Statistical Reasoning with Uncertainty

Risk and uncertainty are fundamentally different notions. Risk means there is a probability some event will occur, and it usually implies that the probability is known or at least estimated with some level of confidence. For example, betting on a coin toss, betting on a lottery ticket, or insuring a person’s health.

Uncertainty, on the other hand, is not quantifiable. In economics we generally conceive of uncertainty as knightian uncertainty, and it exists in two forms. The first form is a known event that occurs with an unknown probability. The second form is an unknown event. Interestingly, it might be the case that unknown events occur with relative predictability. This seems to be the case with the economics of innovation. We don’t know what the inventions will be, but we know the rate at which innovation is happening. That is a bit of a rabbit trail.

My point in this article is that the sort of a priori statistical reasoning we have developed can be used to estimate uncertainty in a remarkably straightforward way. Ceteris paribus, the probability of an uncertain event should be taken as equal to the probability of a risky event. Moreover, again assuming ceteris paribus, the expected cost of uncertainty is equal to the expected risk cost. Why should this be so?

An uncertain event will occur in one of three probability ranges. It might occur with less probability than a comparative risk event, equal probability to a comparative risk event, or greater probability than a comparative risk event. Given the a priori statistical reasoning we spoke of earlier, the rational estimates for the probability of the uncertain event are:

  • Case 1, p(U)*.5 = p(R)
  • Case 2, p(U) = p(R)
  • Case 3, p(U)*2 = p(R)

If we are genuinely uncertain then we have no reason to think any probability case is more likely than another, so a rational estimate would average the three cases. The average yields the expectation that p(U) = p(R).

Let’s bring this to an application. You are considering investing in a security. The security has a known past value history from which we can create a forecast of expected future values. These future forecasts include estimated risk, but not uncertainty. We can now include a rational estimate of uncertainty by simply doubling the expected risk premium (in both directions, of course).

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