Bayesian Data Analysis

Orientation

Joshua Wilson Black
joshua.black@canterbury.ac.nz

Te Kāhui Roro Reo | New Zealand Institute of Language, Brain and Behaviour

Te Whare Wānanga o Waitaha | University of Canterbury

Overview

Overview

  • Plan for these sessions
  • Fit a Bayesian model
  • What’s the big deal?
    • i.e., what has changed from non-Bayesian models
    • A teeny bit about interpretations of probability.

The plan

Bayes sessions

  • Four intro sessions:
    1. Orientation
    2. What does my model say?
    3. What does my model assume?
    4. Is my model healthy?
  • Josh away: 16/03, 23/03.
  • Mid-semester break
  • Six sessions on specific model structuress.
    • Regression
    • Multi-level regression
    • Likert-scale data

&c. &c. Based on your interest.

Fit a Bayesian Model

Data

“What school did you go to?”

Wrangling

trap_sub <- qb2 |> 
  filter(
    school %in% c(
      "Avonside Girls' High School", 
      "St Margaret's College"
    )
  ) |> 
  select(
    age, school, F1_lob2, part_id
  )

A model

trap_fit <- lm(
  F1_lob2 ~ school,  
  data = trap_sub
)
  • This is a very simple ‘analysis of variance’ model
  • We want to see if school (i.e. Avonside vs. St Margarets) makes a difference to trap realisation
    • …without thinking about confounding from other factors 😱
  • How do we ‘make it Bayesian’?

A Bayesian model

library(brms)
trap_fit_b <- brm(
  F1_lob2 ~ school,  
  data = trap_sub
)

🎊 Congratulations, you’re a Bayesian! 🎊

  • brms makes Bayesian methods incredibly accessible.
  • For many models just putting in brm() instead of lm() (lmer(), etc…), will work.

lmer() → brm()

‘Wow! Now my model converges!’

🐉🐉🐉

kia tūpato…

What’s the big deal?

Model interpretation

summary(trap_fit)

Call:
lm(formula = F1_lob2 ~ school, data = trap_sub)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.73538 -0.47347 -0.02352  0.46701  2.95033 

Coefficients:
                            Estimate Std. Error t value Pr(>|t|)    
(Intercept)                  0.24632    0.02946   8.362   <2e-16 ***
schoolSt Margaret's College  0.47782    0.05258   9.087   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.8195 on 1126 degrees of freedom
Multiple R-squared:  0.06832,   Adjusted R-squared:  0.0675 
F-statistic: 82.58 on 1 and 1126 DF,  p-value: < 2.2e-16
  • Point estimates of coefficients
  • Standard error
  • Statistical significance tests

Model interpretation (cont.)

summary(trap_fit_b)
 Family: gaussian 
  Links: mu = identity 
Formula: F1_lob2 ~ school 
   Data: trap_sub (Number of observations: 1128) 
  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
Intercept                    0.25      0.03     0.19     0.30 1.00     3984
schoolStMargaretsCollege     0.48      0.05     0.38     0.58 1.00     4079
                         Tail_ESS
Intercept                    2652
schoolStMargaretsCollege     3128

Further Distributional Parameters:
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     0.82      0.02     0.79     0.86 1.00     3986     3221

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).
  • ‘Draws’?
  • Coefficients
  • Error and intervals
  • ‘Rhat’, ‘Bulk ESS’??
  • No statistical significance tests

Difference: no point estimates

  • Bayesian model gives distributions for all parameters.

No privileged:

  • summary
  • error intervals

summary() gives a mean with 95% interval.

  • Predictions from the model take into account uncertainty about coefficient.

Difference: Bayesian CIs

  • Both Bayesian and non-Bayesian models produce ‘CIs’, but use different words for the ‘C’:
    • ‘Confidence intervals’ vs.
    • Bayesian ‘credible intervals’ (sometimes ‘compatible intervals’)
  • Interpretation: according to our data and model, there is a 95% change the parameter is in the interval.
  • There’s nothing special about 95%, we can use any other number we like (often more than one).

  • This ‘half-eye’ plot shows 50%, 70% and 90% intervals around the mean.
  • More on making these plots next time!

Flexible estimates

  • Let’s predict trap realisation for these schools.
  • Under the hood:
    1. generate an Avonside value from the distribution, then
    2. generate a St Margaret’s value and add to the Avonside value.
  • Repeat lots of times to get a distribution.
  • This is easy, but the models can get much more complicated!

trap realisations

Difference in trap

  • Our model and data are not compatible with saying there is no difference.
  • In this case, this is pretty much identical to the distribution of the ‘St Margarets’ coefficient in our simple model.
  • But models can get more complicated!

