The world is complicated, nearly unmeasurably so. Even the workings of the smallest bits of matter in the universe are incredibly complex, and so to make progress in understanding the world, we have to neglect some of this vast complexity. We must impose simplifications on our mental models of the world in order to make progress in comprehending the world.

Following the definition of William Wimsatt, these simplifications can be called heuristics. By his reckoning, not only are heuristics necessary, but the choice of particular heuristics guides the path of our gaining of knowledge.

Heuristics are necessary because human beings are finite. In the old Enlightenment-era schemes, the world could be perceived as Laplacian in nature, as a finite series of determinate computations. Now that we are aware of the true scope of the universe, the notion of a determinate universe, while perhaps theoretically interesting, lacks utility. Even if we were to possess perfect information, the scale of the universe would not yield to any accurate calculation.

These arguments apply just as well when scaled down to the level of realistic scientific inference. There are too many permutations of gene regulatory networks to model them as the interactions of molecules. Combinatorial explosions abound in dealing with genomes of even a few thousand genes. We must idealize them.

So we require simplifications, and a change in the framework of our thought. Absent a single, unifying conception of the universe, we are allowed a choice of heuristics. For example, one choice of heuristics for gene regulatory networks is to consider the network in the context of graph theory (like Davidson).

You end up with something that looks like this.

In addition to being (sometimes) visually appealing, this heuristic frames the problem in a space of mathematics which seems to apply well to gene regulation. That space is network theory. By considering regulatory interactions (Gene X activates Gene Y) as edges and genes as nodes, we can perhaps gain some useful knowledge by applying the well-understood axioms and practices of network theory to this new, empirical problem of gene regulation. This perspective is intriguing, and also simplifying: under the hood, regulatory interactions are not simple mathematical relationships, they are incredibly complex processes performed by elaborate molecular machinery. Perhaps, however, these incredibly complex machines perform their operations in ways that are similar enough to those simple mathematical relationships that we can, for a moment, neglect the considerable intricacy of those molecular machines.

Supposing that the network theory heuristic is an interesting and useful one, I now want to broaden the scope of this post to a larger, more significant question. If some heuristics are useful, and others aren’t, can we come up with a general theory for which heuristics apply in which situations?

I had this thought while re-reading William Wimsatt’s excellent tome on heuristics. He advocates for them, and for an escape from physics-style thinking (in terms of Lamarckian demons and absolute rule-sets). I agree on both scores, but the trouble with heuristics is that they are frighteningly* arbitrary.

For instance, I could choose to use one set of heuristics to simplify gene regulation, and you could choose a different set. Our two sets would differ in terms of their simplifying assumptions, and, if we advanced the study of each of them sufficiently far, they might produce different predictions about the behavior of some particular gene regulatory systems. How would we know which one was right?

This scenario is something of a false problem, because we could always do experiments to test the predictions of each, and then disregard (or assimilate) the one whose predictions turned out to be correct less often. Even so, as long as there were two seemingly correct systems of thinking about gene regulatory networks, there would be tension between them. And one could envision scenarios in which people proposed wildly inappropriate heuristics to tackle gene regulatory problem-agendas, producing false results but being unprovably wrong (for at least a short time).

In the long run, to avoid these kinds of problems, it would be desirable to have a guidebook of sorts which prescribed the kind of heuristics which are most likely to be useful in each situation. Wimsatt doesn’t provide that guidebook, although he seems to hint at its utility (and perhaps, its future existence).

The Book of Heuristics

In thinking about heuristics, I always tend to come back to the concept of statistical models which, under my reading of Wimsatt, are perfect examples of heuristics. A statistical model describes the way that two or more variables relate to each other. To work properly, it has to assume some structure between the two or more variables. When we then apply the model to actual data, we fit the data into the structure, and in so doing, perhaps learn something about the problem at hand.

For example, linear models assume that two variables are linear functions of each other, that is:

y = ax + b

That equation should be comfortable to the reader, as it’s a fairly straightforward relationship to have. Verbally, it means that for every one unit increase in x, y increases or decreases by some steady amount, a. When we fit a linear model to data, we take a series of y‘s and x‘s and, using some algorithms, impute the values of a and b. By applying this heuristic, we are able to learn something, we think, about the underlying link between x and y (e.g. each unit of x is worth a units of y).

