Sequential Updating & Prior Sensitivity (Ch. 4): incorporating new data over time and seeing how the prior shapes the posterior.
Conjugate Models (Ch. 5): Beta–Binomial, Gamma–Poisson, and Normal–Normal pairs that give closed-form posteriors.
Obtaining the Posterior (Ch. 6 & 7, last lecture): grid approximation and MCMC for the (very common) case where conjugacy isn’t available.
Today, we put posteriors to work, turning a distribution over the parameter into the kind of answer you’d actually report in a paper:
Summarise the posterior with point estimates and credible intervals
Generate forecasts via the posterior predictive distribution
Test interval and point hypotheses using Bayes factors and the Savage–Dickey shortcut
Every code chunk below uses the Dutch Railways running example from last lecture (same NS signal-failure rate \(\lambda\), same prior Gamma(2, 0.5), same data).
A Quick Bayesian Refresher
Let’s revisit the key ideas from Chapters 1–5: the Bayesian toolkit we’re carrying into today’s work.
Four Core Ingredients of Bayesian Analysis
Every Bayesian model is built on four fundamental components:
Prior Model
We begin with a parameter of interest (say, \(\pi\)) and specify what we believe about it before seeing any data: \[
f(\pi)
\]
A common choice for a proportion is the Beta distribution: \[
f(\pi) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} \pi^{\alpha - 1}(1 - \pi)^{\beta - 1}
\]
Data Model
We assume our observed data arise from a known model governed by \(\pi\).
For example, in \(n\) Bernoulli trials: \[
Y \sim \text{Binomial}(n, \pi)
\]
Likelihood Function
After observing \(Y = y\), the likelihood tells us how compatible different values of \(\pi\) are with the data: \[
L(\pi \mid y) \propto \pi^y (1 - \pi)^{n - y}
\] (Note: the data are fixed here, it’s \(\pi\) that varies.)
Posterior Model
Finally, we update our beliefs using Bayes’ Rule: \[
f(\pi \mid y) \propto f(y \mid \pi) \cdot f(\pi)
\]
The posterior distribution is the heart of Bayesian analysis. It tells us what we believe about \(\pi\)after seeing data, and it’s the foundation for all the inferential tools we’ll build today.
But there’s a catch:
Computing the posterior can be hard. Bayes’ Rule is conceptually simple, but that denominator (the marginal likelihood\(f(y)\)) is often intractable.
Conjugate Models to the Rescue
Conjugate Models
To simplify updates, we can use conjugate priors, priors that pair naturally with a likelihood to give a posterior in the same distribution family.
This eliminates the need to compute the normalizing constant directly.
Here are some common conjugate pairs, and one challenge row left blank for you:
Starting from a Gamma(\(2\), \(0.5\)) prior, we observed the four datapoints \(y_1 = (2, 0, 3, 1)\), and updated to a Gamma(\(8\), \(4.5\)) posterior distribution of the failure rate. Now, we observe six more data points \(y_2 = (2, 3, 0, 1, 2, 1)\).
Use the posterior you obtained after observing the first four observations as a new prior, and update with the new data:
# Prior parameterss <-2r <-0.5# Datay1 <-c(2, 0, 3, 1)y2 <-c(2, 3, 0, 1, 2, 1)# Posterior parameters after observing the first four datapointss_post1 <-8r_post1 <-4.5# Posterior parameters after observing the next six datapointss_post2 <- _____ r_post2 <- _____
Meanwhile, your colleague was late to the party.
Starting from the same Gamma(\(2\), \(0.5\)) prior, they now want the posterior using all 10 data points at once: \(y = (2, 0, 3, 1, 2, 3, 0, 1, 2, 1)\)
# Prior parameterss <-2r <-0.5# Datay_all <-c(2, 0, 3, 1, 2, 3, 0, 1, 2, 1)# Posterior parameters after observing all ten datapointss_post_all <- ____r_post_all <- ____
Use the bayesrules package to confirm your result.
