Prior Lab — Bayes Thinking Lab

⟳ Prior Credible Interval → Calculate Parameters

Lower bound

Upper bound

Mass (%)

      The core problem when choosing priors: As a researcher you often have a substantive intuition — e.g. "95% of plausible effects lie between −2 and +5" — but don't know which parameter values correspond to that intuition. For the Normal distribution this is directly computable (μ = midpoint, σ from quantile function). For all other distributions no closed-form formula is available. The solver above finds the parameters numerically.
    

brms Prior Syntax

What is a Prior?

A prior P(θ) is a probability distribution over possible parameter values, before you see the data. It encodes your prior knowledge.

Too vague (e.g. Uniform): you make no assumptions, but the sampler must explore huge spaces → slow, often divergent.

Too tight: you impede learning from the data. The posterior ≈ prior, regardless of the data.

Weakly informative → the golden middle ground: roughly excludes impossible values, but leaves ample room for learning.

Which distribution to choose?

Unconstrained parameters (α, β): NormalStudent-tCauchy

Positive parameters (σ, τ, λ): ExponentialHalf-NormalHalf-tGammaLog-Normal

Proportions / probabilities: Beta

Correlation matrices: LKJ

Rule of thumb: Student-t(3,0,σ) as a robust alternative to Normal, as heavier tails tolerate outliers in the prior.

Gelman's Recommendations

For z-standardised variables (SD=1):
· Intercept α: Normal(0, 2.5)
· Slope β: Normal(0, 1)
· Dispersion σ: Exponential(1) or Half-Normal(0,1)

For raw scales you need to rescale:
σ_prior ≈ 2–3 × SD(y) / SD(x)

Example reaction times (M≈400ms, SD≈80ms):
α: Normal(400, 100), β per SD(x): Normal(0, 160)

Common Pitfalls

Prior scale not matched to the outcome: Normal(0,1) for a slope on raw scales (e.g. kg, €, ms) implies +1 unit x → ±1 unit y — depending on the scale, absurdly tight or far too wide. Always keep the metric of the outcome in mind.

σ as Normal(0,x) in brms: This is correct — brms automatically applies lb=0 (truncation). No special handling needed, no half_normal().

ν for Student-t too small: ν=1 equals Cauchy (no E[x]), ν=2 has no finite variance. Recommended: ν ≥ 3, typically 3–7 for robustness.

Beta prior used despite its own predictors: Beta(α,β) for zi/zoi/coi only holds without its own predictors (no distributional model) — then the parameter lives directly on [0,1]. With predictors (e.g. zi ~ x), it becomes a logistic regression — then a Normal prior on the logit scale for Intercept/Slope, not a Beta prior directly on the proportion anymore.

Comparison Table — Priors for σ and τ

Distribution	brms syntax	E[σ]	Heavy tails	When?
Exponential(1)	`exponential(1)`	1.0	✗	Popular (rethinking package); not a brms default
Half-Normal(1)	`normal(0,1)`	0.80	✗	When σ < 2 is expected
Half-Student-t(3,0,1)	`student_t(3,0,1)`	~1.10	✓	brms default form (there with scale 2.5 instead of 1)
Half-Cauchy(1)	`cauchy(0,1)`	—	✓✓	Very vague; large σ values possible
Gamma(2, 0.1)	`gamma(2,0.1)`	20	✗	brms default for ν in Student-t

GLM Prior — Help

Why GLM Mode?

GLM mode makes visible what otherwise stays invisible: how a prior you set on the model scale (logit/log) actually plays out on the response scale (probability, rate, odds ratio) — and vice versa. That's the core of good prior choice for GLMs: a prior that looks harmless on the model scale can imply completely implausible values on the response scale (examples below).

This side-by-side visualization of model and response scale for the same prior is a feature unique to Prior Lab — also available via the embedded version in the Prior Predictive Check, but not found in any other tool in this lab. Seeing the two scales side by side makes the otherwise abstract scale difference tangible before you enter the prior into brms.

The Core Problem

Priors in brms must be specified on the model scale (logit or log) — but you think in probabilities and rates. These scales are non-linear: Normal(0, 10) on the logit scale sounds vague, but pushes almost all prior mass to the extremes p≈0% and p≈100%.

The solver bridges this gap: you specify your substantive bounds on the outcome scale — the solver returns the matching prior parameters on the model scale, ready to use in brms.

The two plots visualise exactly this relationship:

Top plot — Model scale: The prior distribution as it must be entered in brms (logit or log scale). Not intuitively readable, but technically correct.

Bottom plot — Response scale: What this prior implies on the outcome scale (probability, expected value, OR/RR). This is what we actually care about — but cannot be directly specified in brms. Visual reference only.

Why exp(β) = Multiplier?

