A concrete example

Consider a 5-point Likert scale assessing understanding of Bayesian statistics with options: “nothing,” “a little,” “more or less,” “something,” and “everything.” This scale requires 4 thresholds. The probability for each category corresponds to the area between:

• Category 1 (“nothing”): to
• Category 2 (“a little”): to
• Category 3 (“more or less”): to
• Category 4 (“something”): to
• Category 5 (“everything”): to

Let’s visualize this with specific threshold values for a logistic distribution². We make up the following thresholds

thres <- c(-2, -1, 1.5, 3)

category <- factor(1:5, labels = c("Nothing", "A little", "More or less", 
                                     "Something", "Everything"))
  
x_vals <- seq(-6, 6, length.out = 500)
norm_data <- data.frame(
  x = x_vals,
  y = dlogis(x_vals)
)

category_positions <- c(-Inf, thres, Inf)
category_midpoints <- (category_positions[-1] + category_positions[-length(category_positions)]) / 2
category_midpoints[1] <- max(-6, category_midpoints[1])
category_midpoints[5] <- min(6, category_midpoints[5])

threshold_labels <- data.frame(
  x = thres,
  y = dlogis(thres) + 0.02,
  label = paste0("τ[", 1:4, "]")
)

category_labels <- data.frame(
  x = category_midpoints,
  y = 0.02,
  label = as.character(category)
)

p1 <- ggplot(norm_data, aes(x = x, y = y)) +
  geom_line(linewidth = 1, color = "steelblue") +
  geom_vline(xintercept = thres, linetype = "dashed", color = "red", linewidth = 0.8) +
  geom_text(data = threshold_labels, aes(x = x, y = y, label = label),
            parse = TRUE, size = 4.5, color = "red", hjust = -0.2, vjust = 0) +
  geom_text(data = category_labels, aes(x = x, y = y, label = label),
            size = 3.5, color = "black", hjust = 0.5) +
  labs(title = "Standard Logistic Distribution with Thresholds",
       x = "Latent Variable",
       y = "Density") +
  theme_minimal(base_size = 12)

print(p1)

Computing category probabilities

Now let’s calculate the probability mass between each threshold:

probs <- diff(plogis(c(-Inf, thres, Inf)))

# This above is just a more general case than:
# probs <- c(
#   plogis(thres[1]),  # P(Y = 1)
#   plogis(thres[2]) - plogis(thres[1]),  # P(Y = 2)
#   plogis(thres[3]) - plogis(thres[2]),  # P(Y = 3)
#   plogis(thres[4]) - plogis(thres[3]),  # P(Y = 4)
#   1 - plogis(thres[4])  # P(Y = 5)
# )

prob_data <- data.frame(
  category = category,
  probability = probs
)

prob_data

      category probability
1      Nothing  0.11920292
2     A little  0.14973850
3 More or less  0.54863305
4    Something  0.13499965
5   Everything  0.04742587

ggplot(prob_data, aes(x = category, y = probability, fill = category)) +
  geom_col(alpha = 0.7, color = "black") +
  scale_fill_brewer(palette = "Blues") +
  labs(title = "Category Probabilities (Baseline)",
       x = "Understanding Level",
       y = "Probability") +
  theme_minimal(base_size = 12) +
  theme(legend.position = "none",
        axis.text.x = element_text(angle = 45, hjust = 1)) +
  ylim(0, 1)

We observe that the middle category (“more or less”) receive the highest probability mass.

The effect of shifting thresholds

Now, let’s examine what happens when we shift all thresholds to the right by one unit and a half: simulating less understanding across the board.

thres <- thres + 1.5

      category probability
1      Nothing  0.37754067
2     A little  0.24491866
3 More or less  0.33011480
4    Something  0.03643893
5   Everything  0.01098694

Notice how shifting the thresholds rightward decreases the probability of higher categories (more understanding) while increasing the probability of lower categories.

Simulating data

Let’s generate synthetic data to illustrate this. We’ll simulate a group of students before and after reading our book.

This type of model assumes that, underlyingly, there is a continuous latent “rating”, and we specify a regression on this latent variable that determines how the response categories are affected by our manipulation (note that there is no intercept here; the thresholds will later act as something close to intercepts for each category):

with

where is a treatment indicator (in more realistic settings, one may prefer richer contrast coding; see Chapter 6 of Nicenboim, Schad, and Vasishth (2025)).

In principle, one could allow the predictor to have category-specific effects by estimating regressions for each threshold. However, Bürkner and Vuorre (2019) argue that while this parameterization is unproblematic for some ordinal models (such as sequential or adjacent-category models), it can lead to difficulties in model fitting for cumulative models.

