<commit_before><commit_msg>started a fisher information post!<commit_after>---
title: fisher metric
date: 2016-04-30
---

# The Fisher Metric

The [Cramer-Rao Bound](https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound) is 
a fundamental result in parametric inference. In this post, we are going to begin our 
journey towards a proof which uses tools from differential geometry.

Let $X$ be a random variable and $f(x,\theta)$ a family of possible distrubutions for 
$X$, parameterized by $\theta \in \Theta$, where $\Theta$ is a manifold. If 
$\alpha,\beta \in T_{\theta} \Theta$, then we define
{'fraction_non_alphanumeric': 0.11589403973509933, 'fraction_numerical': 0.024834437086092714, 'mean_word_length': 4.401785714285714, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'dir': 'git-commits-cleaned', 'id': '6950596', 'n_tokens_mistral': 202, 'n_tokens_neox': 187, 'n_words': 75}