Uncertainty about uncertainty

If we are about to launch a brand-new product, uncertainty about future sales is rather different from that in the previous case. Maybe, we know pretty well what may happen, so that the scenarios in Fig. 14.1 are known. However, it is quite hard to assess their probabilities. The following definitions, although not generally accepted, have been proposed:

• Risk is related to uncertainty about the realization of a random variable whose distribution is known.
• Uncertainty, in the strict sense, is related to:
  • The parameters of a probability distribution whose qualitative form is known
  • A probability distribution whose shape itself is unknown

We see that there are increasing levels of uncertainty. We may be pretty sure that the probability distribution of demand is normal, but we are not quite sure about its parameters; this is where we started feeling the need for inferential statistics and parameter estimation. Nonparametric statistics comes into play when we even question the type of probability distribution we should use. But even if we are armed with a formidable array of statistical techniques, we may lack the necessary data to apply them. Probability in this setting tends to be subjective and based on expert opinion.

Whatever the level of uncertainty, if the probabilities π_i are not reliable, a more robust decision making model is needed, as we pointed out in Section 13.5.1; however, there may be something more at play, such as prior opinions and their revision by a learning mechanism. We have taken an orthodox view of statistics, based on the fundamental assumption that parameters are unknown numbers. Then, we try to estimate unknown parameters using estimators; estimators are random variables, and their realized value, the estimate, depends on data collected by random sampling. Given this framework, there are two consequences:

1. There is no probabilistic knowledge associated with parameters, and we never speak about the probability distribution of a parameter.²
2. There is no room for subjective opinions, and collected data are the only sensible information that we should use.

However, when facing new decision making problems, relying on subjective views may be not only appropriate, but also necessary. This is feasible within a Bayesian framework, whereby parameters are regarded as random variables themselves. Within this framework, the distribution that we associate with a parameter depicts our limited state of knowledge, possibly subjective in nature. As we outline in Section 14.7, in the Bayesian approach we start with a prior distribution, which reflects background information and subjective knowledge, or lack thereof; as new information is collected by random sampling, this is reflected by an update of the prior distribution, leading to a posterior distribution by the application of Bayes’ theorem.

Issues surrounding orthodox and Bayesian statistics are quite controversial, but one thing is sure: The probabilities π_i in Fig. 14.1 may not be reliable, and we might move move from decision making under risk to decision-making under uncertainty.