To move from the continuous latent variable to a discrete response, we define the observed outcome as follows:

with boundary conditions and .

We keep the thresholds () defined above as the default cutoff points (brms refers to these as intercepts, though note the difference in sign convention below).

Given a cumulative link, the probability of observing category is

Here, denotes the cumulative distribution function of the error term of . We assume a logistic distribution for computational convenience, which yields a logistic cumulative link (assuming a normal distribution would lead to a probit link, with otherwise very similar expressions).

Thus, under a logistic cumulative link, the probability of observing category is

where denotes the CDF of the logistic distribution (available in R via plogis()). Notice that the regression term, , is subtracted from each threshold.

To simulate data from this model, we treat the resulting category probabilities as defining a categorical distribution and sample responses accordingly:

One important insight is that affects all categories equally: when , increases, increasing the probability of higher response categories at the expense of lower response categories; when , decreases, increasing the probability of lower categories at the expense of higher response categories.

In the following simulation, we use the last thres values we defined (-0.5, 0.5, 3, 4.5) as the cutoff points of the model (; called intercepts in brms), and we assume that reading increases (and also on average) by . We generate data from 1000 participants before and after reading our book:

set.seed(123)

N <- 1000
tau <- thres
beta <- 1.5
eta_noread <- 0 * beta
eta_read  <- 1 * beta

df_sim <- data.frame(
  read = rep(c(0, 1), each = N),
  category = c(
    rcat(N, diff(plogis(c(-Inf, tau - eta_noread, Inf))), labels = category),
    rcat(N, diff(plogis(c(-Inf, tau - eta_read, Inf))), labels = category)
  )
) |>
  mutate(response = as.numeric(category))

# Display summary statistics
table(df_sim$read, df_sim$category)

   
    Nothing A little More or less Something Everything
  0     382      238          329        40         11
  1     114      154          549       138         45

Let’s visualize the distribution of responses in both groups:

ggplot(df_sim, aes(x = response, fill = factor(read, labels = c("Before reading", "After reading")), 
                   group = factor(read))) +
  geom_bar(position = "dodge", alpha = 0.7, color = "black") +
  scale_fill_manual(values = c("Before reading" = "coral", "After reading" = "forestgreen")) +
  labs(title = "Distribution of Responses",
       x = "Understanding Level",
       y = "Count",
       fill = "Group") +
  theme_minimal(base_size = 12)

Now that we have (simulated) data, let’s fit it with brms.

Why tight priors concentrate probability on extremes

Let’s start by examining an intercept-only model with tight priors on the thresholds:

priors_tight <- c(
  prior(normal(0, 0.5), class = "Intercept")
)

With normal(0, 0.5) “Intercept” priors, all thresholds are pulled toward zero with relatively little spread. This concentrates the thresholds in a narrow region. This means that when thresholds cluster tightly around zero, most of the latent distribution’s mass falls outside the middle thresholds: either far below or far above . This assigns most probability to extreme categories (the first and last).

fit_prior_tight <- brm(
  response ~ 1,
  data = df_sim,
  family = cumulative(),
  prior = priors_tight,
  sample_prior = "only",
  cores = 4,
  seed = 123,
  refresh = 0,
  control = list(adapt_delta = 0.9) # avoid divergent transitions
)

ppd_ridgeplot <- function(fit, title = "Prior Predictive Distribution", 
                          subtitle = NULL, ndraws = 500) {
  
  yrep <- posterior_predict(fit, ndraws = ndraws)
  
  proportions_per_draw <- lapply(1:nrow(yrep), function(i) {
    props <- table(factor(yrep[i, ], levels = 1:5)) / ncol(yrep)
    data.frame(
      draw = i,
      response = factor(1:5),
      proportion = as.numeric(props)
    )
  })
  ppd_data <- do.call(rbind, proportions_per_draw)
  ggplot(ppd_data, aes(x = proportion, y = response)) +
    geom_density_ridges(fill = "steelblue", alpha = 0.7, scale = 0.9, stat = "binline") +
    scale_x_continuous(breaks = c(0,.2,.4,.6,.8,1)) +
    geom_vline(xintercept = 0.2, linetype = "dashed") +
    labs(title = title,
         subtitle = subtitle,
         x = "Proportion",
         y = "Response Category") +
    theme_minimal(base_size = 12) +
    coord_flip()
}

ppd_ridgeplot(fit_prior_tight, 
              title = "Prior Predictive Distribution: Tight Priors (SD = 0.5)",
              subtitle = "Most probability mass concentrated in extreme categories")

With tight priors, the model tends to predict extreme responses, as the thresholds cluster near zero and don’t span the full range of the latent distribution.

