# 0.3 Universal coding for classes of sources

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where $Q$ is the prior induced by the coding length function $l$ .

## Minimal redundancy

Note that

$\begin{array}{ccc}\hfill \forall w,l,\phantom{\rule{1.em}{0ex}}\underset{\theta }{sup}{r}_{n}\left(l,\theta \right)& \ge & {\int }_{\Lambda }w\left(d\theta \right){r}_{n}\left(l,\theta \right)\hfill \\ & \ge & \underset{l\in {c}_{n}}{inf}{\int }_{\Lambda }w\left(d\theta \right){r}_{n}\left(l,\theta \right).\hfill \end{array}$

Therefore,

${R}_{n}^{+}=\underset{l}{inf}\underset{\theta }{sup}{r}_{n}\left(l,\theta \right)\ge \underset{w}{sup}\underset{l}{inf}{\int }_{\Lambda }w\left(d\theta \right){r}_{n}\left(l,\theta \right)={R}_{n}^{-}.$

In fact, Gallager showed that ${R}_{n}^{+}={R}_{n}^{-}$ . That is, the min-max and max-min redundancies are equal.

Let us revisit the Bernoulli source ${p}_{\theta }$ where $\theta \in \Lambda =\left[0,1\right]$ . From the definition of [link] , which relies on a uniform prior for the sources, i.e., $w\left(\theta \right)=1,\forall \theta \in \Lambda$ , it can be shown that there there exists a universal code with length function $l$ such that

${E}_{\theta }\left[l\left({x}^{n}\right)\right]\le n{E}_{\theta }\left[{h}_{2},\left(\frac{{n}_{x}\left(1\right)}{n}\right)\right]+log\left(n+1\right)+2,$

where ${h}_{2}\left(p\right)=-plog\left(p\right)-\left(1-p\right)log\left(1-p\right)$ is the binary entropy. That is, the redundancy is approximately $log\left(n\right)$ bits. Clarke and Barron  [link] studied the weighting approach,

$p\left(x\right)={\int }_{\Lambda }dw\left(\theta \right){p}_{\theta }\left(x\right),$

and constructed a prior that achieves ${R}_{n}^{-}={R}_{n}^{+}$ precisely for memoryless sources.

Theorem 5 [link] For memoryless source with an alphabet of size $r$ , $\theta =\left(p\left(0\right),p\left(1\right),\cdots ,p\left(r-1\right)\right)$ ,

$n{R}_{n}^{-}\left(w\right)=\frac{r-1}{2}log\left(\frac{n}{2\pi e}\right)+{\int }_{\Lambda }w\left(d\theta \right)log\left(\frac{\sqrt{|I\left(\theta \right)|}}{w\left(\theta \right)}\right)+{O}_{n}\left(1\right),$

where ${O}_{n}\left(1\right)$ vanishes uniformly as $n\to \infty$ for any compact subset of $\Lambda$ , and

$I\left(\theta \right)\triangleq E\left[\left(\frac{\partial ln{p}_{\theta }\left({x}_{i}\right)}{\partial \theta }\right),{\left(\frac{\partial ln{p}_{\theta }\left({x}_{i}\right)}{\partial \theta }\right)}^{T}\right]$

is Fisher's information.

Note that when the parameter is sensitive to change we have large $I\left(\theta \right)$ , which increases the redundancy. That is, good sensitivity means bad universal compression.

Denote

$J\left(\theta \right)=\frac{\sqrt{|I\left(\theta \right)|}}{{\int }_{\Lambda }\sqrt{|I\left({\theta }^{\text{'}}\right)|}d{\theta }^{\text{'}}},$

this is known as Jeffrey's prior . Using $w\left(\theta \right)=J\left(\theta \right)$ , it can be shownthat ${R}_{n}^{-}={R}_{n}^{+}$ .

Let us derive the Fisher information $I\left(\theta \right)$ for the Bernoulli source,

