代写代考 COMP2610 / COMP6261 Information Theory Lecture 2: First Steps and Basic Pro

COMP2610 / COMP6261 Information Theory Lecture 2: First Steps and Basic Probability
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1 A General Communication System
2 The Role of Uncertainty
3 Basic Concepts In Probability
4 Relating Joint, Marginal and Conditional Probabilities
5 Wrapping Up

A General Communication System
Decoder Destination
: The information source that generates the message to be communicated
: Operates on the message to produce a signal suitable for transmission
: The medium used to transmit the signal
: Reconstructs the message from the signal : The entity that receives the message

Communication over Noisy Channels
A channel is some medium for transmitting messages
A noisy channel is a channel that potentially introduces errors in the
sender’s message
The Problem of Communication
“The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point.” ( , 1948)

Example: Telephone Network
“Hi”% “Hi”%
Source : : Telephone handset Channel : Analogue telephone line Decoder : Telephone handset
Destination : Mark

Examples of Noisy Communication Channels
Other examples of noisy channels: A radio communication link
VDSL NBN connection
The complete link from camera, through editing, Netflix, the rest of the internet, VDSL link to home, wifi to TV screen
Reproducing cells
A magnetic hard disk drive
􏰜 Channel does not need to involve physical movement
What would the other components be for each of these channels?

1 A General Communication System
2 The Role of Uncertainty
3 Basic Concepts In Probability
4 Relating Joint, Marginal and Conditional Probabilities
5 Wrapping Up

Uncertainty in Communication
We can not avoid uncertainty when
1 Dealing with noise (imperfections) in the channel
2 “Compressing” the messages (compare a high-resolution photograph of a manuscript with the typed text that captures the essence; or a transcript of a spoken utterance)

Channel Noise
A noisy channel introduces errors in sender’s message
Thus, receiver is uncertain that the message is what the sender intended How to model, quantify, and mitigate this uncertainty?

Message Compression – I Cover and Thomas, Example 1.1.2
Suppose we’d like to relay the outcome of an 8 horse race to a friend 􏰜 Wewishtoconveyoneof{ A, B, …, H }
Suppose we encode the message as a binary string. A natural coding scheme is
A → 000 B → 001 C → 010 .

Message Compression – II Cover and Thomas, Example 1.1.2
Now say the probabilities of the horses winning are (1/2, 1/4, 1/8, 1/16, 1/64, 1/64, 1/64, 1/64)
Encoding messages based on their probability of the being chosen will give shorter expected lengths:
E → 11110 F → 111100 G → 111101 H → 111111

What is “Information”?
For noise correction and message compression, we will need to quantify the information contained in a message
Roughly, “information” measures how much: Unexpected data a message contains
The receiver’s uncertainty is reduced on seeing the message

The Case for Probability
We run into the notion of uncertainty when trying to pin down:
1 How to deal with channel noise
2 How to compress messages
3 What “information” means
To make progress, we need to formalise uncertainty
We will do this using probability theory
We now commence our review of probability; this will be hard going if you
have not met it before!

1 A General Communication System
2 The Role of Uncertainty
3 Basic Concepts In Probability
4 Relating Joint, Marginal and Conditional Probabilities
5 Wrapping Up

Probability: Example
Quantification and Manipulation of Uncertainty (Bishop, PRML, 2006)

Probability: Example
Quantification and Manipulation of Uncertainty (Bishop, PRML, 2006)
1 Pick a box at random

Probability: Example
Quantification and Manipulation of Uncertainty (Bishop, PRML, 2006)
1 Pick a box at random
2 From the selected box, pick a fruit (apple or orange) at random

Probability: Example
Quantification and Manipulation of Uncertainty (Bishop, PRML, 2006)
1 Pick a box at random
2 From the selected box, pick a fruit (apple or orange) at random
3 Observe the fruit type and return it back to the original box

Probability: Example
Quantification and Manipulation of Uncertainty (Bishop, PRML, 2006) — Cont’d
Identity of the box is a random variable B ∈ {r , b} Identity of the fruit is a random variable F ∈ {a, o}

Probability: Example
Quantification and Manipulation of Uncertainty (Bishop, PRML, 2006) — Cont’d
Identity of the box is a random variable B ∈ {r , b}
Identity of the fruit is a random variable F ∈ {a, o}
Probability of an event: Proportion of times it happens out of a large number of trials

Probability: Example
Suppose we repeat the selection process many times, and ended up picking up the blue box 60% of the time and the red box 40% of the time
p(B =r)=0.4,p(B =b)=0.6

Probability: Basic Properties
By definition, 0 ≤ p(B = b) ≤ 1 and 0 ≤ p(B = r) ≤ 1 Outcomes are mutually exclusive:
p(B = r ANDB = b) = p(B = r,B = b) =0
Outcomes are jointly exhaustive:
p(B = r OR B = b) = p(B = r ) + p(B = b) − p(B = r AND B = b) = p(B = r) + p(B = b)

Probability
What Types of Questions Can We Answer?
What is the probability of picking the red box, and an apple within that box?
What is the (overall) probability of picking up an apple?
Given that we selected a red box, what is the probability of selecting an apple?
We can answer these and more complex questions by using the rules of probability.

Joint Probability
What is the probability of selecting the red box and selecting an apple?

