2 Chapter 2: Advanced Topics
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2.1 Mathematical Equations
You can include mathematical equations using LaTeX syntax:
Inline equation: \(E = mc^2\)
Display equation:
\[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \]
2.2 Custom Macros
This book uses the d-morrison/macros submodule to provide convenient LaTeX shorthand for statistical and mathematical notation.
2.2.1 Probability and Distributions
The normal distribution \(\Normal\paren{\m, \ss}\) can be written using macros as $\Normal\paren{\m, \ss}$, where \m expands to \(\mu\) and \ss expands to \(\sigma^2\).
A random variable \(X \dist \Normal\paren{\mu, \sigma^2}\) has expectation \(\E{X} = \mu\) and variance \(\Var{X} = \sigma^2\).
2.2.2 Regression Notation
In linear regression, we estimate \(\vbeta\) using ordinary least squares. The fitted values are \(\hy = \mX \hb\), where \(\hb = \inv{\mX' \mX} \mX' \vy\).
The standard error of \(\hb\) is \(\hse{\hb} = \hs \sqrt{\inv{\mX' \mX}}\).
2.2.3 Likelihood and Estimation
The log-likelihood function \(\llik(\th)\) is maximized at the MLE \(\hth_{\text{ML}}\).
The score function is \(\score(\th) = \deriv{\th} \llik(\th)\) and the observed information is \(\oinf(\th) = -\hess(\th)\).
2.2.4 Logistic Regression
The logit link function is \(\logitf{\pi} = \logt\paren{\frac{\pi}{1 - \pi}}\) and its inverse is \(\expitf{\eta} = \frac{e^\eta}{1 + e^\eta}\).
2.3 Tables
| Column 1 | Column 2 | Column 3 |
|---|---|---|
| Data 1 | Data 2 | Data 3 |
| Data 4 | Data 5 | Data 6 |
2.4 Figures
You can include images by placing them in the images/ directory:
{#fig-example}Then reference the figure in text using @fig-example.
2.5 Callouts
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