We introduce a theory of sequential causal inference in which learners in a chain estimate a structural model from their upstream teacher and then pass samples from the model to their downstream student. It extends the population dynamics of genetic drift, recasting Kimura's selectively neutral theory as a special case of a generalized drift process using structured populations with memory. We examine the diffusion and fixation properties of several drift processes and propose applications to learning, inference, and evolution. We also demonstrate how the organization of drift process space controls fidelity, facilitates innovations, and leads to information loss in sequential learning with and without memory.
Deep Dive into Structural Drift: The Population Dynamics of Sequential Learning.
We introduce a theory of sequential causal inference in which learners in a chain estimate a structural model from their upstream teacher and then pass samples from the model to their downstream student. It extends the population dynamics of genetic drift, recasting Kimura’s selectively neutral theory as a special case of a generalized drift process using structured populations with memory. We examine the diffusion and fixation properties of several drift processes and propose applications to learning, inference, and evolution. We also demonstrate how the organization of drift process space controls fidelity, facilitates innovations, and leads to information loss in sequential learning with and without memory.
This phrase was heard, it is claimed, over the radio during WWI instead of the transmitted tactical phrase "Send reinforcements we're going to advance" [1]. As illustrative as it is apocryphal, this garbled yet comprehensible transmission sets the tone for our investigations here. Namely, what happens to knowledge when it is communicated sequentially along a chain, from one individual to the next? What fidelity can one expect? How is information lost? How do innovations occur?
To answer these questions we introduce a theory of sequential causal inference in which learners in a communication chain estimate a structural model from their upstream “teacher” and then, using that model, pass along samples to their downstream “student”. This reminds one of the familiar children’s game Telephone. By way of quickly motivating our sequential learning problem, let’s briefly recall how the game works.
To begin, one player invents a phrase and whispers it to another player. This player, believing they have understood the phrase, then repeats it to a third and so on until the last player is reached. The last player announces the phrase, winning the game if it matches the original. Typically it does not, and that’s the fun. Amusement and interest in the game derive directly from how the initial phrase evolves in odd and surprising ways. The further down the chain, the higher the chance that errors will make recovery impossible and the less likely the original phrase will survive.
The game is often used in education to teach the lesson that human communication is fraught with error. The final phrase, though, is not merely accreted error but the product of a series of attempts to parse, make sense, and intelligibly communicate the phrase. The phrase’s evolution is a trade off between comprehensibility and accumulated distortion, as well as the source of the game’s entertainment. We employ a much more tractable setting to make analytical progress on sequential learning, based on computational mechanics [2][3][4], intentionally selecting a simpler language system and learning paradigm than likely operates with children.
Specifically, we develop our theory of sequential learning as an extension of the evolutionary population dynamics of genetic drift, recasting Kimura’s selectively neutral theory [5] as a special case of a generalized drift process of structured populations with memory. This is a substantial departure from the unordered populations used in evolutionary biology. Notably, this requires a new and more general information-theoretic notion of fixation. We examine the diffusion and fixation properties of several drift processes, demonstrating that the space of drift processes is highly organized. This organization controls fidelity, facilitates innovations, and leads to information loss in sequential learning and evolutionary processes with and without memory. We close by describing applications to learning, inference, and evolution, commenting on related efforts.
To get started, we briefly review genetic drift and fixation. This will seem like a distraction, but it is a necessary one since available mathematical results are key.
Then we introduce in detail our structured variants of these concepts-defining the generalized drift process and formulating a generalized definition of fixation appropriate to it. With the background laid out, we begin to examine the complexity of structural drift behavior. We demonstrate that it is a diffusion process within a space that decomposes into a connected network of structured subspaces. Building on this decomposition, we explain how and when processes jump between these subspacesinnovating new structural information or forgetting itthereby controlling the long-time fidelity of the communication chain. We then close by outlining future research and listing several potential applications for structural drift, drawing out consequences for evolutionary processes that learn.
Those familiar with neutral evolution theory are urged to skip to Sec. V, after skimming the next sections to pick up our notation and extensions.
Genetic drift refers to the change over time in genotype frequencies in a population due to random sampling. It is a central and well studied phenomenon in population dynamics, genetics, and evolution. A population of genotypes evolves randomly due to drift, but typically changes are neither manifested as new phenotypes nor detected by selection-they are selectively neutral. Drift plays an important role in the spontaneous emergence of mutational robustness [6,7], modern techniques for calibrating molecular evolutionary clocks [8], and nonadaptive (neutral) evolution [9,10], to mention only a few examples.
Selectively neutral drift is typically modeled as a stochastic process: A random walk that tracks finite populations of individuals in terms of their possessing (or not) a variant of a gene. In the simplest models, the random walk occurs in a space that is
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