# 0.1 Background  (Page 2/2)

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Claim 2 The probability $Q\left(T\left({P}_{x}\right)$ of the type class $T\left({P}_{x}\right)$ obeys,

${\left(n+1\right)}^{-\left(r-1\right)}·{2}^{-nD\left({P}_{x}\parallel {Q}_{x}\right)}\le Q\left(T\left({P}_{x}\right)\right)\le {2}^{-nD\left({P}_{x}\parallel {Q}_{x}\right)}.$

Consider now an event $A$ that is a union over $T\left({P}_{x}\right)$ . Suppose $T\left(Q\right)⊈A$ , then $A$ is rare with respect to (w.r.t) the prior $Q$ . and we have ${lim}_{n\to \infty }Q\left(A\right)=0$ . That is, the probability is concentrated around $Q$ . In general, the probability assigned by the prior $Q$ to an event $A$ satisfies

$\begin{array}{ccc}\hfill Q\left(A\right)& =& {\Sigma }_{x\in A}Q\left(x\right)={\Sigma }_{T\left({P}_{x}\right)\subseteq A}Q\left(T\left({P}_{x}\right)\right)\hfill \\ & \stackrel{˙}{=}& {\Sigma }_{T\left({P}_{x}\right)\subseteq A}{2}^{-nD\left({P}_{x}\parallel Q\right)}\hfill \\ & \stackrel{˙}{=}& {2}^{-n·{min}_{p\in A}D\left(P\parallel Q\right)},\hfill \end{array}$

where we denote ${a}_{n}\stackrel{˙}{=}{b}_{n}$ when $\frac{1}{n}log\left(\frac{{a}_{n}}{{b}_{n}}\right)\to 0$ .

## Fixed and variable length coding

Fixed to fixed length source coding : As before, we have a sequence $x$ of length $n$ , and each element of $x$ is from the alphabet $\alpha$ . A source code maps the input ${x}^{n}\in {r}^{n}$ to a set of ${2}^{Rn}$ bit vectors, each of length $Rn$ . The rate $R$ quantifies the number of output bits of the code per input element of $x$ . We assume without loss of generality that $Rn\in \mathbb{Z}$ . If not, then we can round $Rn$ up to $⌈Rn⌉$ , where $⌈·⌉$ denotes rounding up. That is, the output of the code consists of $nR$ bits. If $n$ and $R$ is fixed, then we call this a fixed to fixed length source code.

The decoder processes the $nR$ bits and yields $\stackrel{ˆ}{x}\in {\alpha }^{n}$ . Ideally we have that $\stackrel{ˆ}{x}=x$ , but if ${2}^{nR}<{r}^{n}$ then there are inputs that are not mapped to any output, and $\stackrel{ˆ}{x}$ may differ from $x$ . Therefore, we want $Pr\left(\stackrel{ˆ}{x}\ne x\right)$ to be small. If $R$ is too small, then the error probability will go to 1. On the other hand, sufficiently large $R$ will drive this error probability to 0 as $n$ is increased.

If $log\left(r\right)>R$ and $Pr\left(\stackrel{ˆ}{x}\ne x\right)$ is vanishing as $n$ is increased, then we are compressing, because ${2}^{log\left(r\right)n}={r}^{n}>{2}^{Rn}$ , where ${r}^{n}$ is the number of possible inputs $x$ and there are ${2}^{Rn}$ possible outputs.

What is a good fixed to fixed length source code? One option is to map ${2}^{Rn}-1$ outputs to inputs with high probabilities, and the last output can be mapped to a “don't care" input.We will discuss the performance of this style of code.

An input $x\in {r}^{n}$ is called $\delta$ -typical if $Q\left(x\right)>{2}^{-\left(H+\delta \right)n}$ . We denote the set of $\delta$ -typical inputs by ${T}_{Q}\left(\delta \right)$ , this set includes the type classes whose empirical probabilities are equal (or closest) to the true prior $Q\left(x\right)$ . Note that for each type class ${T}_{x}$ , all inputs ${x}^{\text{'}}\in {T}_{x}$ in the type class have the same probability, i.e., $Q\left({x}^{\text{'}}\right)=Q\left(x\right)$ . Therefore, the set ${T}_{Q}\left(\delta \right)$ is a union of type classes, and can be thought of as an event $A$ ( [link] ) that contains type classes consisting of high-probability sequences. It is easily seen that the event $A$ contains the true i.i.d. distribution $Q$ , because sequences whose empirical probabilities satisfy ${P}_{x}=Q$ also satisfy

$Q\left(x\right)={2}^{-Hn}>{2}^{-\left(H+\delta \right)n}.$

Using the principles discussed in [link] , it is readily seen that the probability under the prior $Q$ of the inputs in ${T}_{Q}\left(\delta \right)$ satisfies $Q\left({T}_{p}\left(\delta \right)\right)=Q\left(A\right)\to 1$ when $n\to \infty$ . Therefore, a code $\mathcal{C}$ that enumerates ${T}_{Q}\left(\delta \right)$ will encode $x$ correctly with high probability.

