Lprior correlation and parameter pairs

Interfaces brms
,
1

I’ve run a relatively simplistic multivariate outcome with “Imitation” as a common cause of “Motor.Skills” and “Social.Skills” in brms. I had no issues with convergence, no other warnings, and the parameters replicate the frequentist model (in lavaan). Looking at the pairs plot, there is a nearly perfect correlation between lprior and b_MotorSkills_Imitation; my understanding is that this is driven by the narrower prior for b_MotorSkills_Imitation; the lprior reflects. I re-ran the model with each beta parameter’s having equal variance (.1) and now the b_SocialSkills_Imitation parameter correlates strongly with lprior.

My questions are two-fold:

  1. Why are there duplicated intercept terms (2 for each model) in the pairs plot? This seems inefficient, given their extremely strong correlation.
  2. Is a correlation between lprior and a slope parameter necessarily problematic? I’ve looked at the priorsense GitHub, and am trying to get it working on my models.
ME_priors <- c(prior(normal(.22, .05), class="b", coef = "Imitation", resp = "MotorSkills"),
               prior(normal(.46, .12), class="b", coef = "Imitation", resp = "SocialSkills"))

mot.pred <- bf(Motor.Skills ~ Imitation)
soc.pred <- bf(Social.Skills ~ Imitation)

ME_sem_fit_rescor_brms <- brm(
  mot.pred + soc.pred + set_rescor(TRUE), 
  data = data2,
  prior = ME_priors,
  warmup = 1000, iter = 3500, 
  cores = 4, chains = 4,
  file = "./brms_models/ME_rescor_10000.rds"
)

data2 <- structure(c(-1.08, 0.18, 0.18, 1.44, 1.44, -1.08, -1.08, 1.44, 
            0.18, 0.18, -1.08, -1.08, -1.08, -1.08, -1.08, -1.08, 1.44, -1.08, 
            1.44, 0.18, 0.18, 0.18, -1.08, -1.08, 0.18, 0.18, -1.08, -1.08, 
            1.44, -1.08, -1.08, -1.08, -1.08, -1.08, -1.08, 1.44, -1.08, 
            0.18, 1.44, 1.44, 1.44, -1.08, -1.08, -1.08, -1.08, 0.18, 0.18, 
            -1.08, -1.08, -1.08, -1.08, 0.18, -1.08, 0.18, -1.08, 0.18, 0.18, 
            0.18, 0.18, 0.18, -1.08, -1.08, -1.08, 1.44, 1.44, -1.08, 0.18, 
            0.18, -1.08, 0.18, 0.18, -1.08, 0.18, 0.18, 0.18, 0.18, 0.18, 
            0.18, 1.44, 0.18, -1.08, -1.08, -1.08, -1.08, 1.44, 1.44, 1.44, 
            0.18, -1.08, -1.08, 0.18, 0.18, 0.18, -1.08, -1.08, 0.18, -1.08, 
            -1.08, 1.44, -1.08, -0.88, -0.53, -0.31, -0.21, 1.52, -1.07, 
            -1.21, 0.21, 0.14, -0.03, 1.11, -0.16, -0.29, -1.08, 1.3, -1.56, 
            1.99, 0.83, -0.25, 0.13, -0.95, -1.62, -0.52, -0.85, 0.73, 0.96, 
            -0.37, -0.91, -1.46, 0.44, -0.74, -0.75, -1.54, 0.27, -0.82, 
            0.46, 0.68, -0.32, -0.01, 1.11, 0.4, -1.74, 0.26, -1.74, -1.25, 
            -0.67, -0.35, -1.04, 0.13, -1.33, 1.28, -0.07, -0.89, 0.48, 0.5, 
            -0.65, -1.74, -0.61, 0.73, -1.03, -0.35, 0.2, 1.22, 1.98, 1.06, 
            -1.44, -0.56, -1.74, -1.74, -0.74, -0.59, 0.39, 2.18, 1.19, 0.76, 
            0.39, -1.42, -0.14, 0.97, 0.26, -1.07, 0.81, -0.91, -0.69, 0.73, 
            0.67, 0.39, 0.26, -0.68, 0.95, 0.35, -1.25, -0.19, -0.97, -0.5, 
            -0.19, -1.03, -0.91, -0.05, -0.86, -0.96, -0.04, -1.17, 0.78, 
            2.12, -0.92, 0.55, 1.61, 0.07, 0.78, 0.75, -0.13, -0.13, -1.69, 
            -0.95, -1.51, 0.58, 0.04, 1.84, 0.34, -2.12, -1.34, -0.78, -0.39, 
            -0.91, 0.74, 0.38, -0.76, 0.38, -0.52, -0.65, 0.42, -0.28, -0.86, 
            -0.74, 1.56, -0.62, -0.46, 0.64, -0.14, -0.22, -0.52, -0.9, -1.64, 
            -0.52, -0.23, 0.93, -1.03, 0.47, -1.74, -1.83, -1.72, -0.68, 
            -0.56, -0.61, -0.05, 0.65, 0.09, 0.76, -0.77, -0.58, -1.38, 0.31, 
            0.79, 1.44, -0.8, 1.64, 0.49, -1.64, -0.39, -0.76, -0.33, 0.89, 
            -1.23, 0.36, 1.84, -1.44, 0.38, -0.44, 1.3, 0.24, 0.53, -0.04, 
            -0.81, 2.12, 2.12, 0.76, 0.88, -1.21, -1.02, -0.34, 0.18, -0.94, 
            -1.69, -1.02, 0.14, -1.23, -0.81, -0.93, -0.65), dim = c(100L, 
                                                                     3L))

 Family: MV(gaussian, gaussian) 
  Links: mu = identity; sigma = identity
         mu = identity; sigma = identity 
Formula: Motor.Skills ~ Imitation 
         Social.Skills ~ Imitation 
   Data: data2 (Number of observations: 19053) 
  Draws: 4 chains, each with iter = 3500; warmup = 1000; thin = 1;
         total post-warmup draws = 10000

Regression Coefficients:
                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
MotorSkills_Intercept      0.00      0.01    -0.01     0.01 1.00    13776     7568
SocialSkills_Intercept    -0.00      0.01    -0.01     0.01 1.00    13952     8046
MotorSkills_Imitation      0.26      0.01     0.25     0.28 1.00    14713     8124
SocialSkills_Imitation     0.53      0.01     0.52     0.54 1.00    13807     7507

Further Distributional Parameters:
                   Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_MotorSkills      0.96      0.00     0.96     0.97 1.00    15741     7794
sigma_SocialSkills     0.85      0.00     0.84     0.85 1.00    16427     8456

Residual Correlations: 
                                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
rescor(MotorSkills,SocialSkills)     0.26      0.01     0.25     0.27 1.00    13799     8123

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

First plot is from code above

ME_rescor_pairs_main
ME_rescor_pairs_main1920×1280 231 KB

Second plot is from code above with altered priors, such that var is .1 for each

ME_rescor_pairs
ME_rescor_pairs1920×1280 219 KB

  • Operating System: Windows 10 Enterprise
  • brms Version: 2.22.0
2

You can dig in to this by using stancode() in place of brm(), which will show you the underlying stan file. You will see that there is only one intercept per equation in the parameters block, then the second intercept appears in generated quantities. I think that brms is using some centering tricks here to improve sampling efficiency, then converts back to the usual intercept in generated quantities. This means that the correlation between intercepts is harmless.

For the other correlation: lprior is the sum of log-priors for all parameters. If it is correlated with a specific slope, I guess it means that the prior for that slope is relatively informative, and the posterior samples of that slope fall in the high-density part of the prior.