After watching a season of Dirk Gently’s Holistic Detective Agency on Netflix with the kids, I re-read Douglas Adams’ original books. I find them exceptionally funny, but also full of wisdom, especially in the realm of decision theory. In a short series of posts, I want to go through some fine examples of these.

On page 115 of my edition of “The long dark tea-time of the soul” we overhear a psychologist talking to a client over the phone. We only hear the psychologist’s side of the conversation.

“Yes, it is true that sometimes unusually intelligent and sensitive children can appear to be stupid. But, Mrs. Benson, stupid children can sometimes appear to be stupid as well. I think that’s something you might have to consider.”

A bit harsh, of course, but probably true. I especially like the repetition of “sometimes”. We understand what the psychologist is trying to say, of course. But, to make it probably unnecessarily clear, let me sketch a simple decision-theoretic model of the psychologist’s thinking. There is a true state of the world that the child is in: it is either unusually intelligent (UI) or stupid (S). Presumably, the child can also be something in between, but let me ignore this in my simple model. Observing the child for a bit provides us with some information, which can be described by what Blackwell would have called an information structure, or an experiment, or perhaps a signal-generating system. Ultimately, we obtain a signal. The signal is either that the child appears stupid (AS) or that it does not appear stupid (NAS). The probability that each signal is generated depends on the state. These probabilities are known to the expert psychologist. There is the probability that an unusually intelligent child appears stupid, which, the psychologist admits, is positive (the first “sometimes” in the quote above). [In probability theory is often referred to as the probability of appearing stupid (AS) conditional on the child being unusually intelligent.] And there is the probability that a stupid child appears stupid, which, the psychologist claims, is also positive (the second “sometimes” in the above quote). Apparently, it may be less than one.  

When the psychologist says that ”that’s something you have to consider” he means that you should compute the probability of the child being stupid, conditional on the child appearing stupid, and that this depends also heavily on the probability that a stupid child appears stupid, Formally, using Bayes’ law, we get that

Presumably, and the first word in “unusually intelligent” seems to suggest this, the ex-ante (or a priori as Bayesians like to say) probability of a child being unusually intelligent, is low. This means that is probably almost negligible in the calculation relative to which would really suggest an ex-post (or a posteriori as Bayesians like to say) probability of a child being stupid when it appears stupid to be definitely positive, if not even relatively close to one.

The psychologist ends the phone conversation with “I know it’s very painful, yes. Good day, Mrs. Benson.”