Intercept-only model: Medium priors

Now let’s use more dispersed priors:

priors_medium <- c(
  prior(normal(0, 3), class = "Intercept")
)

With normal(0, 3) “Intercept” priors, the thresholds can spread out more widely across the latent scale. This means the middle thresholds are more likely to be well-separated, capturing more of the distribution’s central mass and leading to more balanced predictions across all categories.

fit_prior_medium <- brm(
  response ~ 1,
  data = df_sim,
  family = cumulative(),
  prior = priors_medium,
  sample_prior = "only",
  cores = 4,
  seed = 123,
  refresh = 0
)

ppd_ridgeplot(fit_prior_medium, 
              title = "Prior Predictive Distribution: Medium Priors (SD = 3)",
       subtitle = "More balanced distribution across categories")

Intercept-only model: Diffuse priors

Let’s examine very diffuse priors:

priors_diffuse <- c(
  prior(normal(0, 10), class = "Intercept")
)

fit_prior_diffuse <- brm(
  response ~ 1,
  data = df_sim,
  family = cumulative(),
  prior = priors_diffuse,
  sample_prior = "only",
  cores = 4,
  seed = 123,
  refresh = 0,
  control = list(adapt_delta = 0.9)
)

ppd_ridgeplot(fit_prior_diffuse, 
              title = "Prior Predictive Distribution: Diffuse Priors (SD = 10)",
       subtitle = "Very flat distribution - least informative")

Very diffuse priors allow thresholds to be extremely spread out, often resulting in a very sparse distribution across categories.

Intercept-only model: Nudging priors

We might have substantive information suggesting most people start as beginners. We can encode this by centering the “Intercept” priors at a positive value:

priors_mid_inf <- c(
  prior(normal(1, 3), class = "Intercept")
)

fit_mid_inf <- brm( 
  response ~ 1,
  data = df_sim,
  family = cumulative(),
  prior = priors_mid_inf,
  sample_prior = "only",
  cores = 4,
  seed = 123,
  refresh = 0
)

ppd_ridgeplot(fit_mid_inf, 
              title = "Prior Predictive Distribution: Nudged Priors (Mean = 1, SD = 3)",
       subtitle = "Prior belief that lower understanding is more common")

By centering the”Intercept” priors at 1 instead of 0, we shift thresholds rightward, making lower categories more probable a priori. On the other hand, if we had substantial information that suggests the opposite (e.g. that most people start advanced), we would center the “Intercept” priors at a negative value, shifting the thresholds leftward, making higher categories more probable a priori.

Prior predictive distribution for the slope

Let’s also examine what our prior beliefs about the treatment effect imply:

priors_full <- c(
  prior(normal(0, 3), class = "Intercept"),
  prior(normal(0, 1.5), class = "b")
)

fit_slope_prior <- brm(
  response ~ read,
  data = df_sim,
  family = cumulative(),
  prior = priors_full,
  sample_prior = "only",
  cores = 4,
  seed = 123,
  refresh = 0,
  control = list(adapt_delta = 0.9)
)

ppd_ridgeplot(fit_slope_prior,
              title = "Prior Predictive Distribution: Full Model with Slope Prior",
       subtitle = "normal(0, 1.5) prior on treatment effect")

The prior normal(0, 1.5) on the slope is weakly informative: it allows for moderate effects in either direction while gently regularizing against implausibly large effects.

Fitting to the simulated data

Now let’s fit the model to our simulated data:

fit_full <- brm(
  response ~ read,
  data = df_sim,
  family = cumulative(),
  prior = priors_full,
  cores = 4,
  seed = 123,
  warmup = 1000,
  iter = 2000,
  refresh = 0
)

fit_full

 Family: cumulative 
  Links: mu = logit 
Formula: response ~ read 
   Data: df_sim (Number of observations: 2000) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Regression Coefficients:
             Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept[1]    -0.50      0.06    -0.62    -0.38 1.00     4501     3153
Intercept[2]     0.50      0.06     0.37     0.62 1.00     4802     3359
Intercept[3]     2.98      0.09     2.79     3.16 1.00     3939     3068
Intercept[4]     4.55      0.15     4.26     4.85 1.00     3991     2905
read             1.50      0.09     1.32     1.66 1.00     3854     2924

Further Distributional Parameters:
     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
disc     1.00      0.00     1.00     1.00   NA       NA       NA

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).

