Figuring out Bayesian statistics

statistics
tutorial
Bayesian statistics
regression
Bayesian statistics seems pretty cool, but I don’t really know how to apply it yet. In this blog post, I try to setup a Bayesian workflow that teaches both you and me how to do it.
Published

November 24, 2022

This post is about figuring out how Bayesian statistics works and about developing a workflow to conduct Bayesian analyses. Specifically, I want to go through the process of running a Bayesian analysis and visualizing the different steps of that analysis in order to make sure I know what I’m doing. With a bit of luck this is educational and we’ll both end up learning how this works!

If you want to follow a long, run the following setup code.

Code
# Load packages
library(tidyverse)
library(viridis)
library(brms)
library(tidybayes)
library(ggrepel)

# Load data
data <- read_csv("Howell1.csv")

# Set seed
set.seed(4)

# Set options
theme_set(theme_minimal())

The data we will use to play with is the same data Richard McElreath uses in Chapter 4 of his amazing book called Statistical Rethinking. The data consists of partial census data of the !Kung San, compiled from interviews conducted by Nancy Howell in the late 1960s. Just like in the book, we will focus only on people 18 years or older, so in the code below we create a subset of the data and store the result in a data frame called data.

Code
# Only select people older than 18
data <- filter(data, age >= 18)

# Show the first rows
head(data)
Partial census data for the Dobe area !Kung San compiled by Nancy Howell in the late 1960s.
height weight age male
151.765 47.82561 63 1
139.700 36.48581 63 0
136.525 31.86484 65 0
156.845 53.04191 41 1
145.415 41.27687 51 0
163.830 62.99259 35 1

The general idea behind Bayesian statistics is that you start with some prior beliefs about the parameters of interest and then update those beliefs with the data. Note that this doesn’t mean that you have to personally accept those beliefs. You could simply postulate a belief to serve a particular purpose, such as assuming that a null effect is most likely even though you personally believe that there should be an effect. It does mean that when we want to analyze the data, we should start with defining our beliefs, rather than immediately jumping into running an analysis.

Let’s focus our first question on the heights in the data. We should begin by defining a belief that describes the different heights, which is based on our a priori knowledge of the heights of the Dobe area !Kung San. In other words, we have to describe what we believe their heights to be. This is unlike what you have to do with frequentist statistics, so this part might be a bit tricky.

To make it easier, we will use the amazing brms package to both define and inspect our beliefs, as well as use the data to update those beliefs.

An intercept-only model

If we start with analyzing only the heights data then we’ll be constructing an intercept-only model. You may be familiar with the R formula for this type of model: height ~ 1.

With this formula and the data we can actually use brms to figure out which priors we need to set by running the get_prior() function. This is probably the easiest way to figure which priors you need when you’re just starting out using brms.

Code
get_prior(height ~ 1, data = data)
prior class coef group resp dpar nlpar lb ub source
student_t(3, 154.3, 8.5) Intercept default
student_t(3, 0, 8.5) sigma 0 default

The output shows us that we need to set two priors, one for the Intercept and one for sigma. brms also already determined a default prior for each, but we’ll ignore that for now.

This is not the best way to think about which priors we need, though. Using the function will give you the answer, but it doesn’t really improve our understanding of why we need these two priors. In this case we also omitted an important specification of the heights, which is that we think they are normally distributed (the default assumption in get_prior()). So let’s instead write down our model in a different way, which explicitly specifies how we think the heights are distributed and which parameters we need to set priors on. If we think the heights are normally distributed, we define our model like this:

\[heights_i ∼ Normal(\mu, \sigma)\]

We explicitly note that the heights come from a normal distribution, which is determined by the parameters \(\mu\) and \(\sigma\). This then also immediately tells us that we need to set two priors, one on \(\mu\) and one on \(\sigma\).

In our intercept-only model, the \(\mu\) parameter refers to our intercept and the \(\sigma\) parameter refers to, well, sigma. It’s not often discussed in the literature I’m familiar with, but we’ll figure it out below. In fact, let’s discuss each of these parameters in turn and figure out what kind of prior makes sense.

