Typical set

(Weakly) typical sequences (weak typicality, entropy typicality)

If a sequence x₁, ..., x_n is drawn from an i.i.d. distribution X defined over a finite alphabet X , then the typical set, A_ε⁽ⁿ⁾ ∈ X ⁽ⁿ⁾ is defined as those sequences which satisfy:

Where

is the information entropy of X. The probability above need only be within a factor of 2^nε.

It has the following properties if n is sufficiently large, ε > 0 can be chosen arbitrarily small so that:

1. The probability of a sequence from X being drawn from A_ε⁽ⁿ⁾ is greater than 1 − ε, i.e. P r [ x ( n ) ∈ A ϵ ( n ) ] ≥ 1 − ϵ
2. | A ε ( n ) | ⩽ 2 n ( H ( X ) + ε )
3. | A ε ( n ) | ⩾ ( 1 − ε ) 2 n ( H ( X ) − ε )
4. Most sequences are not typical. If the distribution over X is not uniform, then the fraction of sequences that are typical is

For a general stochastic process {X(t)} with AEP, the (weakly) typical set can be defined similarly with p(x₁, x₂, ..., x_n) replaced by p(x₀^τ) (i.e. the probability of the sample limited to the time interval [0, τ]), n being the degree of freedom of the process in the time interval and H(X) being the entropy rate. If the process is continuous-valued, differential entropy is used instead.

Counter-intuitively, the most likely sequence is often not a member of the typical set. For example, suppose that X is an i.i.d Bernoulli random variable with p(0)=0.1 and p(1)=0.9. In n independent trials, since p(1)>p(0), the most likely sequence of outcome is the sequence of all 1's, (1,1,...,1). Here the entropy of X is H(X)=0.469, while − 1 n log 2 ⁡ p ( x ( n ) = ( 1 , 1 , … , 1 ) ) = − 1 n log 2 ⁡ ( 0.9 n ) = 0.152

So this sequence is not in the typical set because its average logarithmic probability cannot come arbitrarily close to the entropy of the random variable X no matter how large we take the value of n. For Bernoulli random variables, the typical set consists of sequences with average numbers of 0s and 1s in n independent trials. For this example, if n=10, then the typical set consist of all sequences that has a single 0 in the entire sequence. In case p(0)=p(1)=0.5, then every possible binary sequences belong to the typical set.

Strongly typical sequences (strong typicality, letter typicality)

If a sequence x₁, ..., x_n is drawn from some specified joint distribution defined over a finite or an infinite alphabet X , then the strongly typical set, A_ε,strong⁽ⁿ⁾ ∈ X is defined as the set of sequences which satisfy

where N ( x i ) is the number of occurrences of a specific symbol in the sequence.

It can be shown that strongly typical sequences are also weakly typical (with a different constant ε), and hence the name. The two forms, however, are not equivalent. Strong typicality is often easier to work with in proving theorems for memoryless channels. However, as is apparent from the definition, this form of typicality is only defined for random variables having finite support.

Jointly typical sequences

Two sequences x n and y n are jointly ε-typical if the pair ( x n , y n ) is ε-typical with respect to the joint distribution p ( x n , y n ) = ∏ i = 1 n p ( x i , y i ) and both x n and y n are ε-typical with respect to their marginal distributions p ( x n ) and p ( y n ) . The set of all such pairs of sequences ( x n , y n ) is denoted by A ε n ( X , Y ) . Jointly ε-typical n-tuple sequences are defined similarly.

Let X ~ n and Y ~ n be two independent sequences of random variables with the same marginal distributions p ( x n ) and p ( y n ) . Then for any ε>0, for sufficiently large n, jointly typical sequences satisfy the following properties:

1. P [ ( X n , Y n ) ∈ A ε n ( X , Y ) ] ⩾ 1 − ϵ
2. | A ε n ( X , Y ) | ⩽ 2 n ( H ( X , Y ) + ϵ )
3. | A ε n ( X , Y ) | ⩾ ( 1 − ϵ ) 2 n ( H ( X , Y ) − ϵ )
4. P [ ( X ~ n , Y ~ n ) ∈ A ε n ( X , Y ) ] ⩽ 2 − n ( I ( X ; Y ) − 3 ϵ )
5. P [ ( X ~ n , Y ~ n ) ∈ A ε n ( X , Y ) ] ⩾ ( 1 − ϵ ) 2 − n ( I ( X ; Y ) + 3 ϵ )

Typical set encoding

In information theory, typical set encoding encodes only the typical set of a stochastic source with fixed length block codes. Asymptotically, it is, by the AEP, lossless and achieves the minimum rate equal to the entropy rate of the source.

Typical set decoding

In information theory, typical set decoding is used in conjunction with random coding to estimate the transmitted message as the one with a codeword that is jointly ε-typical with the observation. i.e.

where w ^ , x 1 n ( w ) , y 1 n are the message estimate, codeword of message w and the observation respectively. A ε n ( X , Y ) is defined with respect to the joint distribution p ( x 1 n ) p ( y 1 n | x 1 n ) where p ( y 1 n | x 1 n ) is the transition probability that characterizes the channel statistics, and p ( x 1 n ) is some input distribution used to generate the codewords in the random codebook.