Log-link (Poisson/Gamma):
log(μ) = α + β·X → μ = exp(α) · exp(β·X)

Comparing two groups (X: 0 → 1):

X=0: μ₀ = exp(α)
X=1: μ₁ = exp(α + β) = exp(α) · exp(β)
Rate Ratio = μ₁ / μ₀ = exp(β)

A β of e.g. 0.69 means: exp(0.69) ≈ 2 → rate doubles. Adding β on the log scale = multiplying μ by exp(β). That is why we think about the prior for β in terms of ratios, not differences.

Logit-link (Bernoulli/Binomial) — exactly the same structure, just with odds instead of rates:
logit(P) = log(P/(1−P)) = α + β·X

X=0: odds₀ = exp(α)
X=1: odds₁ = exp(α + β) = exp(α) · exp(β)
Odds Ratio = odds₁ / odds₀ = exp(β)

Intercept — Baseline Level

The intercept determines the baseline level when all predictors equal 0.

Logit-link: sigmoid(α) = baseline probability

"Without therapy, 10–40% of patients recover"
→ Input: 0.10 to 0.40
→ logit(0.10) ≈ −2.20 | logit(0.40) ≈ −0.41
→ Solver: Normal(−1.30, 0.46) on logit scale
μ < 0, because the baseline is below 50% ✓

Log-link: exp(α) = baseline expected value

"Baseline: 3–15 hospital admissions per year"
→ Input: 3 to 15
→ log(3) ≈ 1.10 | log(15) ≈ 2.71
→ Solver: Normal(1.90, 0.41) on log scale
μ ≈ 1.90, because exp(1.90) ≈ 6.7 — midpoint of 3–15 ✓

Slope / β — Effect per Unit of X

Logit-link → Odds Ratio (OR) = exp(β):

"Therapy might slightly reduce or strongly increase odds
— OR between 0.7 and 4.0 is plausible"
→ Input: OR 0.70 to 4.0
→ log(0.70) ≈ −0.36 | log(4.0) ≈ 1.39
→ Solver: Normal(0.51, 0.44) on logit scale
μ > 0, because the prior leans slightly toward positive effects ✓

Log-link → Rate Ratio (RR) = exp(β):

"Treatment increases event rate by 50%–5-fold"
→ Input: RR 1.5 to 5.0
→ log(1.5) ≈ 0.41 | log(5.0) ≈ 1.61
→ Solver: Normal(1.01, 0.31) on log scale
μ > 0, because the prior expects only positive effects ✓

Rule of Thumb for Slopes

      Normal(0, 0.5) → 95%-CrI OR/RR ≈ 0.37–2.7  (moderate)

      Normal(0, 1.0) → 95%-CrI OR/RR ≈ 0.14–7.4  (large)

      Normal(0, 2.0) → 95%-CrI OR/RR ≈ 0.02–55   (very vague)

Note on μ = 0: Only when the ratio bounds are symmetric around 1 (product = 1, e.g. 0.5 × 2.0 = 1), the solver necessarily returns μ = 0. This is the classic "no-effect prior". Asymmetric bounds — as in the examples above — yield μ ≠ 0 and encode substantive expectations.

ℹ Prior Lab — Help

What will I learn here?

The Prior Lab shows all important prior distributions for brms — interactively, with CI-solver and direct brms syntax.

Which distribution fits which parameter?
How to translate substantive intuitions (e.g. 95%-CrI) into parameter values?
What is the brms syntax for the chosen prior?
What are Gelman's recommendations for standard situations?

Workflow

Choose a distribution (left): pay attention to category — unbounded / positive / [0,1].
Use the CI-solver: enter substantive bounds (e.g. "95% between −2 and +5") → parameters are calculated automatically and transferred to the sliders.
Copy brms syntax: the code below updates live and can be pasted directly into R.

Prior Types

Vague (uniform, Cauchy): almost no prior assumptions, but sampler sluggish and often divergent.
Weakly informative: roughly excludes impossible values, leaves ample room for learning — the recommended approach.
Strongly informative: only when genuine prior knowledge is available (e.g. from meta-analysis).
Typical defaults for common parameters can be found in the "Load & explore brms defaults" section in the sidebar — click to load parameters directly.

Common Decisions

Intercept α: Student-t(3, 0, 2.5) — brms default
Slope β (z-std.): Normal(0, 1) — Gelman
Dispersion σ / τ: Exponential(1) or Half-Normal(0,1) — or Half-Student-t(3, 0, 2.5), the brms default
Probability / Prop.: Beta(1,1) to Beta(2,8)
Correlation: LKJ(2) — regularised around 0
Reaction times: Log-Normal(μ, σ) on log scale

Why this matters for Bayes

The prior is not a nuisance — it is part of the model. Posterior ∝ Likelihood × Prior. A prior that is too tight dominates the model, one that is too vague slows down the sampler. The CI-solver helps justify prior parameters substantively rather than choosing them arbitrarily.

Next → Prior Predictive Check: verify whether the chosen priors imply sensible predictions — before you see the data