Understanding uncertainty, and even more so explaining it, is hard is because it requires a foundational framework — the reason that most explanations feel isolated is that they avoid this initial step. Today’s Uncertainty Wednesday post is my first cut at presenting the framework, so YMMV. I have found on other difficult topics, such as those covered in my book, it has taken me years of writing and thinking to achieve even moderate clarity. With that caveat in place, here we go.

The framework I will use consists of three entities: reality, explanations and observations. Uncertainty arises from how these three relate to each other.

Reality is what’s “out there.” It is the “actual state of the world and the actual mechanisms by which that state changes over time.” Observations are our measurements of the current state and explanations are our understanding of the mechanisms of change.

Let me make this more concrete using the weather as an example. The "real” weather is the totality of the state of the atmosphere and its change over time. Observations of weather are measurements such as humidity, air pressure, wind speed, rainfall, etc. Explanations are our understanding of weather mechanisms, such as how differences in air pressure result in wind, or how humidity can turn into precipitation, and so on.

Now what does it mean for the weather to be “uncertain”? Well clearly this cannot be a reference to the “real” weather, as the “real” weather, as in the totality of the state of the atmosphere and its changes, just exists. Instead, uncertainty means that we don’t know enough about the “real” weather. Our limited knowledge expresses itself in two forms: we are uncertain about future observations (e.g., will it rain tomorrow?) and we are uncertain about the reason for past observations (e.g., why did it rain yesterday?).

The reasons that we are uncertain are that our measurements are incomplete and our explanations are partial. Together they give us an approximation of the “real” weather, but it is just that — an approximation. The bigger the difference between our approximation to weather and the “real” weather, the more uncertain we are about the weather.

In general: The more limited our observations and the weaker our explanations of reality, the more uncertainty we are facing.

This framework raises many questions that I will attempt to cover in the future, including important ones about the relationship between observations and explanations — how does one inform the other? For today though I want to focus on the question of whether observations and explanations are in fact useful as separate concepts.

Here is a fairly extreme example — I am presenting you with a sequence of observations in the form of digits

8 8 6 0 6 1 3 4 0 8 4 1 4 8 6 3 7 7 6 7 0 0 9 6 1 2 0 7 1 5 1 2 4 9 1 4 0 a b c d

where a b c d are future observations. Further I am telling you that these are precise observations, meaning when there is a 7 in the sequence it is not a 6 or an 8 that accidentally got recorded as a 7. But I am not giving you any explanation at all.

In the absence of an explanation, there is a lot of uncertainty about why we have observed this specific sequence and what the observations a b c d will be when we get those. Each observation could really be any digit!

As it turns out though, I just picked a sequence of digits from the first 100,000 digits of pi. Once you know this explanation, there is no uncertainty left about why we got these observations and which ones will come next. This sequence appears exactly once among the first 100,000 digits and is followed by 4 3 0 2 (i.e. a = 4, b = 3, c = 0, and d = 2).

Now consider the other extreme. Here we are looking at a perfect explanation, meaning I am telling you that the following formula perfectly describes the reality, but we have no observations:

X[n+1] = 3.67 * X[n] * (1- X[n])

This is a super simple deterministic expression. If I give you X[1] you can determine X[2], X[3] and so on, including say X[100] without any uncertainty. But in the absence of observations, X[100] could really be any number! Now let’s say I told you that X[1] has to be between 0 and 1. Then I am reducing the uncertainty somewhat. Any subsequent X[n] now will also be between 0 and 1 (you can convince yourself of that by studying the formula).

The situation is more interesting though. The formula represents a so-called logistic map and I have intentionally chosen the parameter 3.67 to have the map exhibit so-called chaotic behavior. I will explain chaos in more detail at some future point in this series, but for now just see how massively the result for X[100] changes for small variations in X[1]

X[1]         X[100]
0.6900    0.8597542583
0.6990    0.8910200311
0.7000    0.7364501229
0.7001    0.8600081945
0.7100    0.8763037147

So even though we have a precise explanation, a small observational difference on X[1] turns into a large uncertainty about the subsequent value X[100] despite the existence of a perfect explanation. In fact if you look at the numbers you can see that there isn’t even any relationship in the size of the error. Say that 0.7 is the real value of X[1] but if my measurement is 0.699 the error is greater than if I measure 0.69 (which is significantly further from the real value).

I hope these two examples convince you that (a) explanations and observations are two separate concepts and (b) that the more limited our observations and the weaker our explanations of reality the more uncertainty we are facing. Conversely as our knowledge increases in the form of better observations and stronger explanations we can reduce uncertainty.

The weather is a good example of that. Compared to say 100 years ago, today we have much better data on the atmosphere (including for example satellite data) and we have stronger explanations. As a result we have a lot less uncertainty about why we are seeing specific weather observations and we also have much better weather forecasts.

Before I decide what to write about next Wednesday, I would love to hear from everyone reading what questions you have about this post and the concepts in it.