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    Area of Science:

    • Statistics
    • Bayesian Inference
    • Computational Statistics

    Background:

    • Non-Gaussian statistical models often involve complex data structures (e.g., bounded, unit-length vectors).
    • Estimating posterior distributions in these models is analytically intractable with standard methods.
    • Variational Inference (VI) is a common Bayesian estimation framework, with Extended VI (EVI) offering improvements.

    Purpose of the Study:

    • To compare two approximation strategies for Extended Variational Inference (EVI): Multiple Lower Bounds (MLB) and Single Lower Bound (SLB).
    • To analyze the convergence properties and theoretical differences between MLB and SLB approximations under weak and strong conditions.
    • To evaluate the practical performance of SLB versus MLB using EVI-based non-Gaussian models and real-world data.

    Main Methods:

    • Implementation of Extended Variational Inference (EVI) using both Multiple Lower Bounds (MLB) and Single Lower Bound (SLB) approximations.
    • Discussion and theoretical analysis of weak and strong conditions influencing EVI convergence.
    • Empirical comparison of MLB and SLB through extensive experiments on non-Gaussian statistical models and real datasets.

    Main Results:

    • Convergence of EVI is dependent on the lower bound selection, irrespective of the weak or strong condition used.
    • Theoretical analysis reveals distinct convergence properties between MLB and SLB approximations.
    • Experimental results indicate that the SLB approximation offers advantages over the MLB approximation for EVI-based non-Gaussian models.

    Conclusions:

    • The Single Lower Bound (SLB) approximation is more advantageous than the Multiple Lower Bounds (MLB) approximation in practical Extended Variational Inference (EVI) applications.
    • Understanding the convergence properties of different lower bound approximations is crucial for effective Bayesian estimation in non-Gaussian models.
    • EVI, particularly with the SLB strategy, provides a viable approach for analytically tractable solutions in complex statistical modeling.