The Intercept prior (\(\mu\))

The prior for the intercept indicates what we believe the average height of the !Kung San to be.

brms has set the default Intercept prior as a Student t-distribution with 3 degrees of freedom, a mean of 154.3 and a standard deviation of 8.5. That means brms starts off with a ‘belief’ that the average of the heights is 154.3, but with quite some uncertainty reflected in the standard deviation of 8.5 and the fact that the distribution is a Student t-distribution. A Student t-distribution has thicker tails compared to a normal distribution, meaning that lower and higher numbers are considered more likely compared to a normal distribution, at least when the degrees of freedom are low. At higher degrees of freedom, the t-distribution becomes more and more like the normal distribution. So, the thicker tails of the t-distributions means smaller and taller average heights are relatively more plausible.

But this is the default prior. brms determines this automatic prior by peeking at the data, which is not what we want to do. Instead, we should create our own.

So what do I believe the average height to be? As a Dutch person, I might be under the impression that the average height is around 175 centimeters. This is probably too tall to use as an average for the !Kung San because we’re known for being quite tall. So I think the average should be lower than 175, perhaps 170. I am not very sure, though. After all, I am far from an expert on people’s heights; I am only using my layman knowledge here. An average of 165 seems possible to me too. So let’s describe my belief in the form of a distribution in which multiple averages are possible, to varying extents. We should use a Student t-distribution with small degrees of freedom if we want to allow for the possibility of being very wrong (remember, it has thicker tails, so it assigns more probability to a wider range of average heights). We’re not super uncertain about people’s heights, though, so let’s use a normal distribution.

As we saw in defining our height model, a normal distribution requires that we set the \(\mu\) and the \(\sigma\). The \(\mu\) we already covered (i.e., 170), so that leaves \(\sigma\). Let’s set this to 10 and see what happens by visualizing this prior. Below I plot both the default brms prior and our own with \(\mu\) = 170 and \(\sigma\) = 10.

Code
height_prior_intercept <- tibble(
  height_mean = seq(from = 100, to = 250, by = 0.1),
  ours = dnorm(height_mean, mean = 170, sd = 10),
  default = dstudent_t(height_mean, df = 30, mu = 154.3, sigma = 8.5),
)

height_prior_intercept <- pivot_longer(
  height_prior_intercept, 
  cols = -height_mean, 
  names_to = "prior"
) 

ggplot(
    height_prior_intercept, 
    aes(x = height_mean, y = value, linetype = fct_rev(prior))
  ) +
  geom_line() +
  labs(x = "Average height", y = "", linetype = "Prior") +
  scale_x_continuous(breaks = seq(100, 250, 20))

Two priors for \(\mu\)

Our prior indicates that we believe the average height to be higher than the default prior. In terms of the standard deviation, we both seem to be about equally uncertain about this average. To be fair, I think this prior of ours is not very plausible. Apparently we assign quite a chunk of plausibility to an average of 180 cm, or even 190 cm, which is very unlikely. An average of 160 cm is more plausible to me than an average of 180, so I should probably lower the mu, or use more of a skewed distribution. This is one of the benefits of visualizing the prior, it lets you think again about your prior so that you may improve on it. Regardless, we can keep the prior like this for now. We’ll see later that our data easily overshadows our prior.

The sigma prior (\(\sigma\))

What about the standard deviation? I find setting the standard deviation of the distribution of heights (not the mean of the heights) quite difficult. There are parts that are easy, such as the fact that the standard deviation has to be 0 or larger (it can’t be negative), but exactly how large it should be, I don’t know.

I do know it is unlikely to be close to 0, and unlikely to be very large. That’s because I know people’s heights do vary, so I know the sigma can’t be 0. I also know it’s not super large because we don’t see people who are taller than 2 meters very often. This means the peak should be somewhere above 0, with a tail to allow higher values but not too high. We can use a normal distribution for this with a mean above 0 and a particular standard deviation, and ignore everything that’s smaller than 0 (brms automatically ignores negative values for \(\sigma\)).

