Family of distributions
Distributions that can be described by the same set of parameters make up a family of distributions. For example the normal/Gaussian family includes all of the possible distributions that can be described by the two parameters \(\mu\) and \(\sigma\) alone.
In brms you set the family of the probability distribution of the outcome variable with the argument family. For example, family = gaussian(), family = bernoulli().
In practice, you can think of the family as the prior probability distribution of the outcome variable.
Likelihood
This is the probability distribution of the outcome variable conditional on the prior(s).
Notation: \(p(d|\theta)\).
Outcome/response/dependent variables
These are the variables that appear on the left-hand side of a model formula. For example, f0 in f0 ~ attitude; F1 and F2 in c(F1, F2) ~ stress.
Parameter
A parameter in a statistical model (for example intercept/mean, standard deviation, slope/\(\beta\), etc).
A parameter used to describe a probability distribution (for example \(\mu\) and \(\sigma\) for normal/Gaussian distributions).
Predictors, independent variables
These are the variable that appear on the right-hand side of a model formula. For example, novel_word in reaction_t ~ novel_word; s(longitude, latitude) in temperature ~ s(longitude, latitude).
Prior probability distribution or simply prior
This is the probability distribution of the values a parameter can take, based on prior knowledge/belief, domain expertise, previous research, pilot data.
Notation: \(p(\theta)\).