$\begin{array}{ccc}& & {p}_{\theta }\left(x\right)={\theta }^{{n}_{x}\left(1\right)}·{\left(1-\theta \right)}^{{n}_{x}\left(0\right)}\hfill \\ & ⇒& ln{p}_{\theta }\left(x\right)={n}_{x}\left(1\right)ln\theta +{n}_{x}\left(0\right)ln\left(1-\theta \right)\hfill \\ & ⇒& \frac{\partial ln{p}_{\theta }\left(x\right)}{\partial \theta }={n}_{x}\left(1\right)\frac{1}{\theta }-{n}_{x}\left(0\right)\frac{1}{1-\theta }\hfill \\ & ⇒& {\left(\frac{\partial ln{p}_{\theta }\left(x\right)}{\partial \theta }\right)}^{2}=\frac{{n}_{x}^{2}\left(1\right)}{{\theta }^{2}}+\frac{{n}_{x}^{2}\left(0\right)}{{\left(1-\theta \right)}^{2}}-\frac{2{n}_{x}\left(1\right){n}_{x}\left(0\right)}{\theta \left(1-\theta \right)}\hfill \\ & ⇒& E\left[{\left(\frac{\partial ln{p}_{\theta }\left(x\right)}{\partial \theta }\right)}^{2}\right]=\frac{\theta }{{\theta }^{2}}+\frac{1-\theta }{{\left(1-\theta \right)}^{2}}-\frac{2}{\theta \left(1-\theta \right)}E\left[{n}_{x}\left(1\right){n}_{x}\left(0\right)\right]\hfill \\ & & =\frac{1}{\theta }+\frac{1}{1-\theta }-0\hfill \\ & & =\frac{1}{\theta \left(1-\theta \right)}.\hfill \end{array}$

Therefore, the Fisher information satisfies $I\left(\theta \right)=\frac{1}{\theta \left(1-\theta \right)}$ .

Recall the Krichevsky–Trofimov coding, which was mentioned in  [link] . Using the definition of Jeffreys' prior [link] , we see that $J\left(\theta \right)\propto \frac{1}{\sqrt{\theta \left(1-\theta \right)}}$ . Taking the integral over Jeffery's prior,

$\begin{array}{ccc}\hfill {p}_{J}\left({x}^{n}\right)& =& {\int }_{0}^{1}c\frac{d\theta }{\sqrt{\theta \left(1-\theta \right)}}{\theta }^{{n}_{x}\left(1\right)}{\left(1-\theta \right)}^{{n}_{x}\left(0\right)}\hfill \\ & =& c{\int }_{0}^{1}{\theta }^{{n}_{x}\left(1\right)-\frac{1}{2}}{\left(1-\theta \right)}^{{n}_{x}\left(0\right)-\frac{1}{2}}d\theta \hfill \\ & =& \frac{\Gamma \left({n}_{x}\left(0\right)+\frac{1}{2}\right)\Gamma \left({n}_{x}\left(1\right)+\frac{1}{2}\right)}{\pi \Gamma \left(n+1\right)},\hfill \end{array}$

where we used the gamma function. It can be shown that

${p}_{J}\left({x}^{n}\right)=\prod _{t=0}^{n}{p}_{J}\left({x}_{t+1}|{x}_{1}^{t}\right),$

where

$\begin{array}{ccc}\hfill {p}_{J}\left({x}_{t+1}|{x}_{1}^{t}\right)& =& \frac{{p}_{J}\left({x}_{1}^{t+1}\right)}{{p}_{J}\left({x}_{1}^{t}\right)},\hfill \\ \hfill {p}_{J}\left({x}_{t+1}=0|{x}_{1}^{t}\right)& =& \frac{{n}_{x}^{t}\left(0\right)+\frac{1}{2}}{t+1},\hfill \\ \hfill {p}_{J}\left({x}_{t+1}=1|{x}_{1}^{t}\right)& =& \frac{{n}_{x}^{t}\left(1\right)+\frac{1}{2}}{t+1}.\hfill \end{array}$

Similar to before, this universal code can be implemented sequentially. It is due to Krichevsky and Trofimov  [link] , its redundancy satisfies   Theorem 5 by Clarke and Barron  [link] , and it is commonly used in universal lossless compression.

## Rissanen's bound

Let us consider – on an intuitive level – why ${C}_{n}\approx \frac{r-1}{2}\frac{log\left(n\right)}{n}$ . Expending $\frac{r-1}{2}log\left(n\right)$ bits allows to differentiate between ${\left(\sqrt{n}\right)}^{r-1}$ parameter vectors. That is, we would differentiate between each of the $r-1$ parameters with $\sqrt{n}$ levels. Now consider a Bernoulli RV with (unknown) parameter $\theta$ .

One perspective is that with $n$ drawings of the RV, the standard deviation in the number of 1's is $O\left(\sqrt{n}\right)$ . That is, $\sqrt{n}$ levels differentiate between parameter levels up to a resolution that reflects the randomness of the experiment.

A second perspective is that of coding a sequence of Bernoulli outcomes with an imprecise parameter,where it is convenient to think of a universal code in terms of first quantizing the parameter and then using that (imprecise) parameter to encode the input $x$ . For the Bernoulli example, the maximum likelihood parameter ${\theta }_{ML}$ satisfies

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