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Knightian Uncertainty, Chance, and Bayesian Prediction

Article Quant Q&A · Author: Drunk Deriving
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Summary

The document considers whether Knightian uncertainty is equivalent to a Bayesian view of probability. Its answer distinguishes uncertainty about an agent or model from chance in the outcome process, using a fair coin tossed by either a skilled controller or an uninformed person as an illustration. In the example, the coin's chance behavior matters only when the toss is not controlled.

The response characterizes Bayesian posterior prediction as integrating over parameter uncertainty, leaving a predictive distribution of outcomes. It argues that this differs from retaining Knightian uncertainty about unknown parameters, while also questioning whether the distinction can be operationalized in practice. These are conceptual claims rather than a formal treatment: the text provides no mathematical framework, empirical evidence, or decision rule for incorporating Knightian uncertainty into investment models.

Key ideas

  • The discussion separates uncertainty about an agent's behavior from chance in an outcome process.
  • A Bayesian posterior predictive distribution integrates over uncertainty about parameters.
  • The response argues that posterior prediction leaves no parameter uncertainty in the decision process.
  • It questions whether Knightian uncertainty can be operationally separated from chance.

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# Knightian Uncertainty Iff Bayesian Probabilistic View Point


# Knightian Uncertainty Iff Bayesian Probabilistic View Point












If an investor operates under knightian uncertainty, does that investor then have a Bayesian viewpoint on probability implicitly, and vice versa? Has this been answered or do I have a poor understanding of one of them?

## Answer by Dave Harris (score 1)

https://quant.stackexchange.com/a/39363

You have a correct understanding. There is a subtle difference in that Knight effectively distinguishes uncertainty from chance. It could be argued that there is no such thing as chance in the Bayesian posterior density. To understand why, imagine that you were holding a strictly fair coin and you were going to gamble with an unknown stranger who would flip the coin.

Upon doing some research on the stranger you discover it is either Mandrake the Magician who will always be capable of controlling which side comes up, or his brother Chuck the Clueless, his identical twin brother and who know nothing about tossing coins. The chance properties of the coin matter if Chuck tosses the coin, but there are no chance properties if Mandrake does. In the absence of a mistake by Mandrake, the outcome is known. If Mandrake can make mistakes, that still isn't chance, but an incorrect application of policy.

The Bayesian posterior predictive distribution integrates out the uncertainty leaving only chance effects, but unlike Knightian uncertainty, there is no uncertainty remaining at all because the true value of the parameter is no longer a determinant of the decision process.

Now the problem with Knightian uncertainty is that I do not believe you can really operationalize it. I think you are either trapped with uncertainty or chance. I think you are either in the parameter space or the sample space. I don't think you get to be in both at the same time. I am not sure what it would mean to try and split them the way he does, except as a mental exercise. After all, Chuck might also be a magician, but he is purposefully behaving as if chance were the decider.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.