Bayesian Models for Ecologists

Conventions for Statistical Notation

May 26, 2022

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There are conventions for writing statistical models that exist for a reason: they make it easier to read equations and understand them. Failing to adhere to these conventions will raise eyebrows with reviewers of your papers. (Never give a reviewer a reason to knit his or her brow.) The following points outline the most important rules for writing mathematical expressions of statistical ideas.

• Brackets, i.e. \([\,]\) are used to indicate stochastic relationships and for no other purpose. As you will learn, \([a \mid b,c]\) means the probability or probability density of \(a\) conditional on \(b\) and \(c\). Do not use brackets in the same way you would use parentheses. For example, \(\alpha((b-c)+(d+f))\) is correct; \(\alpha[(b-c)+(d+f)]\) is incorrect. Do not use brackets for named distributions: \(\text{normal}(y\mid\mu,\sigma^2)\) and \(y \sim \text{normal}(\mu, \sigma^2)\) are correct; \(\text{normal}[y\mid\mu,\sigma^2]\) and \(y \sim \text{normal}[\mu, \sigma^2]\) are incorrect.

• Notate scalars as lowercase, math font, for example: \(y,x,\sigma, \omega\)

• Notate vectors as lowercase, math bold font, for example: \(\mathbf{y,x,}\,\boldsymbol{\sigma, \omega}\)

• Notate matrices as uppercase, math bold font, \(\mathbf{Y,X,}\,\boldsymbol{\Sigma,\Omega}\)

• Notate distribution names using plain text font, for example: \(\text{normal}(\mu, \sigma^2)\)

• When possible, use single letters for response and predictor variables in models. First choice for scalar responses is always \(y\); for predictor variables is \(x\); and for latent quantities, \(z\). Vector and matrix forms use the same letters with appropriate bolding and capitalization. Other letters can be used as needed.

• Use Greek letters for model parameters, following the case and bolding conventions described above for scalars, vectors, and matrices.

• Avoid using variable names like \(precip\) or \(temp\). These make for cluttered equations and can be confused with multiplication. You can use subscripts to give greater meaning to single variables, for example, \(\sigma_p\) or \(x_{precip}\). If you want to be strict, you should get in the habit of using a plain text font for these descriptive subscripts, for example \(\sigma_{\text{p}}\) to conform to style for ESA journals.

• The only time you should use multiplication symbols is at the left end of multi-line equations, as illustrated above in the equation array. Never use * in mathematical expressions; * is a coding symbol, not a mathematical one. Write \(\beta x_i\) not \(\beta\times x_i\)

• Use expanding brackets when needed. For example: \(\left(\frac{\mu^2}{\sigma^2} \right)\) not \((\frac{\mu^2}{\sigma^2})\).