If this were real data, based on the read coefficient, we would conclude that reading our book affects understanding. The positive coefficient read ≈ 1.5 indicates that reading makes higher categories (more understanding) more probable.

Interpreting predicted probabilities

To understand what the model estimates mean in practical terms, let’s examine the predicted probabilities:

ce <- conditional_effects(fit_full, effects = "read", categorical = TRUE)

plot(ce, plot = FALSE)[[1]] +
  scale_fill_brewer(palette = "RdYlGn", direction = 1) +
  labs(title = "Predicted Response Probabilities by Reading Status",
       subtitle = "Model predictions with 95% credible intervals",
       x = "Read Book",
       y = "Predicted Probability",
       fill = "response") +
  theme_minimal(base_size = 12) +
  theme(legend.position = "right")

The conditional effects plot shows how reading the book shifts the entire distribution toward higher understanding categories. Before reading (read = 0), lower categories have higher probabilities; after reading (read = 1), the distribution shifts substantially toward “something” and “everything.”

We can also examine specific fitted values:

new_data <- data.frame(read = c(0, 1))
fitted_probs <- fitted(fit_full, newdata = new_data)

fitted_df <- data.frame(
  read = rep(c("Before", "After"), each = 5),
  category = rep(1:5, times = 2),
  probability = c(fitted_probs[1, "Estimate", ], fitted_probs[2, "Estimate",]),
  lower = c(fitted_probs[1, "Q2.5",], fitted_probs[2 , "Q2.5",]),
  upper = c(fitted_probs[1, "Q97.5",], fitted_probs[2 , "Q97.5",])
)

ggplot(fitted_df, aes(x = factor(category), y = probability, fill = read)) +
  geom_col(position = "dodge", alpha = 0.7, color = "black") +
  geom_errorbar(aes(ymin = lower, ymax = upper), 
                position = position_dodge(0.9), width = 0.2) +
  scale_fill_manual(values = c("Before" = "coral", "After" = "forestgreen")) +
  labs(title = "Model-Predicted Probabilities with 95% Credible Intervals",
       x = "Understanding Category",
       y = "Predicted Probability",
       fill = "Reading Status") +
  theme_minimal(base_size = 12)

The fitted probabilities show concrete estimates: for instance, before reading, the probability of responding “nothing” is about 0.38, whereas after reading it drops to about 0.12. Conversely, the probability of responding “everything” increases from about 0.01 to 0.05.

Posterior predictive check

Finally, let’s verify that the model fits the observed data well:

pp_check(fit_full, type = "bars", ndraws = 100) +
  labs(title = "Posterior Predictive Check: Full Model",
       subtitle = "Model predictions compared to observed data") +
  theme_minimal(base_size = 12)

The posterior predictive distribution closely matches the observed data patterns, indicating good model fit.

A ranking simulation using pizza toppings

toppings <- corpora("foods/pizzaToppings")$pizzaToppings
N_toppings <- length(toppings)
toppings

 [1] "anchovies"        "artichoke"       
 [3] "bacon"            "breakfast bacon" 
 [5] "Canadian bacon"   "cheese"          
 [7] "chicken"          "chili peppers"   
 [9] "feta"             "garlic"          
[11] "green peppers"    "grilled onions"  
[13] "ground beef"      "ham"             
[15] "hot sauce"        "meatballs"       
[17] "mushrooms"        "olives"          
[19] "onions"           "pepperoni"       
[21] "pineapple"        "sausage"         
[23] "spinach"          "sun-dried tomato"
[25] "tomatoes"        

Let’s say that we want to know the underlying order of pizza toppings. For the modeling, I’m going to assume that the toppings are ordered according to an underlying value, which also represents how likely it is for each topping to be the exemplar of their category.

To get a known ground truth for the ranking, I’m going to simulate an order of pizza toppings. I assign probabilities that sum up to one to the toppings by drawing a random sample from a Dirichlet distribution. The Dirichlet distribution is the generalization of the Beta distribution. It has a concentration parameter, usually , which is a vector as long as the probabilities we are sampling (25 here). When the vector is full of ones, the distribution is uniform: All probabilities are equally likely, so on average each one is ( here). By setting all the concentration parameters below one (namely ), I’m enforcing sparsity in the random values that I’m generating, that is, many probability values close to zero.

