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Comparing two Bayes methods based on the free energy functions in Bernoulli mixtures
Keisuke Yamazaki1, Daisuke Kaji
1Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, G5-19, 4259 Nagatsuta, Yokohama, 226-8503, Japan. k-yam@math.dis.titech.ac.jp
Summary
Variational Bayes methods approximate complex hierarchical models. This study clarifies their accuracy in Bernoulli mixture models, finding similar learning types but different transition points compared to exact Bayes methods.
Area of Science:
- Information Science
- Data Engineering
- Statistical Learning
Background:
- Hierarchical models are common in data science but pose challenges for Bayesian inference due to complex posterior distributions.
- Variational Bayes (VB) offers a tractable approximation using variational free energy, though its accuracy and phase transitions remain unclear.
- Previous studies observed phase transitions in hierarchical models, but precise understanding of VB approximation accuracy is lacking.
Purpose of the Study:
- To precisely analyze the Bayes free energy function in Bernoulli mixture models.
- To compare the free energy functions of exact Bayes and Variational Bayes methods.
- To determine the approximation accuracy and elucidate parameter learning behavior in these models.
Main Methods:
- Derivation of the exact form of asymptotic variational Bayes energy in Bernoulli mixture models.
- Analysis of the phase diagram to identify types of parameter learning.
- Comparative analysis of exact Bayes free energy and variational Bayes free energy functions.
Main Results:
- The asymptotic behavior of variational Bayes in Bernoulli mixture models exhibits phase transitions.
- A detailed analysis of the Bayes free energy function for Bernoulli mixtures was performed.
- Comparison revealed that both exact Bayes and variational Bayes methods exhibit the same types of parameter learning.
Conclusions:
- While both exact Bayes and variational Bayes methods share similar parameter learning types in Bernoulli mixtures, their transition points differ.
- This clarifies the approximation accuracy and behavior of parameter learning in hierarchical models.
- The findings contribute to a better understanding of Bayesian inference approximations in complex data structures.
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