An Introduction to Bayesian Hierarchical Modeling for Data Science

Key takeaways 

Bayesian hierarchical modeling is a statistical approach for analyzing data structured in layers, modeling within-group and between-group variation. Here are some important facts to know:

• Reverend Thomas Bayes first introduced Bayesian statistics when his writings were posthumously published in 1763, two years after his death [1]. It has become increasingly popular over time due to its flexibility and ability to account for uncertainty. 

• Bayesian hierarchical models allow you to combine data from multiple sources, represent uncertainty, and account for variability across groups. 

• You can use Bayesian hierarchical modeling to approach research questions and inform policies in fields such as public health, environmental science, and education. 

Learn how to use Bayesian hierarchical models to model dependencies between variables and make data-driven decisions based on complex, layered data. To build your statistics foundation, consider completing the IBM Data Science Professional Certificate. In as little as four months, you’ll have the opportunity to master up-to-date, practical skills and knowledge used by data scientists in their daily roles. 

What is a Bayesian hierarchical model?

At a fundamental level, a Bayesian hierarchical model is a type of statistical framework that uses prior distributions to estimate the parameters of your model at multiple levels. To understand this model, it’s necessary to understand two concepts: Bayesian models and hierarchical models.

A hierarchical model uses multiple “levels” of data. For example, when looking at health care data, you might have patients within a unit (level one), units within a hospital (level two), and even hospitals within a health care system (level three). Hierarchical models allow you to capture variabilities between groups (such as patients between units) and assess complex, real-world data more accurately. 

Bayesian models incorporate prior beliefs (existing knowledge) with new data using a theorem known as Bayes’ theorem. By using this theorem, you generate a full probability distribution for each parameter, reflecting a range of plausible values based on your evidence.

By combining hierarchical modeling and Bayesian techniques, you can incorporate prior knowledge to continually update your model as new information becomes available while accounting for the nested, or hierarchical, structure of real-world data.

Learn more: What Is Bayesian Statistics? 

What is the Bayesian approach to modeling?

The Bayesian approach to modeling, first introduced by Reverend Thomas Bayes in 1763, combines prior knowledge with observed data using Bayes’ theorem [1]. In contrast with the frequentist approach, which uses point estimates and relies on observed data, Bayesian methods use a more flexible framework that can incorporate historical information and update results as you obtain new data to determine the conditional probability of a given event.

Why use Bayesian hierarchical models?

When your data is naturally grouped, Bayesian hierarchical models allow for patterns in the overall population to inform group-level estimates. Traditional models often struggle when data points aren’t independent or naturally cluster (e.g., voters in different states or learners in different classrooms), either analyzing each group in isolation or treating them as identical.

When you use hierarchical Bayesian models, you allow each group to have its own characteristics, such as one classroom having higher baseline test scores than another, while still generating population-level insights across groups. The layered design of this model captures both group differences and broader patterns. For example, a hierarchical Bayesian model would show how much variation comes from individual learners, how much comes from differences between classrooms, and how much comes from differences between schools overall. You can use this model to better understand what is driving differences between learner performance and inform policies or programs more tailored to your learners. 

Another reason to use this type of model is because of the value Bayesian outputs provide. Instead of a single point estimate, you’ll have a range of the most likely values—and the uncertainty around them—for each parameter. This helps you generate a more accurate picture of your data and assess how much confidence to put in each result. Because your Bayesian model will update as you gather more data, the model remains flexible and reflective of the most recent information.

What is the Bayesian model structure?

To perform Bayesian hierarchical modeling, you’ll have a few building blocks that work together to create your model: posterior, prior, and likelihood. At a base level, you need to know the following equation for Bayes’ theorem:

Posterior = Prior * Likelihood

In this equation, the “posterior” is the updated probability distribution for the parameter after seeing the data. This is a function of the “prior,” which represents where your parameters likely fall before observing your data, and the “likelihood,” which describes how likely the observed data are, given a particular value of your parameter.