# Prior parameterss <-2r <-0.5# Datay1 <-c(2, 0, 3, 1)y2 <-c(2, 3, 0, 1, 2, 1)y_all <-c(2, 0, 3, 1, 2, 3, 0, 1, 2, 1)# Posterior parameters after observing the first four datapointss_post1 <-8r_post1 <-4.5# Posterior parameters after observing the next six datapointss_post2 <- s_post1 +sum(y2)r_post2 <- r_post1 +length(y2)# Check your posteriorsummarize_gamma_poisson(shape = s_post1,rate = r_post1,sum_y =sum(y2),n =length(y2))# Posterior parameters after observing all ten datapointss_post_all <- s +sum(y_all)r_post_all <- r +length(y_all) # Check their posteriorsummarize_gamma_poisson(shape = s,rate = r,sum_y =sum(y_all),n =length(y_all))
These two exercises reinforce two key ideas:
Conjugate models simplify posterior computation
Bayesian updating is coherent: results are consistent whether we learn all at once or in stages
With that foundation in place, we now turn to the next question: What can we do with the posterior?
We’ll begin by learning how to summarize posterior uncertainty and test hypotheses using tools like credible intervals and Bayes factors.
Posterior Inference & Hypothesis Testing
We’ve now built and updated posterior distributions. But how do we use them to draw conclusions?
In this section, we’ll explore two main applications of the posterior: estimation and hypothesis testing.
What’s Ahead
We’ll cover:
- How to summarize a posterior with point estimates and credible intervals
- How to test both interval and point hypotheses using tail areas and Bayes factors
Posterior Estimation (Ch. 8.1)
After this section, you will be able to:
Summarize a posterior distribution using key statistics such as the mean, mode, and standard deviation
Construct central credible intervals using quantile functions
Interpret credible intervals in a Bayesian framework as statements about parameter uncertainty
When we finish a Bayesian analysis, we don’t just want a plot, we want answers.
What’s our best guess for the parameter?
How uncertain are we?
What values are most plausible?
To answer these questions, we turn to posterior summaries: key numbers that condense the full posterior distribution into interpretable insights.
Measures of Location
Let’s start with where the posterior “centers.” What’s our best guess?
The posterior mean, mode, and median all describe central tendency, but they don’t always agree. For symmetric posteriors, they often align. But when the posterior is skewed, they can differ, sometimes significantly.
When might the median be preferred over the mean?
Often, in skewed posteriors, the mean gets pulled toward the tail. You might even suspect a bug in your code, but it’s just the structure of the posterior density. In such cases, the median can give a more representative summary of what’s typical.
Measures of Spread
But summaries aren’t just about the center, we also want to express uncertainty.
A common measure of spread is the posterior standard deviation. It tells us how tightly our beliefs are concentrated around the mean.
A small SD = tight concentration = high certainty
A large SD = wide spread = greater uncertainty
This works well for symmetric, unimodal posteriors, but breaks down when things get skewed or heavy-tailed. That’s why we often complement with credible intervals.
Posterior Credible Interval
A credible interval gives us a direct probabilistic interpretation:
“There’s a 95% posterior probability the parameter lies between X and Y.”
Formally, for a parameter \(\theta\) with posterior distribution \(f(\theta \mid y)\), a \(100(1 - \alpha)\%\)central credible interval is:
You can compute it using quantiles:
qnorm(c(0.025, 0.975), mean = posterior_mean, sd = posterior_sd)
Let’s revisit the railroad failure example and see this in action.
You can also compare them side-by-side with plots; central interval vs. HPD:
Figure 1: ?(caption)
Posterior Prediction
Summarising the posterior is one thing; predicting future observations is another. The posterior tells us about \(\lambda\), the latent rate, but if you’re standing in front of an NS engineer, the question they actually care about is:
How many signal failures should we plan for next month?
That’s a question about a new observation \(\tilde Y\), not the parameter. Bayesian prediction handles it cleanly: average the data-generating distribution \(f(\tilde Y \mid \lambda)\) over our current posterior beliefs about \(\lambda\).
The posterior predictive distribution
\[
f(\tilde y \mid y) \;=\; \int f(\tilde y \mid \lambda)\, f(\lambda \mid y)\, d\lambda
\]
Read this as: “the predictive probability of \(\tilde y\) is its sampling probability under each value of \(\lambda\), weighted by how plausible that value is given the data we’ve already seen.”
For the Gamma–Poisson model, this integral has a closed-form solution: \(\tilde Y \mid y \sim \text{Negative Binomial}\!\bigl(\text{size} = s_{\text{post}}, \text{prob} = r_{\text{post}}/(r_{\text{post}}+1)\bigr)\). But we can also just simulate, and the simulation works for any model, conjugate or not.
