Economic interpretation of probabilities estimated by maximum likelihood or score
The conventional method of estimating a probability prediction model by maximum likelihood (MLE) is a form of maximum score estimation with economic meaning. Of all the probabilities that a given model might have produced, those obtained by MLE yield maximum in-sample betting return to a log utility investor. Recognition of this affinity between MLE and log utility begs the wider methodological question of whether different decision makers benefit in different degrees from different probabilities. Probabilities produced by MLE can be either too conservative or too bold relative to those found by maximizing utility under more risk-tolerant or risk-averse score functions. A very (not very) risk-averse user, who bets characteristically small (large) fractions of wealth based on a conservative forecast, is bound to make a rapidly (slowly) increasing bet as the forecast probability becomes progressively bolder or more distant from the market probability. The effect of this interaction between risk aversion and forecast is that a highly risk-averse user may need a much bolder forecast to obtain the same certainty equivalent as a more risk-tolerant investor. It follows more broadly that professional forecasters should anticipate how a client with given risk aversion expects to gain from any given forecast, or forecast revision, before committing resources toward making a better informed (but still honest) forecast.
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