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Typical set
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Typical set
In information theory, the typical set is a set of sequences whose probability is close to two raised to the negative power of the entropy of their source distribution. That this set has total probability close to one is a consequence of the asymptotic equipartition property (AEP) which is a kind of law of large numbers. The notion of typicality is only concerned with the probability of a sequence and not the actual sequence itself.
This has great use in compression theory as it provides a theoretical means for compressing data, allowing us to represent any sequence Xn using nH(X) bits on average, and, hence, justifying the use of entropy as a measure of information from a source.
The AEP can also be proven for a large class of stationary ergodic processes, allowing typical set to be defined in more general cases.
Additionally, the typical set concept is foundational in understanding the limits of data transmission and error correction in communication systems. By leveraging the properties of typical sequences, efficient coding schemes like Shannon's source coding theorem and channel coding theorem are developed, enabling near-optimal data compression and reliable transmission over noisy channels.
If a sequence x1, ..., xn is drawn from an independent identically-distributed random variable (IID) X defined over a finite alphabet , then the typical set, Aε(n)(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 2n ε. Taking the logarithm on all sides and dividing by -n, this definition can be equivalently stated as
For i.i.d sequence, since
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Typical set
In information theory, the typical set is a set of sequences whose probability is close to two raised to the negative power of the entropy of their source distribution. That this set has total probability close to one is a consequence of the asymptotic equipartition property (AEP) which is a kind of law of large numbers. The notion of typicality is only concerned with the probability of a sequence and not the actual sequence itself.
This has great use in compression theory as it provides a theoretical means for compressing data, allowing us to represent any sequence Xn using nH(X) bits on average, and, hence, justifying the use of entropy as a measure of information from a source.
The AEP can also be proven for a large class of stationary ergodic processes, allowing typical set to be defined in more general cases.
Additionally, the typical set concept is foundational in understanding the limits of data transmission and error correction in communication systems. By leveraging the properties of typical sequences, efficient coding schemes like Shannon's source coding theorem and channel coding theorem are developed, enabling near-optimal data compression and reliable transmission over noisy channels.
If a sequence x1, ..., xn is drawn from an independent identically-distributed random variable (IID) X defined over a finite alphabet , then the typical set, Aε(n)(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 2n ε. Taking the logarithm on all sides and dividing by -n, this definition can be equivalently stated as
For i.i.d sequence, since