for the Gaussian symmetric setting, no 

benefit from the feedback. 

41


An increase in the achievable rate, for the 

Gaussian setting: by not requiring the 

destination to decode ( 

n n 

U ) 1 

, U 

2 , at all. 

This means an improved compression, on 

the expense of wasted bandwidth. 

Nomadic setting is not improved even by 

considering Layering (since, again 

Gaussian WZ equals conditional MI). 

42


Achievable rate: 

R 

< max I( X ; Z , Z ) 

With the constraints: 

( 1; 1 

| 

2 ) 

( ; | ) 

I U Y Y 

2 2 1 

≤ 

C 

I U Y Y ≤ C 

f 

f 

1 2 

( ) 1 

; 

1, 2 

| 

2 

≤ − ( 1; 1 

| 

2 ) 

I Z Y U Z C I U Y Y 

( ) 2 

; 

2, 1 

| 

1 

≤ − ( 2; 2 

| 

1) 

I Z Y U Z C I U Y Y 

⎫ ⎪⎬ 

⎪⎭ 

Standard WZ 

( , ; , , , ) ≤ 2 ( ; | ) 

I Z Z Y Y U U C − ∑ I U Y Y − 

1 2 1 2 1 2 i i 3 i 

i= 

1,2 

WZ on 

remaining 

BW 

43

Achievable with feedback – 

numerical example 

U’s s not decoded by Destination. 

Cf required for 

the best rate 

enhancement. 

2.2 

2 

1.8 

1.6 

Achievable rate w/o feedback, along with the optimal feedback 

R with no feedback 

R with feedback 

C f 

: Optimal feedback BW 

R [bits/sec] 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

44 

0.2 

-5 0 5 10 

44 

SNR [dB]

Conclusions 

A framework for communication schemes which includes 

a nomadic transmitter communicating through agents. 

The Gaussian case is solved provided the agents do not 

know/use the code book of the nomadic transmitter. 

Explicit solution is given for two equivalent agents. 

Simple and Wyner-Ziv 

compression schemes for 

nomadic multi antenna transmitter communicating 

through agents. 

The full multiplexing gain is demonstrated for both 

compression schemes, no CSI at transmitter! 

45

Conclusions – Cont. 

Upper bounds for average rate (block fading). 

Numerical results demonstrate the tightness of 

the bounds and the effectiveness of the WZ 

approach. 

For the single transmit antenna: the impact of a 

feedback link is investigated, in three variations. 

A nomadic setting substantially limits the gain 

from the feedback link. 

For efficient use of the feedback, the agents 

need decoding ability. 

46

Outlook 

Distributed MIMO: Improved upper bounds, 

based on EPI vector versions. 

Distributed MIMO: Multiple antennas agents. 

Distributed MIMO: Including decoding agents 

(broadcast MIMO channel). 

Distributed MIMO: Multi cell sites joint 

processing. 

The impact of feedback with decoding agents. 

47

Thank you!

Decentralized Processing: An Information Theoretic Perspective

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