This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: Why do some groups of organisms contain thousands of species while others have only a handful? Evolutionary biologists have spent decades trying to answer this question using mathematical models that estimate how biological traits and environmental factors influence the formation and extinction of species.
These models have become a cornerstone of modern biology and have been used in more than 1,000 scientific studies. Yet the models carry a known weakness: They can sometimes lead scientists to the wrong conclusions. For years, no one fully understood why.
Several years ago, researchers found that many evolutionary models can generate exactly the same observations even when they rest on entirely different assumptions about evolutionary history. This meant that scientists could unknowingly reach different conclusions that were all equally consistent with the same data. Whether the same ambiguity also affected the more sophisticated models used to study how traits shape biodiversity remained unclear because their mathematics was too complex to analyze directly.
Sergei Tarasov at the Finnish Museum of Natural History and Josef Uyeda at Virginia Tech approached the problem from a different angle. Their path to the solution started with a simple but unusual question: Imagine three apples—one red, one light green and one dark green. Should the two green apples be grouped together or treated as different colors?
The researchers ran into the same classification puzzle while studying beetle anatomy. Searching for an answer led them to lumpability, a mathematical concept introduced in the 1960s that defines when different states of a Markov model can be safely grouped together without changing how a system behaves. Building on it, they unexpectedly discovered a new way of representing Markov models, one of the most fundamental classes of stochastic models used across science.
They showed that every discrete-state Markov model can be rewritten as an equivalent hidden-state model, a decomposition they call Hidden Expansion. Although the rewritten model looks larger, it is built from simple, identical mathematical components. This representation exposed previously hidden mathematical symmetries and turned an intractable problem into a solvable one.
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