Code
set.seed(2026)# Posterior parameters from earlier: Gamma(8, 4.5)s_post <-8r_post <-4.5# Step 1: draw lambda from the posteriorlambda_draws <-rgamma(1e4, shape = s_post, rate = r_post)# Step 2: for each lambda, draw a single future observation y_tilde ~ Poisson(lambda)y_tilde <-rpois(1e4, lambda = lambda_draws)cat("Predictive mean :", round(mean(y_tilde), 3), "\n")
df_pred <-data.frame(y_tilde = y_tilde)ggplot(df_pred, aes(x = y_tilde)) +geom_bar(aes(y = ..prop..), fill ="#0b7285", colour ="white") +scale_x_continuous(breaks =0:max(y_tilde)) +labs(title ="Posterior predictive distribution of future monthly failures",x =expression(tilde(Y)), y ="Predictive probability") +theme_minimal(base_size =13)
Why predict, not estimate?
A credible interval for \(\lambda\) tells you about our knowledge of the mean rate of failures. It’s narrow when you have lots of data.
A predictive interval for \(\tilde Y\) tells you about the next single month. It is always wider, because it folds Poisson sampling noise on top of posterior uncertainty.
Exercise: predictive vs estimation intervals
Note
Compute a 90% credible interval for \(\lambda\) using the Gamma(8, 4.5) posterior.
Compute a 90% predictive interval for \(\tilde Y\) using the simulated draws above.
Sketch what each is telling you in words. Which is wider, and why?
The credible interval is narrow because we have moderate information about \(\lambda\). The predictive interval is much wider because even if we knew \(\lambda\) exactly, the next month’s failure count would still be a Poisson draw.
Now, with both estimation and prediction in our toolkit, we can take the third step: testing specific claims about \(\lambda\). We start with interval claims like “Is the failure rate below 2/month?”
Hypothesis Testing
After this section, you will be able to:
Distinguish between interval and point hypotheses, and understand when each is appropriate
Evaluate both types of hypotheses using Bayes factors, computed from prior/posterior odds or the Savage–Dickey method
Interpret Bayes factors as measures of evidence that quantify how data shift our beliefs
Bayesian inference is not only about estimating parameters; it also allows us to evaluate specific claims about those parameters.
Broadly speaking, Bayesian hypothesis testing addresses two types of questions:
What is the probability that a parameter lies within a certain range? (e.g., Is the treatment effect greater than 0?)
How much support do the data provide for a specific parameter value? (e.g., Is the effect exactly 0?)
We refer to these as:
Interval hypotheses: claims about a range of values
Point hypotheses: claims about a precise value
Both can be evaluated using Bayes factors, but the way we compute them differs:
For interval hypotheses, we compare posterior and prior probabilities of a region to form odds.
For point hypotheses, we use the Savage–Dickey density ratio, which compares the density of the prior and posterior at a single value.
Let’s begin with interval hypotheses.
Interval Hypotheses (Ch. 8.2)
Let’s return to our running example. We’re modeling the monthly failure rate \(\lambda\) on a stretch of railway, and we want to evaluate the claim:
“The average failure rate is below 2 failures per month.”
This is a one-sided hypothesis test:
\(H_0\): \(\lambda \geq 2\)
\(H_a\): \(\lambda < 2\)
Earlier, we used the prior:
\[
\lambda \sim \text{Gamma}(2, 0.5)
\]
After observing the first four months of data (\(\sum y = 6\)), we updated to:
To do this, we’ll compute the Bayes factor, which compares the posterior odds to the prior odds, quantifying how the data have updated our belief in favor of \(H_a\) relative to \(H_0\).
We begin by calculating the prior odds: Our belief in \(H_a\) versus \(H_0\)before seeing any data:
This tells us how strongly we favored \(H_a\) over \(H_0\) based on the prior alone. In this case, a prior odds value of 0.36 means that before seeing any data, we believed \(H_a\) was less than half as likely as \(H_0\).
Next, we calculate the posterior odds: Our updated belief after seeing the data:
This reflects how much more likely we now believe \(H_a\) is relative to \(H_0\), based on the observed data. A posterior odds value of 2.09 means that, after seeing the first 4 months of data, we believe \(H_a\) is about twice as likely as \(H_0\).
Finally, we compute the Bayes factor, which quantifies how much the data have updated our odds in favor of \(H_a\) over \(H_0\).