These is the true order that I’m assuming here:

# all elements of the vector are 0.2
alpha <- rep(.2, N_toppings)
# Generate one draw from a Dirichlet distribution
P_toppings <- c(rdirichlet(1, alpha)) %>%
  # Add names
  setNames(toppings) %>%
  # Sort from the best exemplar
  sort(decreasing = TRUE)
P_toppings %>%
  round(3)

 breakfast bacon          chicken             feta 
           0.294            0.241            0.087 
       anchovies sun-dried tomato           olives 
           0.087            0.077            0.057 
       pepperoni        artichoke           cheese 
           0.056            0.049            0.010 
  Canadian bacon            bacon              ham 
           0.008            0.008            0.006 
       meatballs    chili peppers           garlic 
           0.004            0.004            0.004 
     ground beef         tomatoes        hot sauce 
           0.003            0.003            0.002 
          onions          sausage        pineapple 
           0.000            0.000            0.000 
         spinach        mushrooms   grilled onions 
           0.000            0.000            0.000 
   green peppers 
           0.000 

• Given these values, if I were to ask a participant “What’s the most appropriate topping for a pizza?” I would assume that 29.37 percent of the time, I would get breakfast bacon.

Essentially, we expect something like this to be happening:

With representing the different probabilities for each topping. The probability mass function of the categorical distribution is absurdly simple: It’s just the probability of the outcome.

where breakfast bacon, chicken, feta, anchovies, sun-dried tomato, olives, pepperoni, artichoke, cheese, Canadian bacon, bacon, ham, meatballs, chili peppers, garlic, ground beef, tomatoes, hot sauce, onions, sausage, pineapple, spinach, mushrooms, grilled onions, green peppers.

We can simulate this with 100 participants as follows:

response <- rcat(100, P_toppings, names(P_toppings))

And this should match approximately P_toppings.

table(response)/100

response
 breakfast bacon          chicken             feta 
            0.26             0.19             0.16 
       anchovies sun-dried tomato           olives 
            0.07             0.15             0.06 
       pepperoni        artichoke           cheese 
            0.01             0.08             0.00 
  Canadian bacon            bacon              ham 
            0.02             0.00             0.00 
       meatballs    chili peppers           garlic 
            0.00             0.00             0.00 
     ground beef         tomatoes        hot sauce 
            0.00             0.00             0.00 
          onions          sausage        pineapple 
            0.00             0.00             0.00 
         spinach        mushrooms   grilled onions 
            0.00             0.00             0.00 
   green peppers 
            0.00 

It seems that by only asking participants to give the best topping we could already deduce the underlying order…

True, but one motivation for considering ranking data is the amount of information that we gather with a list due to their combinatorial nature. If we ask participants to rank items, an answer consists in making a single selection out of possibilities. Ordering 7 pizza toppings, for example, constitutes making a single selection out of 5040 possibilities!

If we don’t relay on lists and there is sparcity, it requires a large number of participants until we get answers of low probability. (For example, we’ll need a very large number of participants until we hear something else but hammer as an exemplar of tools).

• Now, what happens if we ask about the second most appropriate topping for a pizza?

Now we need to exclude the first topping that was given, and draw another sample from a categorical distribution. (We don’t allow the participant to repeat toppings, that is, to say that the best topping is pineapple and the second best is also pineapple). This means that now the probability of the topping already given is zero, and that we need to normalize our original probability values by dividing them by the new total probability (which will be lower than 1).

Here, the probability of getting the element (where ) is

where represents the probabilities of all the outcomes except of , which was the first one.

• We can go on with the third best topping, where we need to normalize the remaining probabilities by dividing by the new sum of probabilities (e.g., we remove elements and ).

• We can do this until we get to the last element, which will be drawn with probability 1.

And this is the exploded logit distribution.

This process can be simulated in R as follows:

rexploded <-  function(n, ranked = 3, prob, labels = NULL){
  # run n times
  lapply(1:n, function(nn){
    res <- rep(NA, ranked)
    if(!is.null(labels)){
      res <- factor(res, labels)
    } else {
      # if there are no labels, just 1,2,3,...
      labels <- seq_along(prob)
    }
    for(i in 1:ranked){
      # normalize the probability so that it sums to 1
      prob <- prob/sum(prob)
      res[i] <- rcat(1, prob = prob, labels = labels)
      # remove the choice from the set:
      prob[res[i]] <- 0
    }
    res
  })
}

If we would like to simulate 50 subjects creating a ranking of the best 7 toppings, we would do the following:

res <- rexploded(n = 50,
                 ranked = 7,
                 prob = P_toppings,
                 labels = names(P_toppings))
# subject 1:
res[[1]]

[1] sun-dried tomato artichoke        olives          
[4] breakfast bacon  chicken          pepperoni       
[7] anchovies       
25 Levels: breakfast bacon chicken feta ... green peppers

We have simulated ranking data of pizza toppings, can we recover the original probability values and “discover” the underlying order?