Understanding Joint Distributions in Population Synthesis

Joint vs. Marginal Distributions

Joint distributions capture how variables co-occur in a population, while marginal distributions only show individual variable statistics.

Example with Age, Gender, and Education

Marginal Distributions (Single Variables)

P(Age = 18-25) = 30%
P(Gender = Female) = 55%
P(Education = Degree) = 40%

Joint Distributions (Combinations)

Age	Gender	Education	Probability
18-25	Female	Degree	12%
26-35	Male	School	8%
18-25	Male	Degree	5%

How Methods Handle Joint Distributions

Method	Joint Distribution Preservation	Visualization
Deterministic Reweighting	Only matches single-variable marginals well May distort natural relationships	✅ Age ✅ Gender ✅ Education ❌ Combinations
Iterative Proportional Fitting (IPF)	Better at preserving 2-way relationships 3+ way interactions might still be off	✅ Age×Gender ✅ Gender×Education ⚠️ Age×Gender×Education
Conditional Probabilities	Explicitly models multi-way dependencies Best preserves realistic combinations	✅ Age×Gender×Education ✅ Natural clustering

Real-World Example

A real population might show:

70% of 18-25 year-old females pursue Degree
But only 40% of 26-35 year-old males do

Conditional probability methods will maintain these natural relationships, while simpler methods might artificially flatten them to hit marginal targets.

Key Takeaways

Joint distributions reflect real-world correlations between variables
Simple reweighting can create unrealistic combinations even when marginals match
Choose methods based on your need for relationship preservation vs marginal accuracy