As I mentioned before, there is a downside of using a normal distribution, though. Normal distributions have long tails, but there is actually very little density in those tails. If we are quite uncertain about our belief about sigma, we should use a t-distribution, or perhaps even a cauchy distribution (actually, the cauchy distribution is a special case of the Student t-distribution; they are equivalent if the degree of freedom is 1). The lower the degrees of freedom, the more probability we assign to higher and lower values.

So, a t-distribution requires three parameters: \(\mu\), \(\sigma\), and the degrees of freedom. I set \(\mu\) to 5, \(\sigma\) to 5, and the degrees of freedom to 1. Below I plot this prior and brms’s default prior.

Code
height_prior_sigma <- tibble(
  height_sigma = seq(from = 0, to = 50, by = .1),
  default = dstudent_t(height_sigma, df = 3, mu = 0, sigma = 8.5),
  ours = dstudent_t(height_sigma, df = 1, mu = 5, sigma = 5) 
)

height_prior_sigma <- pivot_longer(
  height_prior_sigma, 
  cols = -height_sigma, 
  names_to = "prior"
)

ggplot(
    height_prior_sigma, 
    aes(x = height_sigma, y = value, linetype = fct_rev(prior))
  ) +
  geom_line() +
  labs(x = "Standard deviation of heights", y = "", linetype = "Prior")

Two priors for \(\sigma\)

As you can see, both distributions have longish tails, allowing for the possibility of high standard deviations. There are some notable differences between the two priors, though. Our prior puts more weight on a standard deviation larger than 0, while the default prior reflects a belief in which a standard deviation of 0 is most likely. However, both priors are quite weak. We’ll see that the data easily overshadows these priors.

Before we run the analysis, we can also check the results of both our priors on the distribution of heights.

A prior predictive check

Before we run our model, we should check what the effect is of both priors combined. Because we have set the priors we can simulate what we believe the data to be. This is one way to see whether our priors actually make sense. It is called a prior predictive check.

We can use brms to do this by running the brm() function. However, instead of running the actual model, we tell it to only sample from the prior.

Code
model_height_prior <- brm(
  height ~ 1,  
  data = data, 
  family = gaussian,
  prior = c(
      prior(normal(170, 10), class = "Intercept"),
      prior(cauchy(5, 5), class = "sigma")
    ), 
  cores = 4,
  seed = 4, 
  sample_prior = "only",
  file = "models/model_height_prior.rds"
)

We then use the tidybayes package to draw samples from the prior and plot these draws.

Code
predictions_prior <- tibble(distribution = "prior")

predictions_prior <- add_predicted_draws(
  newdata = predictions_prior, 
  object = model_height_prior, 
  value = "predicted_height"
)

ggplot(predictions_prior, aes(x = predicted_height)) +
  geom_histogram(binwidth = 1, alpha = .85) +
  xlim(100, 250) +
  labs(x = "Height", y = "")

Prior predictive check

So, our priors result in a normal distribution of heights ranging from about 125 cm to 225 cm. That is too wide, but let’s run the model to see what happens.

Running the model

We run the model with the code below. Notice that we sample from the prior so we can not only visualize our posterior later, but also the priors we have just defined.

Code
model_height <- brm(data = data, 
  family = gaussian,
  height ~ 1,
  prior = c(
    prior(normal(170, 10), class = "Intercept"),
    prior(cauchy(5, 5), class = "sigma")
  ),
  cores = 4,
  seed = 4,
  sample_prior = TRUE,
  file = "models/model_height.rds"
)

After running the model, we first check whether the chains look good.

Code
plot(model_height)

It seems like they do. The distributions look normal and the chains look like caterpillars, which means they’re sampling from the distribution space and that’s what we want.

We can call up the estimates and the 95% confidence intervals by printing the model object.

Code
summary(model_height)
 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: height ~ 1 
   Data: data (Number of observations: 352) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept   154.63      0.41   153.80   155.42 1.00     3674     2697

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     7.77      0.30     7.19     8.39 1.00     3720     2921

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).

