---
date: '2024-01-11'
description: in information theory
id: Entropy
modified: 2026-06-05 15:08:29 GMT-04:00
tags:
  - seed
title: Entropy
created: '2024-01-11'
published: '2024-01-11'
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slug: thoughts/Entropy
permalink: https://aarnphm.xyz/thoughts/Entropy.md
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---
<blockquote class="twitter-tweet" data-lang="fr" data-dnt="true"><p lang="en" dir="ltr">In particular, "good, aligned, conversational AI" is just one of many possible different rollouts. Finetuning / alignment tries to "collapse" and control the entropy to that region of the simulator. Jailbreak prompts try to knock the state into other logprob ravines.</p>&mdash; Andrej Karpathy (@karpathy) <a href="https://x.com/karpathy/status/1632800082679705600?ref_src=twsrc%5Etfw">6 mars 2023</a></blockquote>



idea: quantifies average level of uncertainty information associated with a variable’s potential states or possible outcome.

> \[!definition\] Definition 1.
>
> Given a discrete random variable $\mathcal{X}$, which takes a value in a set $\mathcal{X}$ distributed according to $p : \mathcal{X} \to [0,1]$, the entropy $H(x)$ is defined as
>
> $$
> H(X) \coloneqq - \sum_{x \in \mathcal{X}} p(x) \log p(x)
> $$

Base 2 gives unit of “bits” (or “shannons”), while natural base gives “natural units” (or “nat”), and base 10 gives unit of “dits” (or “bans”, or “hartleys”)

## The Waluigi Effect

[_source_](https://www.lesswrong.com/posts/D7PumeYTDPfBTp3i7/the-waluigi-effect-mega-post)

talks more about the semiotic phenomena which arise within LLMs, and how hallucination come to be.

## joint

_measure of uncertainty associated with a set of variables_

namely, the joint _Shannon entropy_,

> \[!definition\] Definition 2. joint _Shannon entropy_
>
> in bits, of two discrete random variable $X$ and $Y$ with images $\mathcal{X}$ and $\mathcal{Y}$ is defined as:
>
> $$
> H(X,Y) = - \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} P(x,y) \log_2 [P(x,y)]
> $$
>
> where $P(x,y)$ is the joint probability of both $X$ and $Y$ occurring together.

For more than two random variables, this expands onto:

$$
H(X_{1},\ldots,X_{n}) = - \sum_{x_{1} \in \mathcal{X}_{1}} \cdots \sum_{x_{n} \in \mathcal{X}_{n}} P(x_{1},\ldots,x_{n}) \log_2 [P(x_{1},\ldots,x_{n})]
$$