Background: probability

  • The alternative to Bayesianism is ‘frequentism’.
  • Core difference: how they use probability
  • Frequentists: probability quantifies the reliability of the method.
  • Bayesians: probability quantifies uncertainty.
  • Both are well motivated ideas!

Background: probability (cont.)

  • Frequentist probability is connected to proportions of samples, e.g.:
    • ‘If this analysis were repeated over and over again, we would draw the wrong conclusion 5% of the time.’
    • ‘If you sampled NZE speakers born in 1937, 10% of would regularly use “kia ora” as a greeting.’
  • Bayesian’s use probability for uncertainty:
    • ‘Given the data and model, we are 90% sure that the effect of this variable is greater than zero’
    • ‘The probability that language X and language Y are in the same family is 65%’ (…what would you sample?)
  • It’s very hard to understand the Bayesian examples in terms of sampling.
    • ‘If universes were as plenty as blackberries…’ (CS Peirce)

Confidence vs. credible (again)

  • Standard 95% confidence intervals: i.e. ‘frequentist’
    • If we repeat this method, the parameter we’re estimating will be inside the confidence interval 95% of the time.
    • We’re ‘sampling’ applications of the method.
    • It’s not a claim about this exact confidence interval!
  • Bayesian 95% credible intervals:
    • Given the data and model, we reckon, with 95% confidence, that the parameter is somewhere in here!
    • It’s a claim about this exact interval.
    • Easier to interpret, but we’ve changed the topic.

NB

  • Bayesianism is not ‘Statistics 2.0’
  • There’s nothing especially subjective about Bayesianism.
    • Bayesian methods use probability to express some subjective elements.
    • Frequentists often put subjective elements into model assumptions etc.
  • You don’t have to be Bayesian
    • …but you should understand it enough to read and review articles.

Now and next time:

  1. Make sure you have brms and tidybayes packages installed.
  2. Clone or download the github repository at https://github.com/nzilbb/ws-bayes-1.
  3. Next time:
    • interpreting Bayesian models in more detail, esp:
    • plotting coefficients and predictions.

References

Allaire, JJ, Yihui Xie, Christophe Dervieux, Jonathan McPherson, Javier Luraschi, Kevin Ushey, Aron Atkins, et al. 2025. rmarkdown: Dynamic Documents for r. https://github.com/rstudio/rmarkdown.
Bürkner, Paul-Christian. 2017. “brms: An R Package for Bayesian Multilevel Models Using Stan.” Journal of Statistical Software 80 (1): 1–28. https://doi.org/10.18637/jss.v080.i01.
———. 2018. “Advanced Bayesian Multilevel Modeling with the R Package brms.” The R Journal 10 (1): 395–411. https://doi.org/10.32614/RJ-2018-017.
———. 2021. “Bayesian Item Response Modeling in R with brms and Stan.” Journal of Statistical Software 100 (5): 1–54. https://doi.org/10.18637/jss.v100.i05.
Fromont, Robert. 2025. nzilbb.labbcat: Accessing Data Stored in “LaBB-CAT” Instances. https://doi.org/10.32614/CRAN.package.nzilbb.labbcat.
Kay, Matthew. 2024. tidybayes: Tidy Data and Geoms for Bayesian Models. https://doi.org/10.5281/zenodo.1308151.
Müller, Kirill. 2025. here: A Simpler Way to Find Your Files. https://doi.org/10.32614/CRAN.package.here.
R Core Team. 2025. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.
Wickham, Hadley, Mara Averick, Jennifer Bryan, Winston Chang, Lucy D’Agostino McGowan, Romain François, Garrett Grolemund, et al. 2019. “Welcome to the tidyverse.” Journal of Open Source Software 4 (43): 1686. https://doi.org/10.21105/joss.01686.
Wilson Black, Joshua, James Brand, Jen Hay, and Lynn Clark. 2023. “Using Principal Component Analysis to Explore Co‐variation of Vowels.” Language and Linguistics Compass 17 (1): e12479. https://doi.org/10.1111/lnc3.12479.
Xie, Yihui, J. J. Allaire, and Garrett Grolemund. 2018. R Markdown: The Definitive Guide. Boca Raton, Florida: Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown.
Xie, Yihui, Christophe Dervieux, and Emily Riederer. 2020. R Markdown Cookbook. Boca Raton, Florida: Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown-cookbook.