_notation.qmd

{{< include latex-macros/macros.qmd >}}

Mathematical notation is not standardized.
This section states the conventions these notes use,
and the alternatives you may meet in other sources.

| symbol | meaning | LaTeX
|-------------|--------------------------------------------------|-----------
| $\neg$      | [not](proof-writing.qmd#def-not)                   | `\neg`
| $\forall$   | [for all](proof-writing.qmd#def-universal-quantifier)                       | `\forall`
| $\exists$   | [there exists](proof-writing.qmd#def-existential-quantifier)                  | `\exists`
| $\cup$      | union, "or"                                      | `\cup`
| $\cap$      | intersection, "and"                              | `\cap`
| $\mid$      | given, [conditional on](https://morrison-lab.github.io/pds/probability-basics.html#def-conditional-prob) | `\mid`, `|`
| $\sum$      | [sum](algebra.qmd#sec-summation)                 | `\sum`
| $\prod$     | product                                          | `\prod`
| $\mu$       | population mean, $\Expp[X]$, of a [random variable](https://morrison-lab.github.io/pds/random-variables.html#def-random-variable) $X$ | `\mu`
| $\Expp$     | [expectation](https://morrison-lab.github.io/pds/expectation.html#def-expectation)   | `\mathbb{E}`
| $x\'$  | transpose of $x$                                 | `x\'`
| $'$         | transpose or [derivative](calculus.qmd#def-derivative)[^3] | `'`
| $\ind$      | [independent](https://morrison-lab.github.io/pds/independence.html#def-indpt)         | `\perp\!\!\!\perp`
| $\tf$       | [therefore](proof-writing.qmd#def-logical-entailment), thus       | `\therefore`
| $\eta$      | [linear predictor][eta] of a generalized linear model[^4] | `\eta`
| $\floor{x}$ | floor of $x$: largest [integer](number-systems.qmd#def-integers) less than or equal to $x$ | `\lfloor x \rfloor`
| $\ceil{x}$  | ceiling of $x$: smallest [integer](number-systems.qmd#def-integers) greater than or equal to $x$ | `\lceil x \rceil`
| $\indic{A}(x)$, $\indicp{P}$ | indicator function ([definition](indicator-functions.qmd#def-indicator-function)): $1$ if condition holds, $0$ otherwise | `\indic{A}(x)`, `\indicp{P}` |

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

The links for conditioning, random variables, expectation, and independence
go to the Morrison Lab's probability notes,
which define those terms.

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

[^3]: depending on whether it is applied to a matrix or a function

[^4]: A generalized linear model (GLM) is a regression model
    in which a chosen function $g$, the link function,
    of the population mean $\mu$ of the outcome
    equals the linear predictor $\eta = x\' \beta$,
    a [linear combination](linear-algebra.qmd#def-linear-combination)
    of the predictor values $x$ with coefficients $\beta$:
    $g(\mu) = \eta$.

## Writing math in Quarto {#sec-writing-math-quarto}

The third column of @tbl-notation-collected
gives the LaTeX command for each symbol.
Quarto and R Markdown documents write math in LaTeX syntax:
put inline math between single dollar signs, as in `$x^2$`,
and displayed equations between double dollar signs, as in `$$x^2$$`.
The [equations section of Quarto's Markdown guide](https://quarto.org/docs/authoring/markdown-basics.html#equations)
shows the details.
These notes also use shorthand macros,
such as `\floor{x}` for $\floor{x}$,
defined in the `latex-macros` submodule's `macros.qmd`.

## Subscripts and parentheses {#sec-subscripts-parentheses}

{{< include mds/_subfiles/notation/_rem-subscript-parentheses-convention.qmd >}}

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{{< include mds/_subfiles/notation/_tbl-subscript-parentheses.qmd >}}

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{{< include mds/_subfiles/notation/_exm-process-notation.qmd >}}

## 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 run 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 [several competing notation systems](https://en.wikipedia.org/wiki/Notation_for_differentiation).
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).