An algorithm that reliably averages mismatched probability distributions from any starting point.

Comparing data shapes that have unequal amounts of mass is difficult. Standard math tools often assume both sides have equal totals. The researchers studied this issue by looking for a balanced middle shape, called a barycenter, using Gaussian shapes, which look like bell curves. To solve this, they profiled the mass directly, turning the calculation into a smooth formula with internal weights. They built a step-by-step solving method that successfully reaches a stable answer starting from any normal bell curve. They proved an optimal shape always exists. When the mismatch penalty is large, there is one unique solution that matches classic shapes. When the penalty is small, the solutions approach the peak of a specific overlap measure.

Why it matters

This method enables accurate comparisons between datasets that have different total amounts or scales, which is common in data analysis. It ensures that no dataset unfairly dominates the average, leading to more reliable conclusions when data is incomplete.

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