BF <- post_odds / prior_odds#> BF#[1] 5.812208
This tells us how strongly the data have shifted our beliefs in favor of \(H_a\) over \(H_0\). A Bayes factor of 5.81 means the observed data have made \(H_a\) nearly six times more plausible than it was under our prior beliefs.
This represents substantial evidence (Example) in favor of the alternative hypothesis, according to conventional interpretive scales.
Bayes Factor
The Bayes factor compares how much more the data support one hypothesis over another, by comparing how our odds changed from prior to posterior:
This plot shows the prior and posterior distributions. The shaded regions under each curve represent the probability mass supporting the alternative hypothesis (\(\lambda < 2\)). The embedded pie charts show those probabilities visually.
Was there a shift in beliefs from prior to posterior?
Let’s now see how our evidence changes when we update with additional data.
In the next exercise, you’ll work with the full 10 months of data: \(\sum y = 15\), leading to the posterior:
\[
\lambda \mid y \sim \text{Gamma}(17, 10.5)
\]
How does the Bayes factor change? Has the evidence for \(H_a\) strengthened?
The Bayes factor based on the full dataset is approximately 14.3, indicating strong support for the alternative hypothesis (\(\lambda < 2\)) over the null (\(\lambda \geq 2\)).
Notably, this strengthens the conclusion we reached after just 4 months of data, when the Bayes factor already modestly favored \(H_a\).
In this case, the full dataset provides substantial to strong evidence that the failure rate is indeed below the target threshold. We now turn to a different type of hypothesis: one that asks whether a parameter equals a specific value.
Testing Point Hypotheses
In the previous section, we tested interval hypotheses like:
“Is the failure rate \(\lambda\) below 2?”
We did this by computing posterior and prior odds, and comparing them via the Bayes factor. But what if we want to evaluate a point hypothesis like:
“Is the average number of failures per month exactly 2?”
This question falls outside the scope of the tools we’ve used so far. Under a continuous prior and posterior, the probability that a parameter equals any exact value is zero.
So how do we assess a point hypothesis?
A Shift in Perspective
Testing point hypotheses requires a different mindset. Instead of assessing the posterior probability of a value (which is zero), we reframe the problem as a model comparison:
\(H_0\): The parameter equals a specific value (e.g., \(\theta = \theta_0\))
\(H_1\): The parameter is unknown and follows a continuous prior (e.g., \(\theta \sim \text{Normal}(m_0, s_0^2)\))
We then use the Bayes factor to compare how well each model explains the data.
Enter the Savage–Dickey Density Ratio
The Savage–Dickey method provides an elegant shortcut for computing the Bayes factor without calculating marginal likelihoods.
If:
\(H_0\) is nested within \(H_1\) (e.g., \(\theta = \theta_0\) is a special case of \(\theta \sim \text{Normal}\)),
The prior is continuous at \(\theta_0\), and
We can compute both the prior and posterior densities at \(\theta_0\),
This approach may seem surprising: in interval testing, we emphasized that the probability of a single point under a continuous distribution is zero. But here, we’re comparing densities, not probabilities, which is a crucial distinction.
Let’s visualize what this really means.
Savage-Dickey Density Ratio
Savage-Dickey Density Ratio: Diffuse Prior
Savage-Dickey Density Ratio: More Peaked Posterior
Why It Works (Don’t worry, this won’t be on the test)
We start with the posterior density under \(H_1\):
Your safety team tracks monthly signal failures across a critical 10 km stretch. Historically, the failure rate has been assumed to be exactly 2 failures per month, which serves as a threshold in staffing models.
After recent maintenance upgrades, you want to test whether this long-standing benchmark still holds:
In our example, the Bayes factor computed via the Savage–Dickey method is approximately 3, indicating modest evidence in favor of the point hypothesis\(\lambda = 2\) over the continuous alternative.
Thus, we can assess point hypotheses by comparing the posterior and prior densities at a specific parameter value. The Savage–Dickey method provides a direct, interpretable way to quantify evidence for precise claims, all without computing full marginal likelihoods.
Bayes factors, whether for interval or point hypotheses, offer a unified framework for evaluating claims in light of data. As always, interpreting them requires context, not just a threshold, but an understanding of the models and prior assumptions involved.
Understanding Bayes Factors
Bayes factors quantify how the data have updated our beliefs between two competing hypotheses. Specifically, the Bayes factor is the ratio of how likely the observed data are under one hypothesis compared to another:
A Bayes factor of 5 means the data are 5 times more likely under \(H_0\) than under \(H_1\). A Bayes factor of 0.2 means the data are 5 times more likely under \(H_1\) than under \(H_0\).