Linear models work great in all sorts of places, but they also fail when applied to some datasets. If one were to attempt to model the amount of sunlight as a function of the hour of the day (with a 10-year dataset of each) using a linear model, one would get something like a flat line. Moreover, the fit of the data would be terrible. We know that there isn’t a linear relationship between these two variables, and so to assume that there is defies common sense and good statistical practice.

The answer to the question I posed earlier (could we build a guidebook for heuristics?) seems to hinge on whether we could identify, in advance, whether a certain heuristic would be likely to fail when applied to a given problem.

To a certain degree maybe that objective is possible, in that we can let intuition and the “eye test” guide us. If, for example, we were to plot hour of the day vs. amount of sunlight, it would be very plain that the dataset is not amenable to a linear fit, although there is clearly a pattern present. But this ‘eye test’ seems really just like mental model fitting.

I wonder if we couldn’t do better still, and without gathering the data beforehand. As a parting quandary, I pose the following question. Is it possible that, by leveraging our intuition about the architecture of a system, we could rule out certain heuristics as being likely to apply poorly to that system?

I think the answer is yes, and I think this kind of heuristic choice is exercised successfully all the time (albeit without a firm theoretical basis for its use). For example, imagine some highly complex system, in which different parts of the system are strongly interconnected. The system functions in such a fashion that changes to any single part ripple outwards in unpredictable ways, sometimes being buffered by the other parts of the system, sometimes causing catastrophic deviations from the system’s normal functioning.

Intuitively, when I envisage such a system, I think of it as being a poor fit for any simple heuristic like a linear model, at least for most questions. Because of the strong interdependence between parts of the system, and the overall intricacy of the structure as a whole, I imagine most perturbations to such a system are unlikely to result in linear effects on any appreciable scale. Again, as a matter of intuition, I would prescribe the use of (for example) more sophisticated statistical models, those with fewer constraints and more flexibility, in order to better agree with the inherent characteristics of the system.

If my intuition is correct (perhaps it’s not!), then it suggests that there could yet be a guidebook of heuristics, a way to tell in advance which heuristic approaches are likely to be most fruitful. In this way, we could build a set of heuristics of heuristics, i.e. meta-heuristics, which shaped not only the particulars of the simplifications we applied to models of complex inquiry, but the kinds of simplifications.

*Frightening only to some. I think we shouldn’t be frightened of their inconsistency with each other; as the quote goes, “Consistency is the hobgoblin of small minds.” Instead, I think it’s one of the universes most forgiving and helpful properties that multiple schemes of thinking about a problem can converge on the same correct answer. We should embrace and enjoy the chaotic, diverse world of heuristics, and set about determining which ones are the best and why.

Information theory is a discipline of computer science which purports to describe how ‘information’ can be transferred, stored, and communicated. This science has grand ambition: merely to define information rigorously is difficult enough. Information theory takes several steps further than that.

The most important single metric in information theory, and arguably the basis for the whole discipline, is entropy. Nobody agrees precisely on what entropy means; whereas entropy has been called a quantification of information, Norbert Wiener famously thought that information was better defined as the inverse of entropy (‘negentropy’).

Whatever entropy is, it’s important. It keeps appearing in science–in statistics, as a fundamental limit on what is learnable–in physics, as a potentially conserved quantity like matter/energy–in chemistry, where it was first discovered, as a law of thermodynamics. Entropy is everywhere, so let’s see what it is.

Some History

One of the most famous and important scientists you may never have heard of is a man named Claude Shannon. A mathematician by training, Shannon is best thought of as the inventor, and simultaneously the primary explorer within, information theory. He holds a status within information theory akin to what Charles Darwin is to evolutionary biology: a mythical figure who at once created an entire discipline and then figured out most of that discipline, before anyone else had a chance to so much as finish reading.

In 1949, Shannon wrote a paper called “A Mathematical Theory of Communication“. I don’t think there are many more ambitiously titled papers, although, to further my analogy, On the Origin of Species is certainly a contender. In this paper, now a book, Shannon begins by defining entropy. Of note, he writes:

The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point… The signiﬁcant aspect is that the actual message is one selected from a set of possible messages.

Thus the stage is immediately set for entropy. It relates to the idea of sets of signals, one of which may be chosen at a time. A priori, it seems some sets may be more ‘informational’ than others. Books communicate more information than traffic lights. To this point, he says,

If the number of messages in the set is ﬁnite then this number [the number of possible messages] or any monotonic function of this number can be regarded as a measure of the information produced when one message is chosen from the set, all choices being equally likely.

