Bayesian Full Factorial Design — Bayesian Full Factorial Design of Experiments
Bayesian Full Factorial Design of Experiments · Also known as: Bayesian FFD, Bayesian complete factorial experiment, Bayesian full factorial experiment, Bayesian all-combinations design
Bayesian full factorial design combines the complete combinatorial structure of classical full factorial experiments — running every combination of factor levels — with a Bayesian inferential framework that incorporates prior knowledge about factor effects and yields full posterior distributions over main effects, interactions, and model parameters, rather than point estimates and p-values.
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When to use it
Use Bayesian full factorial design when you need to estimate all main effects and interactions without aliasing AND you have meaningful prior information about effect magnitudes that should be incorporated formally. It is especially valuable when the number of runs is small relative to the number of parameters — the prior regularizes estimation — or when decision-making requires probability statements rather than binary reject/fail-to-reject conclusions. Do NOT use it when the number of factors is large (four or more factors at two levels already yields 16 runs; five factors 32 runs), as the full factorial layout grows exponentially and a Bayesian fractional or screening design becomes more efficient. Also avoid when the team has no credible basis for prior specification and the posterior will be entirely data-driven — in that case a classical full factorial with ANOVA may be simpler and equally valid.
Strengths & limitations
- Estimates every main effect and every interaction without aliasing, providing a complete picture of the factor space.
- Prior information from past experiments or physical models is formally incorporated, improving estimation efficiency especially with small run counts.
- Outputs are probability distributions over effects, enabling direct probability statements such as 'there is an 87% chance this factor changes the mean yield by more than 5 units.'
- Credible intervals have a direct probabilistic interpretation that confidence intervals do not, making communication to decision-makers more intuitive.
- Sequential updating is natural: posterior from one experiment becomes the prior for the next, allowing adaptive experimental programs.
- Run count grows as L^k — even modest factor numbers lead to large experiments; not practical for more than four or five factors.
- Requires specification of prior distributions, which demands subject-matter expertise and introduces analyst subjectivity.
- MCMC-based computation is more complex than ANOVA; requires statistical software (Stan, JAGS, R-INLA, or equivalent) and diagnostic checks for chain convergence.
- Results can be sensitive to prior choice, particularly when sample sizes are small; sensitivity analysis adds additional analytical burden.
Frequently asked
How is Bayesian full factorial design different from classical full factorial design?
The experimental layout — running every factor-level combination — is identical. The difference is inferential: classical analysis uses ANOVA and F-tests to produce p-values and confidence intervals; Bayesian analysis combines a likelihood with prior distributions to produce posterior distributions over effects. The Bayesian approach supports direct probability statements and formal incorporation of prior knowledge.
What prior distributions should I use for factorial effects?
For factorial effects (contrast coefficients), weakly informative normal or half-normal priors are common defaults. When prior information from previous experiments is available, elicit the prior mean and variance from that data. For variance components, half-Cauchy or inverse-gamma priors are frequently used. Always perform prior predictive checks to ensure the prior implies plausible response values before observing data.
Can I use standard software like R or Python for Bayesian full factorial analysis?
Yes. In R, the brms package (wrapping Stan) fits Bayesian linear models, including factorial designs, with minimal code. Python users can use PyMC. Both provide MCMC sampling, posterior summaries, and diagnostics. For conjugate normal models the rstanarm package offers faster computation. Contrast coding for main effects and interactions follows the same principles as in classical ANOVA.
When should I prefer a Bayesian fractional factorial design over a Bayesian full factorial design?
When the number of factors is large enough that a full factorial would require prohibitively many runs — typically more than four factors at two levels — a Bayesian fractional factorial design is preferable. The Bayesian framework can partially compensate for the aliasing in a fractional design by placing priors that down-weight higher-order interactions, making fractional designs more defensible than in the classical setting.
How do I report Bayesian full factorial results in a publication?
Report the prior specifications and justification, the MCMC sampler used and convergence diagnostics (R-hat, effective sample size), posterior means and credible intervals for each main effect and interaction, any model comparison metric used (Bayes factor, LOO-CV, WAIC), and a sensitivity analysis showing how conclusions change under alternative reasonable priors.
Sources
- Chaloner, K., & Verdinelli, I. (1995). Bayesian experimental design: A review. Statistical Science, 10(3), 273–304. DOI: 10.1214/ss/1177009939 ↗
- Box, G. E. P., Hunter, J. S., & Hunter, W. G. (2005). Statistics for Experimenters: Design, Innovation, and Discovery (2nd ed.). Wiley-Interscience. ISBN: 978-0471718130
How to cite this page
ScholarGate. (2026, June 3). Bayesian Full Factorial Design of Experiments. ScholarGate. https://scholargate.app/en/experimental-design/bayesian-full-factorial-design
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Bayesian Design of ExperimentsExperimental design↔ compare
- Bayesian Fractional Factorial DesignExperimental design↔ compare
- Central Composite DesignExperimental design↔ compare