Here we see the Intercept and sigma estimates. Apparently our posterior estimate for the Intercept is 154.63 and the estimate for \(\sigma\) is 7.77. We also see the 95% CIs, but let’s visualize these results instead.

Comparing the prior and posterior distributions

Inspecting the chains also showed us the posterior distributions of the two parameters, but let’s create our own graphs that compare both the prior and posterior distributions.

Code
results <- model_height %>%
  gather_draws(b_Intercept, sigma, prior_Intercept, prior_sigma) %>%
  mutate(
    parameter = if_else(str_detect(.variable, "sigma"), "sigma", "intercept"),
    distribution = if_else(str_detect(.variable, "prior"), "prior", "posterior")
  )

results_intercept <- filter(results, parameter == "intercept")
results_sigma <- filter(results, parameter == "sigma")

ggplot(results_intercept, aes(x = .value, fill = fct_rev(distribution))) +
  geom_histogram(binwidth = 1, position = "identity", alpha = .85) +
  xlim(145, 195) +
  labs(x = "Average height", y = "", fill = "Distribution") +
  scale_fill_viridis(option = "mako", discrete = TRUE, begin = .25, end = .75)

Here we see that the posterior distribution of average heights is now much more narrow and centered around 155 cm. So not only should we switch from thinking the average is a lot lower than 170, we can also be much more confident about the mean.

How about sigma?

Code
ggplot(results_sigma, aes(x = .value, fill = fct_rev(distribution))) +
  geom_histogram(binwidth = 0.25, position = "identity", alpha = .85) + 
  xlim(0, 25) +
  labs(x = "Height standard deviation", y = "", fill = "Distribution") +
  scale_fill_viridis(option = "mako", discrete = TRUE, begin = .25, end = .75)

Similarly, we see that the posterior for sigma is also much more narrow and around 8.

A final step is to visualize the posterior distribution of all heights (posterior predictive check) and compare it the distribution of heights based on our priors (the prior predictive check).

Code
predictions_posterior <- tibble(distribution = "posterior")

predictions_posterior <- add_predicted_draws(
  newdata = predictions_posterior,
  object = model_height, 
  value = "predicted_height"
)

predictions <- bind_rows(predictions_prior, predictions_posterior)

ggplot(predictions, aes(x = predicted_height, fill = distribution)) +
  geom_histogram(binwidth = 1, alpha = .85, position = "identity") +
  xlim(100, 250) +
  labs(x = "Height", y = "", fill = "Distribution") +
  scale_fill_viridis(option = "mako", discrete = TRUE, begin = .25, end = .75)

Prior and posterior predictive check

Adding a predictor

Now let’s add a predictor to our model. Besides heights, the data set also contains their weights. We can create a model in which we regress heights onto weights. The formula syntax for a model like that in R is height ~ weight. We can use this formula again in get_prior() to see which priors we need to specify.

Code
get_prior(height ~ weight, data = data)
prior class coef group resp dpar nlpar lb ub source
b default
b weight default
student_t(3, 154.3, 8.5) Intercept default
student_t(3, 0, 8.5) sigma 0 default

The output is a bit trickier this time. We see the Intercept and sigma priors from our previous model, as well as two extra rows referring to a class called b. These two rows actually refer to the same prior, one refers specifically to the weight predictor and one refers to all predictors. If you run a model with many more predictors, you could set one prior that applies to all predictors. In this case though, we only have 1 predictor so it actually doesn’t matter, both refer to the same prior.

Given that this is a bit trickier, and given that I said writing down your model explicitly is better, we should go ahead and do that.

\[ heights_i ∼ Normal(\mu_i, \sigma)\\ \mu_i = \alpha + \beta x_i \]

We again specify that the heights are normally distributed, so we still have a \(\mu\) and \(\sigma\), but this time the \(\mu\) is no longer a parameter we will estimate. Instead, it’s constructed from other parameters, \(\alpha\), \(\beta\), and an observed variable \(x_i\) (the weight observations).

