Specify a normal prior

Construct a normal prior for passing to mlumr() via prior_intercept, prior_beta, or prior_sigma. The resulting list is consumed by the Stan models.

Usage

prior_normal(mean = 0, sd = 10, autoscale = FALSE)

Arguments

mean

Prior mean (default 0).

sd

Prior standard deviation (default 10). The default matches the historical "very weak" scale; explicit tighter values are recommended for regression coefficients (see Details).

autoscale

If TRUE and this prior is passed as prior_beta, the scale is divided by each covariate's empirical SD so the prior is weakly-informative regardless of predictor scaling. Default FALSE to preserve backward-compatible behavior; set to TRUE explicitly when passing unstandardized predictors. Ignored for prior_intercept and prior_sigma.

Value

A list with components distribution, mean, sd, df, autoscale.

Choosing a scale

The Stan community's prior-choice wiki (Vehtari et al., 2025) describes five broad categories, from least to most informative:

1. Flat prior — not recommended.

2. Super-vague proper prior, e.g., normal(0, 1e6) — not recommended.

3. Weakly informative, very weak, e.g., normal(0, 10).

4. Generic weakly informative, e.g., normal(0, 1).

5. Specific informative, e.g., normal(0.4, 0.2).

Those scales assume parameters are on roughly unit scale. In ML-UMR models the natural scales are:

Treatment intercepts

On the linear-predictor (link) scale. For a binary outcome with logit link, the intercept is a baseline log odds; normal(0, 10) spans ±20 log-odds at 95 percent and is "very weak". It is the default because the data usually constrain the intercept strongly. Tightening to normal(0, 5) is reasonable when the expected event rate is far from the extremes.

Regression coefficients (beta)

On the link scale, per unit of covariate. For logistic regression with predictors on unit scale, Gelman et al. (2008) and the Stan wiki recommend student_t(df, 0, 2.5) with df in 3:7, or — as a practical approximation — normal(0, 2.5). That is the default used by mlumr(). Use normal(0, 1) if you expect small effects (e.g., standardized predictors in a normal-outcome model). If predictors are on very different scales, set autoscale = TRUE so the scale is divided by each covariate's SD.

Residual SD (sigma, normal family only)

prior_sigma is interpreted as a half-normal via the Stan <lower=0> constraint. The default normal(0, 2.5) (i.e., half-normal(0, 2.5)) is weakly informative for residual SDs on the scale of the outcome. Scale to the outcome if it is far from unit scale, or use prior_exponential().

Prior sensitivity is especially important for the relaxed model, where beta_comparator is identified only by the AgD likelihood. Run prior_sensitivity() to quantify how much conclusions move under alternative scales; see vignette("mlumr-models").

References

Gelman, A., Jakulin, A., Pittau, M. G., & Su, Y.-S. (2008). A weakly informative default prior distribution for logistic and other regression models. Annals of Applied Statistics, 2(4), 1360–1383.

Vehtari, A. et al. Prior Choice Recommendations (Stan wiki): https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations.

Examples

# Default weakly-very-weak intercept prior
prior_normal(mean = 0, sd = 10)
#> $distribution
#> [1] "normal"
#> 
#> $mean
#> [1] 0
#> 
#> $sd
#> [1] 10
#> 
#> $df
#> [1] NA
#> 
#> $autoscale
#> [1] FALSE
#> 

# Gelman 2008 default for logistic-regression coefficients
prior_normal(mean = 0, sd = 2.5)
#> $distribution
#> [1] "normal"
#> 
#> $mean
#> [1] 0
#> 
#> $sd
#> [1] 2.5
#> 
#> $df
#> [1] NA
#> 
#> $autoscale
#> [1] FALSE
#> 

# Autoscaled coefficient prior (dividing 2.5 by each covariate's SD)
prior_normal(mean = 0, sd = 2.5, autoscale = TRUE)
#> $distribution
#> [1] "normal"
#> 
#> $mean
#> [1] 0
#> 
#> $sd
#> [1] 2.5
#> 
#> $df
#> [1] NA
#> 
#> $autoscale
#> [1] TRUE
#>