Setting Priors in brms

Bayesian Mixed Effects Models with brms for Linguists

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• 1 Setting Priors in brms (20 min)
  • 1.1 Default vs. Weakly Informative Priors
    • 1.1.1 What is a Prior?
    • 1.1.2 The Intercept Adapts to Your Data!
  • 1.2 Default brms Priors
  • 1.3 Setting Weakly Informative Priors for Reaction Times
    • 1.3.1 Why These Numbers?
    • 1.3.2 Comparison: Normal vs. Student-t
  • 1.4 Summary

1 Setting Priors in brms (20 min)

1.1 Default vs. Weakly Informative Priors

brms uses weakly informative priors by default (not completely flat). However, for psycholinguistics, domain-specific priors are even better.

1.1.1 What is a Prior?

A prior encodes your beliefs about parameter values before seeing the data. In Bayesian inference:

\[\text{posterior} \propto \text{likelihood} \times \text{prior}\]

Types of priors:

• Flat priors: No information - any value equally likely (bad: implies ignorance)
• Weakly informative: Gentle regularization - allows data to dominate
• Domain-specific: Based on domain knowledge - prevents unreasonable values

1.1.2 The Intercept Adapts to Your Data!

This is important: The default intercept prior depends on mean(y). Let’s see this in action:

library(brms)
library(tidyverse)

# Create example datasets with different scales
set.seed(123)

# RT data: log-transformed, mean ≈ 6 (≈ 400ms)
rt_data_typical <- data.frame(
  subject = factor(rep(1:20, each = 10)),
  item = factor(rep(1:10, times = 20)),
  condition = factor(rep(c("A", "B"), each = 5, times = 20)),
  log_rt = rnorm(200, mean = 6, sd = 0.5)
)

# RT data: extreme scale, mean ≈ 10 (≈ 22,000ms - unrealistic)
rt_data_extreme <- data.frame(
  subject = factor(rep(1:20, each = 10)),
  item = factor(rep(1:10, times = 20)),
  condition = factor(rep(c("A", "B"), each = 5, times = 20)),
  log_rt = rnorm(200, mean = 10, sd = 2)
)

Now let’s check what default priors brms suggests:

cat("\n=== DEFAULT PRIORS: Typical RT data (mean log-RT ≈ 6) ===\n\n")

=== DEFAULT PRIORS: Typical RT data (mean log-RT ≈ 6) ===

rt_priors_typical <- get_prior(
  log_rt ~ condition + (1 + condition | subject) + (1 | item),
  data = rt_data_typical, 
  family = gaussian()
)
print(rt_priors_typical)

                prior     class       coef   group resp dpar nlpar lb ub tag
               (flat)         b                                             
               (flat)         b conditionB                                  
               lkj(1)       cor                                             
               lkj(1)       cor            subject                          
 student_t(3, 6, 2.5) Intercept                                             
 student_t(3, 0, 2.5)        sd                                     0       
 student_t(3, 0, 2.5)        sd               item                  0       
 student_t(3, 0, 2.5)        sd  Intercept    item                  0       
 student_t(3, 0, 2.5)        sd            subject                  0       
 student_t(3, 0, 2.5)        sd conditionB subject                  0       
 student_t(3, 0, 2.5)        sd  Intercept subject                  0       
 student_t(3, 0, 2.5)     sigma                                     0       
       source
      default
 (vectorized)
      default
 (vectorized)
      default
      default
 (vectorized)
 (vectorized)
 (vectorized)
 (vectorized)
 (vectorized)
      default

cat("\n\n=== DEFAULT PRIORS: Extreme RT data (mean log-RT ≈ 10) ===\n\n")