If you’re used to linear regression equations, this notation should not surprise you. \(\alpha\) refers to the intercept and \(\beta\) to the slope.

We need to set priors on these parameters. We previously already discussed the intercept prior so we could reuse that prior, although that means we need to center the data so the intercept refers to the average height of someone with an average weight rather than someone with 0 weight. So let’s first mean center the weight observations.

Code
data <- mutate(data, weight_mc = weight - mean(weight))

Now we can use the same prior as before, which was a normal distribution with a mean of 170 and a standard deviation of 10.

Next is the prior for the slope. This represents the relationship between weights and heights. For every 1 increase in weight, how much do we think that the height will increase or decrease? We could begin with an agnostic prior in which we do not specify the direction and instead just add some uncertainty so the slope can go in either direction. For example, let’s put a normal distribution on the slope with a mean of 0 and a standard deviation of 10.

Code
model_height_weight_prior <- brm(
  height ~ weight_mc,  
  data = data, 
  family = gaussian,
  prior = c(
      prior(normal(170, 10), class = "Intercept"),
      prior(cauchy(5, 5), class = "sigma"),
      prior(normal(0, 10), class = "b")
    ), 
  cores = 4,
  seed = 4, 
  sample_prior = "only",
  file = "models/model_height_weight_prior.rds"
)

We can again create a prior predictive check to see whether our priors actually make sense. However, instead of plotting the predicted distribution of heights, we’re mostly interested in the relationship between weight and height, so we should plot a check of that relationship instead. This can be done by drawing sets of intercepts and slopes from the model results.

Below we draw intercepts and slopes from the model result and plot 100 of them. To help make sense of the sensibility of the slopes I’ve added the average weight to the weights so we’re back on the normal scale and not the mean centered scale and I’ve added two dashed lines to indicate the minimum and maximum height we can expect.

Code
draws <- spread_draws(
  model_height_weight_prior, b_Intercept, b_weight_mc
)

weight_mean <- data %>%
  pull(weight) %>%
  mean()

ggplot(data, aes(x = weight_mc, y = height)) +
  geom_blank() +
  geom_hline(yintercept = 0, linetype = "dashed") +
  geom_hline(yintercept = 272, linetype = "dashed") +
  geom_abline(
    data = filter(draws, .draw <= 100),
    mapping = aes(intercept = b_Intercept, slope = b_weight_mc),
    alpha = .25
  ) +
  geom_label(x = 15, y = 260, label = "Tallest person ever") +
  labs(x = "Weight", y = "Height") +
  scale_x_continuous(labels = function(x) round(x + weight_mean))

A prior predictive check of the relationship between weight and height

The plot shows a wide range of possible slopes, some of which are definitely unlikely. We should lower our uncertainty by reducing the standard deviation on the prior. In the next model I lower it to 3. Additionally, the negative slopes are all pretty unlikely because we should expect a positive relationship between weight and height (taller people tend to be heavier). We could therefore also change our prior to force it to be positive using the lb argument in our prior for b.

Code
model_height_weight_prior <- brm(
  height ~ weight_mc,  
  data = data, 
  family = gaussian,
  prior = c(
      prior(normal(170, 10), class = "Intercept"),
      prior(cauchy(5, 5), class = "sigma"),
      prior(normal(0, 3), class = "b", lb = 0)
    ), 
  cores = 4,
  seed = 4, 
  sample_prior = "only", 
  file = "models/model_height_weight_prior_lb.rds",
  control = list(adapt_delta = 0.9)
)

When I first ran this model I received the warning that there was 1 divergent transition after warmup. The Rhat values did not show this was problematic but I wanted to get rid of the warning anyway so I increased the adapt_delta, as suggested in the documentation, from .8 to .9.