To interpret Bayes factors in practice, we often use this scale from Kass & Raftery (1995):
Bayes Factor (BF)
Evidence for \(H_0\)
1 to 3
Barely worth mentioning
3 to 10
Substantial
10 to 30
Strong
30 to 100
Very strong
> 100
Decisive
Reference: Kass, R. E., & Raftery, A. E. (1995). Bayes Factors. Journal of the American Statistical Association, 90(430), 773–795.
These are guidelines, not rules. The strength of evidence needed to shift beliefs can vary by context and domain. Bayesian inference gives us a continuous scale of evidence; these categories help us summarize that evidence clearly.
But context matters. A Bayes factor of 10 might offer compelling evidence in a routine analysis, yet barely register when prior odds are extremely low. In extraordinary claims like precognition, even a seemingly strong BF may be overwhelmed by a skeptical prior. Inference depends on posterior odds, and the Bayes factor is just one part of that update.
Coherence: Sequential Bayes Factors
Bayes factors not only offer a measure of evidence, they also maintain coherence when applied in stages. This means we can accumulate evidence gradually and consistently over time.
Suppose we observe data in two batches and update our beliefs sequentially. The total Bayes factor from the full data is then the product of:
The Bayes factor after the first batch, and
The Bayes factor based on the second batch given the first.
As you can see, the sequential Bayes factor (BF_seq) and the direct computation from the full data (BF_full) match. This confirms that Bayesian inference is coherent, both in terms of belief updating and the accumulation of evidence.
Posterior Analysis from MCMC Draws (Ch. 8.4)
Everything above used the closed-form Gamma(8, 4.5) posterior. But from Chapter 9 onward (and in most real research), the posterior has no clean formula to read summaries off. The reassuring message of Chapter 8.4 is that none of today’s tools actually require one. Given a sample of posterior draws (from MCMC, Lecture 4), we approximate any posterior feature by the corresponding feature of the draws.
The recipe: same questions, computed from draws
Posterior quantity
From the closed form
From MCMC draws
Mean / median
\(s/r\), qgamma(.5, …)
mean(draws), median(draws)
Mode
\((s-1)/r\)
peak of density(draws)
95% credible interval
qgamma(c(.025, .975), …)
quantile(draws, c(.025, .975))
Interval hypothesis \(P(\lambda < c \mid y)\)
pgamma(c, …)
mean(draws < c)
Let’s check it on the Dutch Railways posterior. We take 50,000 draws from a short Metropolis sampler (exactly the kind we built in Lecture 4) and compare every summary to its analytic Gamma(8, 4.5) value.
The MCMC column lands right on top of the analytic values: MCMC works. (The mode is the fussiest to recover, as it depends on the kernel-density bandwidth, but even it is close.) This is exactly what brms/Stan compute for you automatically, and it is why we can trust the more complex models of the coming weeks even though we can never write their posteriors down, provided the Lecture 4 diagnostics check out.
What We Learned Today
In this lecture, we put the posterior to work: summarising it, predicting future observations from it, and using it to test specific scientific claims.
Summary
Posterior summaries (mean, median, mode, ETI, HPD) describe what we have learned about the parameter; each highlights a different aspect of the posterior.
The posterior predictive distribution propagates parameter uncertainty into predictions for new data, and is always wider than the credible interval for the parameter alone.
Posterior probabilities answer interval claims like “the rate is below 2/month” directly from the posterior.
Bayes factors quantify the evidence in the data for one hypothesis over another, computed either from prior + posterior odds or directly via the Savage–Dickey density ratio for point hypotheses.
MCMC draws deliver every one of these summaries via the corresponding feature of the sample (mean, quantile, mean(draws < c)), so the same toolkit carries over to models with no closed-form posterior (Ch. 8.4).
To consolidate your understanding and prepare for the exam and group assignment:
Suggested exercises from the book: 2, 3, 4, 6, 7, 8, 9, 10, & 14
Suggested exercises from the addendum: 25, 26, & 27
Solutions for all suggested exercises are available on the Canvas page for this lecture.
Explore:
Try the Savage–Dickey shortcut with different models and priors
Compare your predictive intervals to credible intervals on a few different datasets. When does the gap widen?