The key question is the size of $\mathcal{C}$ , or the cardinality of ${T}_{Q}\left(\delta \right)$ . Because each $x\in {T}_{Q}\left(\delta \right)$ satisfies $Q\left(x\right)>{2}^{\left(-H+\delta \right)n}$ , and ${\sum }_{x\in {T}_{Q}\left(\delta \right)}Q\left(x\right)\le 1$ , we have $|{T}_{Q}\left(\delta \right)|<{2}^{\left(H+\delta \right)n}$ . Therefore, a rate $R\ge H+\delta$ allows near-lossless coding , because the probability of error vanishes(recall that $Q\left({\left({T}_{p}\left(\delta \right)\right)}^{C}\right)\to 0$ , where ${\left(·\right)}^{C}$ denotes the complement).

On the other hand, a rate $R\le H-\delta$ will not allow lossless coding, and the probability of error will go to 1. We will see this intuitively. Because the type class whose empirical probability is $Q$ dominates, a type class ${T}_{x}$ whose sequences have larger probability, e.g., $Q\left(x\right)>{2}^{-\left(H-\delta \right)n}$ , will have small probability in aggregate. That is,

$\sum _{x:Q\left(x\right)>{2}^{-n\left(H-\delta \right)}}Q\left(x\right)\stackrel{n\to \infty }{\to }0.$

In words, choosing a code $\mathcal{C}$ with rate $R=H-\delta$ that contains the words $x$ with highest probability will fail, it will not cover enough probabilistic mass.We conclude that near-lossless coding is possible at rates above H and impossible below H.

To see things from a more intuitive angle, consider the definition of entropy, $H\left(Q\right)=-{\sum }_{a\in \alpha }Q\left(a\right)log\left(Q\left(a\right)\right)$ . If we consider each bit as reducing uncertainty by a factor of 2,then the average log-likelihood of a length- $n$ input $x$ generated by $Q$ satisfies

$\begin{array}{ccc}\hfill E\left[-log\left(Pr\left(x\right)\right)\right]& =& E\left[-log\left(\prod _{i=1}^{n}Pr\left({x}_{i}\right)\right)\right]\hfill \\ & =& -\sum _{i=1}^{n}E\left[log\left(Q\left({x}_{i}\right)\right)\right]\hfill \\ & =& -\sum _{i=1}^{n}\sum _{a\in \alpha }Q\left(a\right)·log\left(Q\left(a\right)\right)\hfill \\ & =& nH.\hfill \end{array}$

Because the expected log-likelihood of $x$ is $nH$ , it will take $nH$ bits to reduce the uncertainty by this factor.

Fixed to variable length source coding : The near-lossless coding above relies on enumerating a collection of high-probability codewords ${T}_{Q}\left(\delta \right)$ . However, this approach suffers from a troubling failure for $x\notin {T}_{Q}\left(\delta \right)$ . To solve this problem, we incorporate a code that maps $x$ to an output consisting of a variable number of bits. That is, the length of the code will be approximately $nH$ on average, but could be greater or lesser.

One possible variable length code is due to Shannon. Consider all possible $x\in {\alpha }^{n}$ . For each $x$ , allocate $⌈-log\left(Q\left(x\right)\right)⌉$ bits to $x$ . It can be shown that it is possible to construct an invertible (uniquely decodable)code as long as the length of the code $l\left(x\right)$ in bits allocated to each $x$ satisfies

$\sum _{x}{2}^{-l\left(x\right)}\le 1.$

This result is known as the Kraft Inequality. Seeing that

$\begin{array}{ccc}\hfill \sum _{x}{2}^{-l\left(x\right)}& =& \sum _{x}{2}^{-⌈-log\left(Q\left(x\right)\right)⌉}\hfill \\ & \le & \sum _{x}{2}^{-\left(-log\left(Q\left(x\right)\right)\right)}\hfill \\ & =& \sum _{x}Q\left(x\right)=1,\hfill \end{array}$

we see that the length allocation we suggested satisfies the Kraft Inequality. Therefore, it is possible to construct an invertible (and hence lossless) codewith lengths upper bounded by

${l}_{x}=⌈-log\left(Q\left(x\right)\right)⌉\le -log\left(Q\left(x\right)\right)+1,$

and we have

$E\left[l\left(x\right)\right]\le E\left[-log\left(Q\left(x\right)\right)\right]+1=nH+1.$

This simple construction approaches the entropy up to 1 bit.

Unfortunately, a Shannon code is impractical, because it requires to construct a code book of exponential size ${|\alpha |}^{n}$ . Instead, arithmetic codes  [link] are used; we discussed arithmetic codes in detail in class, but they appear in all standard text books and so we do not describe them here.

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