The key point here is that the posterior is the prior distribution weighted by your observed evidence. In hierarchical modeling, you’re extending this model by allowing your parameters to come from higher-level distributions (sometimes called hyperpriors), which create a layered structure. The first level is the distribution of the data, conditioned on the parameter, and the second level is the distribution of the parameter itself, given the prior.

How to do Bayesian hierarchical modeling

Using Bayesian hierarchical modeling relies on a combination of historical and present-day data to estimate your outcome. To use this type of model, you follow three general steps:

1. Specify your priors.

You can choose this based on your beliefs about the parameters before you see any data. With a hierarchical model, this involves group-level and population-level priors.

2. Define your likelihood.

Choose a statistical model that accurately represents your beliefs about the observation given your parameters. For example, you might assume your data follows a normal distribution.

3. Calculate your posterior based on new data.

Apply Bayes’ theorem to combine priors with observed data. From this, you generate the posterior distribution for each parameter.

Challenges of Bayesian hierarchical models

Like any statistical tool, Bayesian hierarchical models come with their own set of challenges. A few to be aware of include:

• Computational intensity: In some cases, Bayesian hierarchical models may require a high level of computational resources due to their complexity.

• Prior specification: Specifying your prior relies on translating subject matter knowledge into mathematical formulas.

• Reliability concerns: Defining priors is subjective, which can lead to differences in opinion on whether posterior distributions are accurately defined.

While these concerns are worth noting, taking advantage of modern computational resources helps to mitigate challenges related to resource intensity, while measures of uncertainty in your models help to quantify how reliable your results are. 

Applications of Bayesian hierarchical models

You can find applications of Bayesian hierarchical models across fields with multisourced, uncertain, or naturally grouped data. Because they allow for a layered structure, professionals often use them to combine data from multiple sources or account for variability across groups. A few fields you might see Bayesian hierarchical models used in include:

Education

Education is a common application for this type of model. For example, researchers can evaluate test scores from learners across multiple classrooms, even if each class starts at a different baseline due to factors like teaching style or learner background. By using a Bayesian hierarchical model, you can model how learner scores vary within each classroom and how classroom averages vary across the school. Smaller classrooms with lower learner volume can “borrow strength” from the overall school distribution using something called partial pooling, helping you to prevent outliers and provide a more reliable overall estimate of classroom performance.

Environmental science

This same logic and type of model extends beyond education to other fields. In environmental health, Bayesian modeling allows researchers to incorporate uncertainty, whether related to uncertainty of measurement or prior relationship with outcomes. For example, ecological researchers can combine several types of environmental data at different levels (e.g., spatial pollutant data, county-level pollutant concentrations, health outcome counts) to estimate the effect of a certain pollutant on a defined health care event, such as heart attacks or strokes. 

Public health

Other common fields for this type of model include public health research. For instance, when responding to the HIV/AIDS pandemic, researchers can estimate the size of certain populations by using Bayesian hierarchical models to combine data sources and produce stable, reliable estimates. Researchers can also use these models in disease screening, as clinicians can combine historical clinical data for reference populations with new patient data to predict disease trajectories and risk.

Tools for Bayesian hierarchical modeling

When using Bayesian hierarchical modeling, choosing the right software and statistical tools can help you set up your model and run complex computations more easily. Popular software and packages to explore when starting include:

• R: Try popular packages like brms to create your models.

• Python: Use tools like PyMC and TensorFlow Probability to define your models.

• Julia: Consider using TuringGLM to build Bayesian models in Julia.

Explore our free resources for statistical modeling

You can discover more exciting statistical modeling methods by subscribing to the Coursera YouTube channel. Or check out the following statistics resources to keep learning more:

• Learn helpful tools: Python Syntax Cheat Sheet 

• Hear from experts: Statistics: Making Sense of Data with Alison Gibbs and Jeffrey Rosenthal 

• Explore course options: Which Data Science Course Should You Take? Find Out in 1 Minute 

Accelerate your career growth with a Coursera Plus subscription. When you enroll in either the monthly or annual option, you’ll get access to over 10,000 courses. 

Build job-ready skills with Coursera Plus

Start 7-day free trial

• 
• 
• 
• 

Start 7-day free trial