factorial.qmd

---
title: "Bayesian Factorial Design"
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  description: "Business Data Science: What Does it Mean to Be Data-Driven?"
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In many scenarios, your goal is to use data to design the optimal treatment.
This is where factorial designs come into play. A factorial design is an
experimental setup where multiple factors are manipulated or varied
simultaneously. Each factor can have multiple levels, and the experiment
involves testing all possible combinations of these factor levels. This allows
us to not only assess the main effects of each factor but also to investigate
the interactions between factors.

Bayesian analysis offers a powerful approach to factorial designs, especially
when dealing with complex experiments with many factors and levels. By
incorporating prior knowledge and using hierarchical models, Bayesian methods
can improve the precision of estimates and control the risk of false positives
from multiple comparisons.

In a Bayesian factorial design, we start with prior distributions for the
effects of each factor level and the interactions between them. These priors can
be based on previous research, expert opinion, or simply reflect our uncertainty
about the effects. As we collect data from the experiment, we update these
priors using Bayes' theorem, resulting in posterior distributions that reflect
our updated beliefs about the effects.

One of the key advantages of Bayesian factorial designs is the ability to
"borrow strength" across different factor levels and interactions. This means
that if the data for one factor level are limited, the model can use information
from other factor levels to improve the estimate for that level. This is
particularly useful in complex experiments where some factor combinations might
have smaller sample sizes.

::: {.callout-tip}
## Learn more
@kassler2018beyond Beyond “Treatment versus Control”: How Bayesian Analysis Makes Factorial Experiments Feasible in Education Research.
::: 

### Example with code

TODO