Setting priors with some assumptions for ordinal cumulative probit brms model

Modeling
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1

I have a (perhaps naive) question about setting priors for the intercept coefficients with some assumptions about the outcome variable for a cumulative probit model with brms.

The outcome variable is a 7-point Likert scale response. Based on this tutorial [Notes on the Bayesian cumulative probit | A. Solomon Kurz], I could set weakly informative priors for the intercept coefficients, assuming a uniform distribution between the Likert items. Say though that one might expect a slight positive bias toward the higher values, assuming larger values have a positive association.

How might I set and model such priors for the intercepts? Would it still be based, e.g., on the proportions of the tibble, or is there another way to think about this?

2

Is it worth the trouble? A Dirichlet prior on the multinomial probabilities, that is then transformed to the link scale inside Stan code, seems to work quote well. RMSb

3

Fair enough. I may be trying to be too precise in that regard. I could go with as is, as visually the distribution of the simulated data sets is roughly what one could expect for the intercepts.

I do have another question related to setting the beta priors for this model:

I have a group predictor, as well as continuous predictors that were log-transformed, scaled and centered. I had thought to set regularizing priors of N(0,1) for the group predictor. I was unsure how to estimate the prior for continuous predictor, so I generated a fake dataset of ordinal and continuous variables with varying degrees of correlations. I then ran those through a simple model looking at the relationship between the two with default priors to get a rough idea of the types of estimates, given this fake data. This would lead me to think I could also use N(0,1). However, my prior predictive plot is rather U-shaped, which I did not expect. Am I approaching the continuous predictor correctly, or is there another way I should consider this?

Related code for reference:

# set contrasts (sum-coding)
contrasts(df$group) = contr.sum(2)

# develop formula
bf.form = brms::bf(rating ~ group + cont1 + cont2 + 
                     group:cont1  + group:cont2)

get_prior(bf.form, data = df, family = cumulative('probit'))

tibble(rating = 1:7) %>% 
  mutate(proportion = 1/7) %>% 
  mutate(cumulative_proportion = cumsum(proportion)) %>% 
  mutate(right_hand_threshold = qnorm(cumulative_proportion))

priors = c(
  prior(normal(-1.07, 1), class = Intercept, coef = 1),
  prior(normal(-0.57, 1), class = Intercept, coef = 2),
  prior(normal(-0.18, 1), class = Intercept, coef = 3),
  prior(normal(0.18, 1), class = Intercept, coef = 4),
  prior(normal(0.57, 1), class = Intercept, coef = 5),
  prior(normal(1.07, 1), class = Intercept, coef = 6),
  prior(normal(0, 1), class = b) 
)

# simulate data
generator <- SBC_generator_brms(bf.form, data = df, 
                                family = cumulative('probit'), init = 0.1, prior = priors,
                                thin = 50, warmup = 10000, refresh = 2000,
                                # Will generate the log density - this is useful, 
                                #but a bit computationally expensive - turned off
                                generate_lp = FALSE)

datasets <- generate_datasets(generator, 100)

This is the prior predictive plot of the simulated datasets given the above noted priors:

beta-priors-01
beta-priors-01828×525 6.09 KB

4

In the rmsb package I specify priors on effects through the use of contrasts. This allows you to use your original scale. For example if x is modeled with a simple regression spline (so is nonlinear) and you want to specify a prior on the odds ratio for the inter-quartile-range effect of x you could do that quite easily. Several examples are here.

1 Like
5

I’d also suggest moving to ordinal logit. Not only do we have it coded in Stan, it’s a lot faster because the cdf for the standard logistic is a lot easier to compute than the cdf for the standard normal.

I don’t see how you could do that because it would require a fixed scale. You’d have to reparameterize everything in terms of differences to make sure you stay within that scale. It’s usually much more performant to put a soft constraint rather than hard constraint (like hard uniform distribution bounds).

In Stan itself, you have an increasing set of cutpoins and you can put a prior on the diffs between them without a Jacobian adjustment (though that’s still one prior short of being proper as you’ve only fixed N - 1 of N degrees of freedom).

6

Yes, using the approach in my blog, you would just change the proportions in the tibble to fit with your prior expectations. Just make sure the proportions sum up to 1.