Joint Probability
What is the probability of selecting the red box and selecting an apple?
Joint Probability of a Set of Events
The proportion of times these events happened together out of the total number of trials.
If we repeated our experiment many (say N = 100) times, and in 10 of the trials we saw B = r and F = a, then we may estimate
p(B=rANDF =a)=p(B=r,F =a) = 10

Marginal Probability
What is the probability of an apple being picked, regardless of the box we selected?

Marginal Probability
What is the probability of an apple being picked, regardless of the box we
Marginal Probability of an Event
The proportion of times that this event happened out of the total number of trials.
Remember that we selected a red box and an apple in 10 out of 100 trials Say that in 45 of the trials, we selected a blue box and an apple
So, irrespective of B, an apple was selected 45 + 10 = 55 times, and:
p(F = a) = 55 = 11 100 20

Conditional Probability
What is the probability of an apple being picked up, given that a red box was selected?

Conditional Probability
What is the probability of an apple being picked up, given that a red box
was selected?
Conditional Probability of an Event
The conditional probability of an event X with respect to an event Y is the proportion of times that X happens out of the times that Y happens.
The trials where we selected a blue box are irrelevant, whether or not an apple was selected
We selected a red box and an apple 10 out of 100 times
We selected a red box (regardless of the fruit) 40 out of 100 times
p(F = aGIVENB = r) = p(F = a|B = r) = 10 = 1 40 4

1 A General Communication System
2 The Role of Uncertainty
3 Basic Concepts In Probability
4 Relating Joint, Marginal and Conditional Probabilities
5 Wrapping Up

Joint, Marginal and Conditional Probabilities: A More General Formulation (1)
Consider the more general case of two random variables:
X ∈ {x1,x2,…,xM} and Y ∈ {y1,y2,…,yL}
N : Total number of trials
nij : ♯(X =xi,Y =yj)=♯oftimesthatxi andyi happen

Joint, Marginal and Conditional Probabilities: A More General Formulation (2)
By definition:
p(X = xi,Y = yj) = nij (Joint) N
p(X = xi ) = ci (Marginal) N
p(Y = yj ) = rj (Marginal) N
p(Y = yj |X = xi ) = nij (Conditional) ci

Joint, Marginal and Conditional Probabilities: A More General Formulation (1)
Bins and fruit example:
Verify the computations from previous section with this table

Joint, Marginal and Conditional Probabilities: A More General Formulation (3)
􏰐j nij p(X=xi)= N
=􏰄p(X =xi,Y =yj) j
p(Y =yj|X =xi)= nij = nij􏱡ci ci NN
=p(X =xi,Y =yj)/p(X =xi)

The Rules of Probability Sum Rule / Marginalization :
p(X =xi)=􏰄p(X =xi,Y =yj) j

The Rules of Probability Sum Rule / Marginalization :
Product Rule :
p(X =xi)=􏰄p(X =xi,Y =yj) j
p(X =xi,Y =yj)=p(Y =yj|X =xi)p(X =xi)

The Rules of Probability Sum Rule / Marginalization :
Product Rule :
p(X =xi)=􏰄p(X =xi,Y =yj) j
p(X =xi,Y =yj)=p(Y =yj|X =xi)p(X =xi)
and by symmetry:
P(Y =yj,X =xi)=p(X =xi|Y =yj)p(Y =yj)

The Rules of Probability Sum Rule / Marginalization :
Product Rule :
p(X =xi)=􏰄p(X =xi,Y =yj) j
p(X =xi,Y =yj)=p(Y =yj|X =xi)p(X =xi)
and by symmetry:
Therefore:
P(Y =yj,X =xi)=p(X =xi|Y =yj)p(Y =yj)
P(X =xi)=􏰄P(X =xi,Y =yj)=􏰄P(X =xi|Y =yj)P(Y =yj) jj

An Illustration of a Distribution over Two Variables

An Illustration of a Distribution over Two Variables
joint marginal

An Illustration of a Distribution over Two Variables
joint marginal
marginal conditional

Joint, Marginal and Conditional Probabilities: An even More General Formulation
Given D random variables X1,…,XD:
p(X1,…,Xi−1,Xi+1,…,XD) = 􏰄p(X1,…,XD) Xi

Joint, Marginal and Conditional Probabilities: An even More General Formulation
Given D random variables X1,…,XD: p(X1,…,Xi−1,Xi+1,…,XD) = 􏰄p(X1,…,XD)
Chain Rule: We can also express:
p(X1, X2) = p(X1)p(X2|X1) What are we using here?
p(X1, X2, X3) = p(X1, X2)p(X3|X1, X2) = p(X1)p(X2|X1)p(X3|X1, X2)

1 A General Communication System
2 The Role of Uncertainty
3 Basic Concepts In Probability
4 Relating Joint, Marginal and Conditional Probabilities
5 Wrapping Up

General architecture for communication systems
Why we need probability
Probability theory: joint, marginal and conditional distribution
Reading: Mackay § 2.1 and § 2.2; Bishop § 1.2

Coming Back to our Original Example
Given: p(B =r)=0.4,p(B =b)=0.6
Assume the fruit are selected uniformly from each box
Write down all conditional probabilities P(F|B) Evaluate the overall probabilities P(F)

More on joint, marginal and conditional distributions
When can we say that X , Y do not influence each other?
What, if anything, does p(X = x|Y = y) tell us about p(Y = y|X = x)?

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