Information theory
Entropy Differential entropy Conditional entropy Joint entropy Mutual information Directed information Conditional mutual information Relative entropy Entropy rate Limiting density of discrete points
Asymptotic equipartition property Rate–distortion theory
Shannon's source coding theorem Channel capacity Noisy-channel coding theorem Shannon–Hartley theorem

In information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the statistical limits to possible data compression for data whose source is an independent identically-distributed random variable, and the operational meaning of the Shannon entropy.

Named after Claude Shannon, the source coding theorem shows that, in the limit, as the length of a stream of independent and identically-distributed random variable (i.i.d.) data tends to infinity, it is impossible to compress such data such that the code rate (average number of bits per symbol) is less than the Shannon entropy of the source, without it being virtually certain that information will be lost. However it is possible to get the code rate arbitrarily close to the Shannon entropy, with negligible probability of loss.

The source coding theorem for symbol codes places an upper and a lower bound on the minimal possible expected length of codewords as a function of the entropy of the input word (which is viewed as a random variable) and of the size of the target alphabet.

Note that, for data that exhibits more dependencies (whose source is not an i.i.d. random variable), the Kolmogorov complexity, which quantifies the minimal description length of an object, is more suitable to describe the limits of data compression. Shannon entropy takes into account only frequency regularities while Kolmogorov complexity takes into account all algorithmic regularities, so in general the latter is smaller. On the other hand, if an object is generated by a random process in such a way that it has only frequency regularities, entropy is close to complexity with high probability (Shen et al. 2017).^[1]

Statements

Source coding is a mapping from (a sequence of) symbols from an information source to a sequence of alphabet symbols (usually bits) such that the source symbols can be exactly recovered from the binary bits (lossless source coding) or recovered within some distortion (lossy source coding). This is one approach to data compression.

Proof: source coding theorem

Given X is an i.i.d. source, its time series X₁, ..., X_n is i.i.d. with entropy H(X) in the discrete-valued case and differential entropy in the continuous-valued case. The Source coding theorem states that for any ε > 0, i.e. for any rate H(X) + ε larger than the entropy of the source, there is large enough n and an encoder that takes n i.i.d. repetition of the source, X^1:n, and maps it to n(H(X) + ε) binary bits such that the source symbols X^1:n are recoverable from the binary bits with probability of at least 1 − ε.

Proof of Achievability. Fix some ε > 0, and let

${\displaystyle p(x_{1},\ldots ,x_{n})=\Pr \left[X_{1}=x_{1},\cdots ,X_{n}=x_{n}\right].}$

The typical set, A^ε
_n, is defined as follows:

${\displaystyle A_{n}^{\varepsilon }=\left\{(x_{1},\cdots ,x_{n})\$ :\ \left|-{\frac {1}{n}}\log p(x_{1},\cdots ,x_{n})-H_{n}(X)\right|<\varepsilon \right\}.}

The asymptotic equipartition property (AEP) shows that for large enough n, the probability that a sequence generated by the source lies in the typical set, A^ε
_n, as defined approaches one. In particular, for sufficiently large n, ${\displaystyle P((X_{1},X_{2},\cdots ,X_{n})\in A_{n}^{\varepsilon })}$ can be made arbitrarily close to 1, and specifically, greater than ${\displaystyle 1-\varepsilon }$ (See AEP for a proof).

The definition of typical sets implies that those sequences that lie in the typical set satisfy:

${\displaystyle 2^{-n(H(X)+\varepsilon )}\leq p\left(x_{1},\cdots ,x_{n}\right)\leq 2^{-n(H(X)-\varepsilon )}}$

• The probability of a sequence ${\displaystyle (X_{1},X_{2},\cdots X_{n})}$ being drawn from A^ε
  _n is greater than 1 − ε.
• ${\displaystyle \left|A_{n}^{\varepsilon }\right|\leq 2^{n(H(X)+\varepsilon )}}$, which follows from the left hand side (lower bound) for ${\displaystyle p(x_{1},x_{2},\cdots x_{n})}$.
• ${\displaystyle \left|A_{n}^{\varepsilon }\right|\geq (1-\varepsilon )2^{n(H(X)-\varepsilon )}}$, which follows from upper bound for ${\displaystyle p(x_{1},x_{2},\cdots x_{n})}$ and the lower bound on the total probability of the whole set A^ε
  _n.

Since ${\displaystyle \left|A_{n}^{\varepsilon }\right|\leq 2^{n(H(X)+\varepsilon )},n(H(X)+\varepsilon )}$ bits are enough to point to any string in this set.

The encoding algorithm: the encoder checks if the input sequence lies within the typical set; if yes, it outputs the index of the input sequence within the typical set; if not, the encoder outputs an arbitrary n(H(X) + ε) digit number. As long as the input sequence lies within the typical set (with probability at least 1 − ε), the encoder does not make any error. So, the probability of error of the encoder is bounded above by ε.

Proof of converse: the converse is proved by showing that any set of size smaller than A^ε
_n (in the sense of exponent) would cover a set of probability bounded away from 1.

Proof: Source coding theorem for symbol codes

For 1 ≤ i ≤ n let s_i denote the word length of each possible x_i. Define ${\displaystyle q_{i}=a^{-s_{i}}/C}$, where C is chosen so that q₁ + ... + q_n = 1. Then

${\displaystyle {\begin{aligned}H(X)&=-\sum _{i=1}^{n}p_{i}\log _{2}p_{i}\\&\leq -\sum _{i=1}^{n}p_{i}\log _{2}q_{i}\\&=-\sum _{i=1}^{n}p_{i}\log _{2}a^{-s_{i}}+\sum _{i=1}^{n}p_{i}\log _{2}C\\&=-\sum _{i=1}^{n}p_{i}\log _{2}a^{-s_{i}}+\log _{2}C\\&\leq -\sum _{i=1}^{n}-s_{i}p_{i}\log _{2}a\\&=\mathbb {E} S\log _{2}a\\\end{aligned}}}$

where the second line follows from Gibbs' inequality and the fifth line follows from Kraft's inequality:

${\displaystyle C=\sum _{i=1}^{n}a^{-s_{i}}\leq 1}$

so log C ≤ 0.

For the second inequality we may set

${\displaystyle s_{i}=\lceil -\log _{a}p_{i}\rceil }$

so that

${\displaystyle -\log _{a}p_{i}\leq s_{i}<-\log _{a}p_{i}+1}$

and so

${\displaystyle a^{-s_{i}}\leq p_{i}}$

and

${\displaystyle \sum a^{-s_{i}}\leq \sum p_{i}=1}$

and so by Kraft's inequality there exists a prefix-free code having those word lengths. Thus the minimal S satisfies

${\displaystyle {\begin{aligned}\mathbb {E} S&=\sum p_{i}s_{i}\\&<\sum p_{i}\left(-\log _{a}p_{i}+1\right)\\&=\sum -p_{i}{\frac {\log _{2}p_{i}}{\log _{2}a}}+1\\&={\frac {H(X)}{\log _{2}a}}+1\\\end{aligned}}}$

Extension to non-stationary independent sources

See also

• Channel coding
• Error exponent
• Noisy-channel coding theorem

References