4.2 A language for describing models

A clear model notation separates the likelihood from the linear model. Writing the model as a list of distributional assumptions and deterministic equations makes it easier to communicate, critique, and extend.

4.2.1 Model notation

The standard notation for a Bayesian model has two parts:

1. Probability statements: Describe how the data are distributed and the priors on parameters.
2. Deterministic statements: Describe how the parameters of the likelihood depend on predictors.

For a simple linear regression:

$y_i \sim \text{Normal}(\mu_i, \sigma)$ $\mu_i = \alpha + \beta x_i$ $\alpha \sim \text{Normal}(0, 10)$ $\beta \sim \text{Normal}(0, 10)$ $\sigma \sim \text{Uniform}(0, 50)$

The first line is the likelihood: each observation $y_i$ is normally distributed around $\mu_i$ with standard deviation $\sigma$. The second line is the linear model: the expected value $\mu_i$ is a linear function of the predictor $x_i$. The remaining lines are the priors.

4.2.2 Why this notation matters

This notation is powerful because:

• Transparency: Every assumption is explicit. You can see the likelihood, the linear model, and the priors at a glance.
• Modifiability: You can change any component independently. Swap the likelihood, add a predictor, change a prior.
• Generality: The same notation works for linear models, generalized linear models, multilevel models, and more.
• Communication: The notation is compact and unambiguous, making it easy to share models with others.

# Model notation in R: the rethinking package uses alist
# A simple linear regression model:
# y_i ~ Normal(mu_i, sigma)
# mu_i = alpha + beta * x_i
# alpha ~ Normal(0, 10)
# beta ~ Normal(0, 10)
# sigma ~ Uniform(0, 50)

# In rethinking notation:

# model <- alist(

# y ~ dnorm(mu, sigma),

# mu <- a + b * x,

# a ~ dnorm(0, 10),

# b ~ dnorm(0, 10),

# sigma ~ dunif(0, 50)

# )

# Demonstrate with simulated data

set.seed(42)
n <- 100
x <- runif(n, 0, 10)
y <- 2 + 1.5 * x + rnorm(n, sd = 2)

# Fit with lm (frequentist equivalent)

fit <- lm(y ~ x)
cat("Intercept (alpha):", round(coef(fit)[1], 3), "\n")
cat("Slope (beta):", round(coef(fit)[2], 3), "\n")
cat("Residual SD (sigma):", round(summary(fit)$sigma, 3), "\n")

# Plot

plot(x, y, pch = 16, col = "steelblue",
main = "Linear regression: y = alpha + beta * x",
xlab = "x", ylab = "y")
abline(fit, lwd = 2, col = "firebrick")

4.2.3 Link functions

For generalized linear models (Chapter 9), we add a link function $g$ that maps the linear model to the parameter of the likelihood:

$y_i \sim f(\theta_i)$ $g(\theta_i) = \alpha + \beta x_i$

For example, for a binomial model with a logit link:

$y_i \sim \text{Binomial}(n, p_i)$ $\text{logit}(p_i) = \alpha + \beta x_i$

The link function ensures that the parameter stays in the valid range (e.g., $p \in [0, 1]$ for the binomial). We will study link functions in detail in Chapter 9.

# Demonstrate a link function: logit
# logit(p) = log(p / (1 - p))
# inverse logit: p = 1 / (1 + exp(-x))

x <- seq(-5, 5, length.out = 100)
logit_vals <- x
prob_vals <- 1 / (1 + exp(-x))

par(mfrow = c(1, 2))
plot(x, logit_vals, type = "l", lwd = 2, col = "steelblue",
main = "Logit link: logit(p) = alpha + beta * x",
xlab = "Linear predictor", ylab = "logit(p)")
abline(h = 0, lty = 2, col = "gray50")

plot(x, prob_vals, type = "l", lwd = 2, col = "firebrick",
main = "Inverse logit: p = 1/(1+exp(-x))",
xlab = "Linear predictor", ylab = "p")
abline(h = 0.5, lty = 2, col = "gray50")
abline(v = 0, lty = 2, col = "gray50")
par(mfrow = c(1, 1))

Summary of Section 4.2

• Model notation separates likelihood, linear model, and priors into explicit statements.
• The notation is transparent, modifiable, general, and communicable.
• Link functions map the linear model to the likelihood parameter for generalized linear models.
• The model notation makes every assumption visible and critiquable.