=== DEFAULT PRIORS: Extreme RT data (mean log-RT ≈ 10) ===

rt_priors_extreme <- get_prior(
  log_rt ~ condition + (1 + condition | subject) + (1 | item),
  data = rt_data_extreme, 
  family = gaussian()
)
print(rt_priors_extreme)

                 prior     class       coef   group resp dpar nlpar lb ub tag
                (flat)         b                                             
                (flat)         b conditionB                                  
                lkj(1)       cor                                             
                lkj(1)       cor            subject                          
 student_t(3, 10, 2.5) Intercept                                             
  student_t(3, 0, 2.5)        sd                                     0       
  student_t(3, 0, 2.5)        sd               item                  0       
  student_t(3, 0, 2.5)        sd  Intercept    item                  0       
  student_t(3, 0, 2.5)        sd            subject                  0       
  student_t(3, 0, 2.5)        sd conditionB subject                  0       
  student_t(3, 0, 2.5)        sd  Intercept subject                  0       
  student_t(3, 0, 2.5)     sigma                                     0       
       source
      default
 (vectorized)
      default
 (vectorized)
      default
      default
 (vectorized)
 (vectorized)
 (vectorized)
 (vectorized)
 (vectorized)
      default

# Compare intercept priors
cat("\n=== Intercept Comparison ===\n")

=== Intercept Comparison ===

cat("Typical data prior:  ", 
    rt_priors_typical[rt_priors_typical$class == "Intercept", "prior"], "\n")

Typical data prior:   student_t(3, 6, 2.5) 

cat("Extreme data prior:  ", 
    rt_priors_extreme[rt_priors_extreme$class == "Intercept", "prior"], "\n")

Extreme data prior:   student_t(3, 10, 2.5) 

cat("→ The intercept prior CHANGES with data scale!\n")

→ The intercept prior CHANGES with data scale!

Key insight: The intercept prior automatically scales with your data. This is convenient but has a problem: if you don’t specify priors, your prior assumptions implicitly depend on how you code your variables!

1.2 Default brms Priors

When you don’t specify priors, brms assigns defaults:

• Intercept: student_t(3, mean(y), 2.5) - DATA-DEPENDENT! Centers at your data mean
  • Adapts to your data scale automatically
  • For RT data with mean(log_rt) = 6: allows roughly 150ms-1100ms range
• Slopes (b): (flat) - improper uniform prior over (-∞, +∞)
  • No information: any effect size equally likely
  • Technically improper (doesn’t integrate to 1)
• Sigma (residual SD): student_t(3, 0, 2.5) with lower bound 0
  • Weakly informative for residual variance
• SD (random effects): student_t(3, 0, 2.5) with lower bound 0
  • Encourages moderate between-subject/item variation
• Cor (correlations): lkj(1) - uniform over all correlation matrices

1.3 Setting Weakly Informative Priors for Reaction Times

For psycholinguistics, it’s better to specify priors based on domain knowledge:

# Define priors explicitly
rt_priors <- c(
  prior(normal(6, 1.5), class = Intercept, lb = 4),  # log(RT) around 400ms, min ~55ms
  prior(normal(0, 0.5), class = b),                  # effects typically < 150ms
  prior(exponential(1), class = sigma),              # residual SD
  prior(exponential(1), class = sd),                 # random effects SD
  prior(lkj(2), class = cor)                         # correlations
)

cat("Our weakly informative priors:\n")

Our weakly informative priors:

print(rt_priors)

          prior     class coef group resp dpar nlpar   lb   ub tag source
 normal(6, 1.5) Intercept                               4 <NA>       user
 normal(0, 0.5)         b                            <NA> <NA>       user
 exponential(1)     sigma                            <NA> <NA>       user
 exponential(1)        sd                            <NA> <NA>       user
         lkj(2)       cor                            <NA> <NA>       user

1.3.1 Why These Numbers?

1.3.1.1 normal(6, 1.5) for Intercept with lb = 4

• Mean = 6 on log scale → exp(6) ≈ 403ms (typical RT)
• SD = 1.5 → 95% prior interval spans [3, 9] on log scale
• Lower bound = 4 → exp(4) ≈ 55ms (minimum physiologically possible motor response)
  • Without lb, the prior puts only ~9% probability below this threshold
  • Adding lb makes our assumption explicit: we don’t entertain impossibly fast RTs
  • In practice, data will dominate anyway, but bounded priors clarify our theoretical assumptions
• But 95% of prior mass is between ±1.96 × 1.5 around the mean
• This puts most probability on reasonable RTs (100-1500ms), while still allowing outliers

1.3.1.2 normal(0, 0.5) for Effects

• Mean = 0 (no directional assumption)
• SD = 0.5 on log scale
• 95% prior interval: [-1, 1] on log scale
• Translates to effect sizes of roughly ±65% of baseline (multiplicative)
• Or approximately ±100-150ms for typical RTs around 400-600ms
• Why not flat? Flat priors prefer extreme effect sizes (counterintuitive!)