So clearly, part of the reason books are more informational is because there’s simply a larger ‘bookspace’, that is, number of possible books, than there is a space of possible traffic lights.

That only works, however, if all of the choices are equally likely. To get a more general formulation of entropy–which to Shannon is information–Shannon creates a list of requirements. They are as follows.

1. Entropy needs to be continuous, so long as the probability of some message is defined from 0 to 1. In other words, picking a message from a set of messages with any probability p ought to produce a continuous metric.

2. As above, if all the possible messages are equally likely, then entropy should increase monotically with the number of possible messages. We already covered this–books are more entropic, and thus informational, than traffic lights.

3. Entropies ought to be additive in a weighted fashion. So if a single choice from three messages is broken down into two sequential choices of the same three choices, the total entropy remains the same. This one is the hardest to grasp, but relates essentially to the idea that information can be translated into different languages. I can translate a signal in an 8-letter language into the same signal in a 2-letter language without the entropy changing, simply by recoding my 2-letter language.

Three (relatively) simple rules, and yet, as it turns out, Shannon proves that there is one and exactly one function that obeys all three rules. That function is entropy, the very same function Gibbs found earlier in reference to chemistry, now recruited in service of information…

There’s something fascinating about not just the formula itself, but the way Shannon derives it. He sets out a series of requirements, detailed above, and realizes that there’s only a single mathematical relationship which obeys all three requirements. All of the requirements are straightforward, common-sense attributes which any characterization of information must obey. Strike out any of the three, and one is no longer discussing information. It’s an elegant way of making an argument.

Some Theory

Shannon calls entropy H before he even invents the formula for it. A couple of pieces of background information are useful in understanding the formula. One is that elements of the set–remember, communication is just choosing from a set of signals–are numbered or indexed with i, going from 1, the first element, to n, the last. Any formulation of entropy must take into account each such element, so we sum (Σ) over all the elements.

p_{i} refers to the probability of the ith element. In practice, we rarely know this probability, but it can be estimated (as the frequency of the ith message divided by the total number of observed messages). If an element occurs 0 times, its probability is estimated as 0, causing it to drop out of the calculation.

For a given element, we multiply p_{i} by the logarithm of p_{i}. Interestingly, the base of the logarithm is arbitrary, meaning that the actual number that entropy assumes is also arbitrary. Traditionally, the base is two, which outputs entropy in terms of bits, which are universally familiar now that computers are ubiquitous. But entropy could just as easily be computed in base 10 or 100 or 42, and it would make no difference. No other number in science so significant is also so fluid.

Which highlights an important point: entropy means relatively little in and of itself. Entropy is most useful when comparing two or more ‘communications’. When applied to a single source, it can be difficult to grasp what a given entropy means. The clearest formulation is in terms of bits, in which a message can be understood as a series of yes/no questions. For example, a message or source with an entropy of 1 bit can be described by a single binary question. But even this is really describing one source of information as another (that is, language).

But back to calculation–one must repeat the p_{i} log p_{i} calculation for each of the i possible elements, and sum the results. One now has a negative number, because log(x) when 0 < x < 1 is negative. To reverse this sorry situation (after all, how could information be negative?*), we multiply by -1. Then we have a positive quantity called entropy that measures the uncertainty of the n outcomes.

Entropy is Magic

When entropy is high, we don’t know what the next in a series of messages will be. When it is low, we have more certainty. When it is 0, we have perfect knowledge of the identity of the next message. When it is maximal–when each of the p_{i}‘s is equiprobable, and equal to 1/n–we have no knowledge as to the next in a series of messages. Were we to guess, we’d perform no better than random chance.

We can see then that entropy is a (relatively) simple, straightforward mathematical expression which measures something like information. It’s not clear, as I mentioned above, what the exact relationship between entropy and information is. There’s something fascinating about the fact that entropy (as a formula) appears vitally important, but is simultaneously difficult to translate back into language.

In any case, entropy is important not only on its own but because of the paths it opens up. With an understanding of entropy, one can approach a whole set of questions related to information which would have appeared unanswerable before. For example, conditional entropy builds on entropy by introducing the concept of dependency between variables–the idea that knowing X could inform one’s knowledge of Y. From there, one can develop ever-more complex measures to capture the relationships between different sources of information.

*That’s a rhetorical question, of course. It could be negative. So much of this (most important) metric is arbitrary, up to the discretion of the user, which reinforces the fact that entropy is only meaningful in relative terms.