# 0.4 Universal compression for context tree sources

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## Semi-predictive approach

Recall that a context tree source is similar to a Markov source, where the number of states is greatly reduced. Let $T$ be the set of leaves of a context tree source, then the redundancy is

$r\lesssim \frac{|T|\left(r-1\right)}{2}\left(log,\left(\frac{n}{|T|}\right),+,O,\left(1\right)\right),$

where $|T|$ is the number of leaves, and we have $log\left(\frac{n}{|T|}\right)$ instead of $log\left(n\right)$ , because each state generated $\frac{n}{|T|}$ symbols, on average. In contrast, the redundancy for a Markov representation of the tree source $T$ is much larger. Therefore, tree sources are greatly preferable in practice, they offer a significant reductionin redundancy.

How can we compress universally over the parametric class of tree sources? Suppose that we knew $T$ , that is we knew the set of leaves. Then we could process $x$ sequentially, where for each ${x}_{i}$ we can determine what state its context is in, that is the unique suffix of ${x}_{1}^{i-1}$ that belongs to the set of leaf labels in $T$ . Having determined that we are in some state $s$ , $Pr\left({x}_{i}=0|s,{x}_{1}^{i-1}\right)$ can be computed by examining all previous times that we were in state $s$ and computing the probability with the Krichevsky-Trofimov approach based on the number of times that the following symbol(after $s$ ) was 0 or 1. In fact, we can store symbol counts ${n}_{x}\left(s,0\right)$ and ${n}_{x}\left(s,1\right)$ for all $s\in T$ , update them sequentially as we process $x$ , and compute $Pr\left({x}_{i}=0|s,{x}_{1}^{i-1}\right)$ efficiently. (The actual translation to bits is performed with an arithmetic encoder.)

While promising, this approach above requires to know $T$ . How do we compute the optimal ${T}^{*}$ from the data?

Semi-predictive coding : The semi-predictive approach to encoding for context tree sources  [link] is to scan the data twice, where in the first scan we estimate ${T}^{*}$ and in the second scan we encode $x$ from ${T}^{*}$ , as described above. Let us describe a procedure for computing the optimal ${T}^{*}$ among tree sources whose depth is bounded by $D$ . This procedure is visualized in [link] . As suggested above, we count ${n}_{x}\left(s,a\right)$ , the number of times that each possible symbol appeared in context $s$ , for all $s\in {\alpha }^{D},a\in \alpha$ . Having computed all the symbol counts, we process the depth- $D$ tree in a bottom-top fashion, from the leaves to the root, where for each internal node $s$ of the tree (that is, $s\in {\alpha }^{d}$ where $d ), we track ${T}_{s}^{*}$ , the optimal tree structure rooted at $s$ to encode symbols whose context ends with $s$ , and $\text{MDL}\left(s\right)$ the minimum description lengths (MDL) required for encoding these symbols.

Without loss of generality, consider the simple case of a binary alphabet $\alpha =\left\{0,1\right\}$ . When processing $s$ we have already computed the symbol counts ${n}_{x}\left(0s,0\right)$ and ${n}_{x}\left(0s,1\right)$ , ${n}_{x}\left(1s,0\right)$ , ${n}_{x}\left(1s,1\right)$ , the optimal trees ${T}_{0s}^{*}$ and ${T}_{1s}^{*}$ , and the minimum description lengths (MDL) $\text{MDL}\left(0s\right)$ and $\text{MDL}\left(1S\right)$ . We have two options for state $s$ .

1. Keep ${T}_{0S}^{*}$ and ${T}_{1S}^{*}$ . The coding length required to do so is $\text{MDL}\left(0S\right)+\text{MDL}\left(1S\right)+1$ , where the extra bit is spent to describe the structure of the maximizing tree.
2. Merge both states (this is also called tree pruning ). The symbol counts will be ${n}_{x}\left(s,\alpha \right)={n}_{x}\left(0s,\alpha \right)+{n}_{x}\left(1s,\alpha \right),\alpha \in \left\{0,1\right\}$ , and the coding length will be
$\text{KT}\left({n}_{x}\left(s,0\right),{n}_{x}\left(s,1\right)\right)+1,$
where $\text{KT}\left(·,·\right)$ is the Krichevsky-Trofimov length  [link] , and we again included an extra bit for the structure of the tree.

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