1.3.1.3 exponential(1) for Sigma and SD

• Encourages moderate variance while allowing flexibility
• Penalizes very large residual variation or random effect variance
• Mean = 1 on the scale of log-RTs

1.3.1.4 lkj(2) for Correlations

• η = 2: slight preference for correlations near 0
• Skeptical of strong correlations (like perfect intercept-slope correlation)
• If you truly expected strong correlations, you could use lkj(1) or lower

1.3.2 Comparison: Normal vs. Student-t

brms defaults use student_t(3, μ, 2.5) which has heavier tails than normal(). Why switch?

# Compare tail behavior
set.seed(456)
n_samples <- 100000
normal_samples <- rnorm(n_samples, 0, 0.5)
studentt_samples <- rt(n_samples, 3) * 0.5

normal_99 <- quantile(normal_samples, c(0.001, 0.999))
studentt_99 <- quantile(studentt_samples, c(0.001, 0.999))

cat("Tails comparison (1st and 99th percentiles):\n")

Tails comparison (1st and 99th percentiles):

cat("Normal(0, 0.5):       ", round(normal_99, 2), "\n")

Normal(0, 0.5):        -1.55 1.56 

cat("Student-t(3) × 0.5:   ", round(studentt_99, 2), "\n")

Student-t(3) × 0.5:    -5.26 5.16 

cat("Student-t ratio:      ", round(studentt_99[2] / normal_99[2], 2), "x wider\n\n")

Student-t ratio:       3.3 x wider

cat("Student-t allows extreme values 2x more likely than normal!\n\n")

Student-t allows extreme values 2x more likely than normal!

# Graphical comparison
library(ggplot2)
comparison_df <- data.frame(
  value = c(normal_samples, studentt_samples),
  distribution = rep(c("Normal(0, 0.5)", "Student-t(3, 0, 0.5)"), each = n_samples)
)

ggplot(comparison_df, aes(x = value, fill = distribution, color = distribution)) +
  geom_density(alpha = 0.3, linewidth = 1) +
  geom_vline(xintercept = normal_99, linetype = "dashed", color = "#F8766D", linewidth = 0.7) +
  geom_vline(xintercept = studentt_99, linetype = "dashed", color = "#00BFC4", linewidth = 0.7) +
  coord_cartesian(xlim = c(-3, 3)) +
  scale_fill_manual(values = c("#F8766D", "#00BFC4")) +
  scale_color_manual(values = c("#F8766D", "#00BFC4")) +
  labs(
    title = "Normal vs. Student-t Prior Distributions",
    subtitle = "Student-t has heavier tails (dashed lines = 0.1% and 99.9% quantiles)",
    x = "Value",
    y = "Density",
    fill = "Distribution",
    color = "Distribution"
  ) +
  theme_minimal(base_size = 12) +
  theme(legend.position = "bottom")

When to use each:

• Student-t: Default choice (conservative, robust to outliers)
• Normal: When you have strong domain knowledge about plausible ranges (better for RT data in controlled experiments)

Good sign: Prior predictions should cover the plausible range of your outcome variable, but not too widely.

1.4 Summary

Key takeaways:

1. Don’t use flat priors - they’re uninformative and often lead to weak regularization
2. Default intercept priors adapt to data - implicit assumptions depend on your coding!
3. Specify priors explicitly based on domain knowledge
4. Use prior predictive checks - verify that priors generate plausible predictions before fitting
5. Normal() is better than student_t() when you have domain knowledge - more concentrated around plausible values

Next steps: - See 02_prior_predictive_checks_rt.qmd for detailed prior validation - See 03_posterior_predictive_checks_rt.qmd for checking if the fitted model makes sense

Setting Priors in brms Setting Priors in brms

Setting Priors in brms Setting Priors in brms Setting Priors in brms Bayesian Mixed Effects Models with brms for Linguists Bayesian Mixed Effects Models with brms for Linguists