Bayesian Modeling
Overview
Bayesian modeling in MixModeler provides a probabilistic approach to Marketing Mix Modeling that quantifies uncertainty in coefficient estimates. Unlike traditional OLS regression that gives point estimates, Bayesian models produce full probability distributions for each parameter, allowing you to make statements about the likelihood of different coefficient values.
This approach is particularly valuable when dealing with limited data, multicollinearity issues, or when you want to incorporate domain knowledge through informative priors.
Key Concepts
What is Bayesian Inference?
Bayesian inference updates prior beliefs about model parameters with observed data to produce posterior distributions. The core principle follows Bayes' theorem:
Posterior ∝ Likelihood × Prior
In practical terms:
Prior Distribution: Your initial beliefs about parameter values before seeing the data
Likelihood: How well different parameter values explain your observed data
Posterior Distribution: Updated beliefs after combining prior knowledge with data
Why Use Bayesian Modeling?
Uncertainty Quantification: Get full probability distributions for coefficients, not just point estimates. This allows you to answer questions like "What's the probability this channel has a positive effect?"
Incorporating Domain Knowledge: Use informative priors to encode business insights, such as "we expect this channel to have a positive effect between 0 and 5."
Better Handling of Small Datasets: Bayesian methods can produce more stable estimates when data is limited by leveraging prior information.
Multicollinearity Robustness: Priors can help stabilize estimates when independent variables are highly correlated.
Credible Intervals: Unlike confidence intervals, credible intervals have a direct probability interpretation.
Bayesian vs OLS
When to Choose Each Approach
Dataset Size
Works well with large datasets (100+ observations)
Excellent for small to medium datasets (26+ observations)
Speed
Very fast (seconds)
Slower (2-10 minutes depending on settings)
Uncertainty
Provides standard errors and confidence intervals
Provides full posterior distributions and credible intervals
Domain Knowledge
Cannot incorporate prior beliefs
Can encode business knowledge through priors
Multicollinearity
Can be unstable with high correlation
More stable with regularizing priors
Interpretability
Single coefficient estimates
Probability distributions for coefficients
Output Differences
OLS Output:
Coefficient: 2.5
Standard Error: 0.8
95% CI: [0.9, 4.1]
P-value: 0.003
Bayesian Output:
Posterior Mean: 2.4
Posterior Std: 0.7
95% HDI: [1.1, 3.8]
Probability > 0: 99.8%
Probability > 2: 65.3%
The Bayesian output allows you to make direct probability statements about coefficients, which is often more intuitive for business decision-making.
Model Selection Workflow
Step 1: Choose Bayesian in Model Builder
When creating or editing a model in the Model Builder:
Select your KPI and features as usual
In the "Modeling Method" dropdown, choose Bayesian
The interface will expand to show additional Bayesian-specific options
Step 2: Initial Model with Default Priors
For your first Bayesian model:
Use the default "weakly informative" priors
These priors are intentionally vague and let the data drive the results
Run the model to see baseline posterior distributions
Step 3: Evaluate and Refine
After reviewing initial results:
Check convergence diagnostics (R-hat, ESS)
Examine posterior distributions for reasonableness
Identify any parameters that may benefit from informative priors
Iterate by adjusting priors and MCMC settings
Model Execution
MCMC Inference
By default, Bayesian models use Markov Chain Monte Carlo (MCMC) sampling to approximate posterior distributions:
Chains: Multiple independent sampling chains (default: 4) Draws: Number of samples per chain (default: 2,000) Tuning: Warm-up samples discarded for adaptation (default: 1,000) Total Samples: Chains × Draws = 8,000 posterior samples
Fast Inference Mode
For large models or quick iterations, enable Fast Inference mode:
Uses Variational Inference (SVI) instead of MCMC
10-20x faster than standard MCMC
Approximate posterior distributions
Best for initial exploration and model comparison
Switch to full MCMC for final production models
Toggle Fast Inference in the Bayesian settings panel before running the model.
Best Practices
Start with Defaults: Begin with weakly informative priors and standard MCMC settings. This provides a baseline that's primarily data-driven.
Check Convergence First: Always verify convergence diagnostics before interpreting results. Poor convergence means unreliable estimates.
Use Informative Priors Judiciously: Only apply strong priors when you have solid business reasons. Document the rationale for any informative priors.
Compare with OLS: Run both OLS and Bayesian versions of your model. Large discrepancies warrant investigation.
Iterate Thoughtfully: If convergence is poor, first try increasing draws/tuning. Only adjust priors if you have domain justification.
Communicate Uncertainty: When presenting results, emphasize the full posterior distribution, not just point estimates. This is the key advantage of Bayesian methods.
Common Use Cases
Limited Historical Data: When you have only 6-12 months of data, Bayesian priors can stabilize coefficient estimates by incorporating reasonable bounds.
New Channel Testing: Use informative priors based on similar channels to get more reliable estimates from limited new channel data.
Seasonal Business: Apply priors that encode seasonal patterns from previous years or industry benchmarks.
Budget Planning: Credible intervals provide natural ranges for scenario planning and budget allocation decisions.
Regulatory Requirements: Full uncertainty quantification can satisfy compliance requirements for marketing attribution.
Technical Notes
Computational Requirements: Bayesian MCMC requires more memory and processing time than OLS. Expect 2-10 minutes for typical models on modern hardware.
Convergence: Not all models will converge perfectly on the first run. This is normal. Follow convergence diagnostic guidance to improve results.
Prior Selection: While defaults work well in most cases, domain expertise significantly enhances Bayesian modeling effectiveness.
Reproducibility: Models use random seeds for reproducibility. The same data, priors, and settings will produce consistent results across runs.
Next Steps: Learn about Prior Configuration to customize the prior distributions for your model parameters, or explore MCMC Settings to optimize sampling performance.
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