# A Lower-Bound for the Maximin Redundancy in Pattern Coding

## Abstract

**:**

## 1. Introduction

#### 1.1. Universal Coding

- First, a deterministic approach judges the performance of ${q}_{n}$ in the worst case by the maximal redundancy ${R}^{+}({q}_{n},\Theta )={sup}_{\theta \in \Theta}R({q}_{n},\theta )$ The lowest achievable maximal redundancy is called minimax redundancy:$${R}^{+}(n,\Theta )=\underset{{q}_{n}}{min}\underset{\theta}{max}R({q}_{n},\theta )$$
- Second, a Bayesian approach consists in providing Θ with a prior distribution π, and then considering the expected redundancy ${\mathbb{E}}_{\pi}\left[R({q}_{n},\theta )\right]$ (the expectation is here taken over θ). Let ${q}_{n}^{\pi}$ be the coding distribution minimizing ${\mathbb{E}}_{\pi}\left[R({q}_{n},\theta )\right]$ The maximin redundancy ${R}^{-}(n,\Theta )$ of class $\mathcal{C}$ is the supremum of all ${\mathbb{E}}_{\pi}\left[R({q}_{n}^{\pi},\theta )\right]$ over all possible prior distributions π:$${R}^{-}(n,\Theta )=\underset{\pi}{max}\underset{{q}_{n}}{min}{\mathbb{E}}_{\pi}\left[R({q}_{n},\theta )\right]$$

#### 1.2. Dictionary and Pattern

- a dictionary $\Delta =\Delta \left(x\right)$ defined as the sequence of different characters present in x in order of appearance; in the example $\Delta =(a,b,r,c,d)$.
- a pattern $\psi =\psi \left(x\right)$ defined as the sequence of positive integers pointing to the indices of each letter in Δ; here, $\psi =12314151231$.

#### 1.3. Pattern Coding

## 2. Theorem

**Theorem 1**For all integers n large enough, the maximin pattern redundancy is lower-bounded as:

## 3. Proof

**Figure 1.**The profile of pattern ψ forms a partition of n that can be “shrunk” to θ, the parameter partition of c, with high probability.

## Acknowledgment

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**MDPI and ACS Style**

Garivier, A. A Lower-Bound for the Maximin Redundancy in Pattern Coding. *Entropy* **2009**, *11*, 634-642.
https://doi.org/10.3390/e11040634

**AMA Style**

Garivier A. A Lower-Bound for the Maximin Redundancy in Pattern Coding. *Entropy*. 2009; 11(4):634-642.
https://doi.org/10.3390/e11040634

**Chicago/Turabian Style**

Garivier, Aurélien. 2009. "A Lower-Bound for the Maximin Redundancy in Pattern Coding" *Entropy* 11, no. 4: 634-642.
https://doi.org/10.3390/e11040634