Last Uncertainty Wednesday, I set up the first example which I called Zoltar as a machine that
only prints out two different fortunes: it either prints “You will have a Good day tomorrow” or “You will have a Bad day tomorrow.” For brevity we will refer to these as G and B respectively.
And I asked how to use the Uncertainty Wednesday framework to think about uncertainty here. This requires us to do two things. We have to think about what our observations are (and how those have clear limits) and then do the same for our explanations.
So let’s get started with observations. We could just take a sequence of B and G as our observations. In doing so let’s understand that we are ignoring some things that might be quite important, such as the sounds the machine makes, or how long the machine takes between printing fortunes. These may or may not include information that could be used to better understand the machine and thus reduce uncertainty. What do I mean? For instance, it could turn out to be the case that the machine takes 1 minute to print a B fortune but 2 minutes to print a G fortune. If that were consistently the case, well, then as soon as more than 1 minute has passed after pressing the “start” button you would know with certainty that the next fortune would be G. Readers familiar with cryptography may recognize this as an example of a Timing Attack.
Now let’s for a moment assume that sound and time (and other potential observations) do not contain information so we can focus just on the series of B and G. What are the limitations on that sequence? The most important one is how many of these we have already collected and can collect going forward. For instance, if we happen upon the machine “at a fair or carnival” (as described in the post introducing the example), we have no idea how long the machine has already been in operating. That is we do not know how many fortunes and what kind have been printed before our first observation. And going forward, observing each new fortune takes time and, let’s assume the machine is coin-operated, also money!
So what about explanations of the Zoltar machine? Well the first thing to recognize is that implicit in my description was in fact an explanation. I stated that the machine only prints two types of fortunes. But how do we know this? Well, we really don’t. I just said it. Even if there were a sign printed on the side of the machine saying that, clearly that sign could be wrong (by mistake or intentionally).
But we have to start somewhere and so we can take B and G as the only two possible fortunes (at least until we observe the machine printing something different, say a “You will have a meh day” fortune). Now when I defined explanation I said it had to be relational. What kind of relation can we state here?
If N is the total number of fortunes and B are Bad fortunes and G good ones, then the following relation *must* hold
B + G = N
Now you might object to this being called an “explanation.” After all, it doesn’t explain anything about *why* the machine selects or prints those fortunes. And while it is true that a “deeper” explanation would be better (in the sense that it would reduce uncertainty), there are many powerful “explanations” that don’t get to that. In an earlier post I mentioned Kepler’s formula for elliptical orbits of planets. It “explains” the orbits but doesn’t answer the *why* planets have that orbit.
So what justifies calling this an explanation? Well, the fact that it is possible to have observations that are consistent or inconsistent with the explanation. If we were to see a “Meh” fortune printed, for example, we would know that our explanation is incorrect. Conversely, the longer the machine runs and the more we see only B and G observations the more confident we will feel that our explanation is correct. Also, as we will see in upcoming posts, even the seemingly trivial equation B + G = N that so far captures our explanation, has power in letting us reason about observations.
