notation.qmd

# Notation

{{< include macros.qmd >}}

| symbol | meaning | LaTeX
|-------------|--------------------------------------------------|-----------
| $\neg$      | not                                              | `\neg`
| $\forall$   | all                                              | `\forall`
| $\exists$   | some                                             | `\exists`
| $\cup$      | union, "or"                                      | `\cup`
| $\cap$      | intersection, "and"                              | `\cap`
| $\mid$      | given, conditional on                            | `\mid`, `|`
| $\sum$      | sum                                              | `\sum`
| $\prod$     | product                                          | `\prod`
| $\mu$       | mean                                             | `\mu`
| $\Expp$     | [expectation](probability.qmd#def-expectation)   | `\mathbb{E}`
| $x^{\top}$  | transpose of $x$                                 | `x^{\top}`
| $'$         | transpose or derivative[^1]                      | `'`
| $\ind$      | [independent](probability.qmd#def-indpt)         | `⫫`
| $\tf$       | therefore, thus                                  | `\therefore`
| $\eta$      | [linear component of a GLM][eta]                 | `\eta`
| $\floor{x}$ | floor of $x$: largest integer smaller than $x$   | `\lfloor x \rfloor`
| $\ceil{x}$  | ceiling of $x$: smallest integer larger than $x$ | `\lceil x \rceil`

: Notation used in this book {#tbl-notation-collected}

[eta]: https://en.wikipedia.org/wiki/Generalized_linear_model#:~:text=The%20linear%20predictor%20is%20the,data%20through%20the%20link%20function "linear predictor notation"

[^1]: depending on whether it is applied to a matrix or a function

## Information matrices

There is no consistency in the notation for observed and expected information matrices 
(see @tbl-info-mat-symbols).

| book                  | observed information | expected information |
|-----------------------| -------------------- | -------------------- |
|@dobson4e              | $U'$                 | $\mathfrak{I}$       |
|@dunn2018generalized   | $\mathfrak{I}$       | $\mathcal{I}$        |
|@mclachlan2007em       | $I$                  | $\mathcal{I}$        |
|@wood2017generalized   | $\hat{I}$            | $\mathcal{I}$        |

: notation for information matrices {#tbl-info-mat-symbols}

These notes currently have a mixture of notations, 
depending on my whims and what reference I had last looked at.
Eventually, I will try to standardize my notation to 
$\oinf$ for observed information
and $\einf$ for expected information.

## Percent sign ("%")

The percent sign "%" is just a shorthand for "$\times \frac{1}{100}$". 
The word "percent" comes from the Latin "per centum"; "centum" means 100 in Latin, so "percent" means "per hundred" (c.f., <https://en.wikipedia.org/wiki/Percentage>)

So, contrary to what you may have learned previously, $10\% = 0.1$ is a true and correct equality.

:::{.proof}

$$
\ba
10\% &= 10 \times \frac{1}{100}
\\ &= \frac{10}{100}
\\ &= 0.1
\ea
$$

:::

## Why is notation in probability and statistics so inconsistent and disorganized?

In grad school, 
we are asked to learn
from increasingly disorganized materials and lectures. 
Not coincidentally, 
as the amount of organization decreases, 
the amount of complexity increases, 
the amount of difficulty increases,
the number of reliable references decreases, 
and the amount of inconsistency in notation and content increases 
(both between multiple references and within single references!). 
In other words, 
as you approach the cutting-edge of most fields, 
you start to encounter into content 
that hasn’t been fully thought through or standardized. 
This lack of clarity 
is unfortunate and undesirable, 
but it is understandable and inevitable.

It's worth noting 
that calculus was formalized in the [1600s](https://en.wikipedia.org/wiki/Leibniz%27s_notation), 
elementary algebra was formalized  around [820](https://en.wikipedia.org/wiki/Al-Jabr), 
and arithmetic [even earlier](https://en.wikipedia.org/wiki/Arithmetic#History).
And calculus still has at least two major notation systems:
[Leibniz's notation](https://en.wikipedia.org/wiki/Leibniz%27s_notation) ($\deriv{y}x$) 
and [Euler's notation](https://en.wikipedia.org/wiki/Notation_for_differentiation#Lagrange's_notation)
([more commonly called](https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy) [Lagrange notation](https://en.wikipedia.org/wiki/Notation_for_differentiation#Lagrange's_notation)).
In contrast, the field of statistics only emerged in the 
[late 1800s and early 1900s](https://en.wikipedia.org/wiki/History_of_statistics#Development_of_modern_statistics), 
so it’s not surprising that the notation and terminology is still developing.
Generalized linear models were only formalized in 1972 (@nelder1972generalized),
which is very recent in terms of the [pace of scientific development](https://en.wikipedia.org/wiki/The_Structure_of_Scientific_Revolutions).