Statistical Biology

A) What is a mathematical model? (WATCH THIS INTERVIEW) There is a lot of philosophy that can go into answering the first question, but for me, a mathematical model is a map that allows me to logically connect a set of assumptions to a set of outcomes. If done well, mathematics imposes a kind of logic to the deductive process that simply using words does not. 

B) Why might a mathematical model be something you want to construct? The reason, I think, why you might want to construct a model is to figure out what is cause and what is effect. Models are almost always trying to make an inference. Step 1: You see some phenomena out there in the world, or in a lab. Step 2: Based on prior work, intuition, and discussions with experimental colleagues, you have a guess as to what the underlying cause is. But usually, its not straightforward to assess whether the mechanism you just guessed would indeed give rise to the phenomena you are observing. So what you do is try to describe your potentially causal mechanism in a mathematical manner. Step 3: Calculate the consequences of your assumed causal mechanism, and Step 4: compare to data. If your predictions and experiment do not match then you are wrong, and you are back at Step 2. That is essentially what a model is and why you might want to construct one. Its because you want to know why something happens. Caution: The wrong assumption could have an effect that looks similar to the phenomena. This is why you typically calculate many independent effects of a single causal assumption/starting point and compare all of them to the observed phenomena. Then the probability of you tricking yourself into thinking you have the right thing, when you really don’t, decreases ... one hopes. 

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What constitutes an explanation? (WATCH THIS INTERVIEW​) This raises an interesting point when considering phenomena that are, in some senses, not fundamental. For example, consider the freezing of water, which is an example of a collective phase transition – note, in case you never have before, that nothing changes about the water molecules when liquid water freezes. It really is something about the packing of these identical particles in space that is manifestly distinct when liquid water freezes. Now there is no doubt that indeed the laws of quantum mechanics underpin the mechanism of freezing/melting. So the problem is solved! In fact all the problems of chemistry/biology are already solved from this point of view. There are indeed people that hold this view point. This brings us to an interesting aspect of theorizing/modeling complex emergent phenomena: Thus the question, what constitutes an explanation? To make it clear, ask yourself why are some material magnetic and some materials not? Why are some insulators, while others are conductors? The answer that its all quantum mechanics doesn’t really help you understand why some material or phenomena has the behavior it does. It just tells me that all materials are made of some similar building blocks and physical laws. It would be like never studying architecture, and just saying its all cement/bricks/stone etc. That really isn’t useful.

C) What kind of model might be appropriate for a given situation? The question of what kinds of models have been used, or might be useful, is a much trickier question than what we have addressed thusfar. There is no “correct” model. There can however be more useful models in a given circumstance. Or perhaps even more agnostic is that different models, alternate mathematical descriptions, for the same phenomena might help address different questions. Now when I mean different models I don’t mean different guesses at causal mechanisms. What I mean is fundamentally different manners in which you can describe the phenomena. For example, a deterministic versus probabilistic approach, an many-bodied approach or a mean-field approach, a continuum approach or a particulate approach etc. etc. A helpful analogy, I hope, is to think about your different senses. For many things out there in the world you can see it, hear it, smell it, touch it, and even taste it. Each one gives you some view/lense into the thing, but no individual one can ever give you the whole pictures. It really depends on what you want to learn about the thing. It’s the same with models. Different kinds of models are like different senses, it’s a different modality through which to view a phenomena.

The phase diagram above is my attempt, surely incomplete and error prone, of trying to describe mathematical models in terms of three features: Degrees of freedom (the number of constitutient things that are playing out some dynamics), the interactions between the degrees of freedom (for example electrostatic forces, hard-sphere repulsion, etc.), and the level of noise/stochasticity in the system. This last one is of course tricky since I have lumped a large number of degrees of freedom into a "noise" source, but lets not worry about this. It turns out this is a useful way of categorizing models.