Code
draws <- spread_draws(
  model_height_weight_prior, b_Intercept, b_weight_mc
)

ggplot(data, aes(x = weight_mc, y = height)) +
  geom_blank() +
  geom_hline(yintercept = 0, linetype = "dashed") +
  geom_hline(yintercept = 272, linetype = "dashed") +
  geom_abline(
    data = filter(draws, .draw <= 100),
    mapping = aes(intercept = b_Intercept, slope = b_weight_mc),
    alpha = .25
  ) +
  geom_label(x = -10, y = 260, label = "Tallest person ever") +
  labs(x = "Weight", y = "Height") +
  scale_x_continuous(labels = function(x) round(x + weight_mean))

A prior predictive check of the relationship between weight and height

This looks a lot better, so let’s run the model for real now.

Code
model_height_weight <- brm(data = data, 
  height ~ weight,
  family = gaussian,
  prior = c(
      prior(normal(170, 10), class = "Intercept"),
      prior(cauchy(5, 5), class = "sigma"),
      prior(normal(0, 3), class = "b", lb = 0)
    ), 
  cores = 4,
  seed = 4,
  sample_prior = TRUE,
  file = "models/model_height_weight.rds",
  control = list(adapt_delta = 0.9)
)

model_height_weight
 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: height ~ weight 
   Data: data (Number of observations: 352) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept   113.91      1.93   110.02   117.68 1.00     4029     2527
weight        0.90      0.04     0.82     0.99 1.00     4054     2520

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     5.11      0.20     4.75     5.51 1.00     3814     2859

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).

We see that the estimate for the weight predictor is 0.9. Let’s plot the entire posterior for the estimate and also compare it to the prior we set for it.

Code
results <- model_height_weight %>%
  gather_draws(prior_b, b_weight) %>%
  mutate(
    distribution = if_else(
      str_detect(.variable, "prior"), "prior", "posterior"
    )
  )

ggplot(results, aes(x = .value, fill = fct_rev(distribution))) +
  geom_histogram(binwidth = 0.05, position = "identity", alpha = .85) +
  xlim(0, 5) +
  labs(x = "Slope", y = "", fill = "Distribution") +
  scale_fill_viridis(option = "mako", discrete = TRUE, begin = .25, end = .75)

Apparently our prior was still very uninformed because the posterior shows we can be a confident in a much narrower range of slopes!

Thinking correlations instead

Maybe one reason our prior was so uninformed was because it’s harder to think of the right prior for a content-specific topic such as weights and heights of the !Kung San. Maybe we can instead standardize both the heights and weights in order to turn the regression model into a simple correlation analysis. That way we can specify a prior on what we think the correlation should be, which may be easier to do because we then think in terms of whether we think the relationship is small or medium or large, or something along those lines.

So, let’s standardize the heights and weights.

Code
data <- mutate(
  data, 
  height_z = (height - mean(height)) / sd(height),
  weight_z = (weight - mean(weight)) / sd(weight)
)

The formula for our correlation analysis is height_z ~ weight_z. Which priors we have to specify remains the same, but what these priors should be did change. For instance, we know that the Intercept has to be 0 now because the heights have been standardized. This means the mean will be 0. In brms, we can specify a constant as a prior using constant().

What should the prior for \(\sigma\) be? Let’s keep that one the same for now and see what the posterior results will be.

The prior for the slope is a lot easier now. We can simply specify a normal distribution with a mean of 0 and a standard deviation equal to the size of the effect we deem likely, together with a lower bound of 0 and upper bound of 1.

Code
model_height_weight_z <- brm(
  height_z ~ weight_z,  
  data = data, 
  family = gaussian,
  prior = c(
      prior(constant(0), class = "Intercept"),
      prior(cauchy(0, 2.5), class = "sigma"),
      prior(normal(0, 1), class = "b", lb = 0, ub = 1)
    ), 
  cores = 4,
  seed = 4, 
  sample_prior = TRUE,
  file = "models/model_height_weight_prior_z.rds"
)

model_height_weight_z
 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: height_z ~ weight_z 
   Data: data (Number of observations: 352) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept    -0.00      0.00    -0.00    -0.00 1.00     3371       NA
weight_z      0.75      0.03     0.68     0.82 1.00     3371     2588

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.66      0.03     0.61     0.71 1.00     3024     2617

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).