Bayes Primer
Ecological Forecasting Workshop, Cape Town, July 2025
Joint and marginal: continuous case
In our penguin example, \(X\) and \(Y\) are discrete.
For continuous \(X\) and \(Y\), the joint and marginal distributions are determined by probability density functions.
joint density \(f(x,y)\),
marginal densities are obtained by “integrating out” the other variable(s)
for example, \[f(y) = \int f(x,y)dx \]
Conditional distributions
In general, conditional distributions are “joint divided by marginal”.
In the continuous setting this applies to the densities.
\[ f(x|y) = \frac{f(x,y)}{f(y)} \hspace{5pt} \text{ and } \hspace{5pt} f(y|x) = \frac{f(x,y)}{f(x)} \]
Geometrically, conditional densities are “rescaled slices” of the joint.
- e.g \(f(y|x=-2)\) and \(f(y|x=2)\)
Conditional probability caution
ambiguity in language: consider “probability of breeding at colony A”
“probability of breeding given that it’s at colony A”
“probability of breeding and doing so at colony A”
What’s being modelled with \(f(x,y)\) vs. \(f(x|y)\)
\(f(x,y)\): both \(X,Y\) are modelled in the sense that there are probabilities associated with both
\(f(x|y)\): only \(X\) is being modelled, \(Y\) is assumed fixed with value \(y\).
conditional probability is easily misunderstood
A distribution from a sample
In most models, we can’t obtain the posterior analytically. Fortunately there is a powerful set of algorithms that draw samples from the posterior. Any distribution can be approximated by a sufficiently large sample from it.
