To scratch or not to scratch - how you do statistics any instance
Stefan Böhringer, 2013|
Does statistics seem remote to you? Let us imagine a balmy summer night barbeque. The fire is crackling, the atmosphere relaxed, suddenly someone talks about mosquitoes. You start feeling an itch on your left foot, you hear a buzz, your body is itching all over - luckily no bite yet. What just happened? Statistics in action.
Our skin contains different types of receptors whose signals are transmitted via nerves and integrated in the brain to form perceptions such as an itch. We are also able to locate the origin of an itch which means that receptors with enough distance in between - say some millimeters - transmit independent information each belonging to a particular location (receptive field). Our thought experiment highlights one important fact about this process: it is random. If the process was deterministic there would not be a need for a varying perception threshold: we would feel an itch or not. Instead, by experience, we know that an itch can just come up without any apparent reason. This is an erroneous perception as we have to search for an itch-reason in vain. On the other hand we do not want to get bitten by mosquitoes and make another error by ignoring an imminent bite. Our body automatically ignores signals below a specific threshold as appropriate for the situation and the barbeque story shows how this threshold can dramatically shift from one instance to another - I guess you didn’t feel an itch when you first kissed. Let us now consider being an elephant - with the very type of skin we have right now. This might turn out to be a problem. We would suddenly have a multiple of receptive fields to contend with and with the same thresholds as before we would be as many times more likely (roughly) to feel an itch. Choosing a higher threshold would not solve this problem, however, because mosquitoes would now have better chances to deliver the bite. With the skin surface large enough there would finally be no way to make a meaningful distinction. On stochastic reasons alone we can therefore conclude that elephants probably have thicker skins which then only allow bigger animals than mosquitoes to make an impact. They would make bigger holes into the skin, which would allow to make receptive fields to be bigger thereby allowing us (as elephants) to re-establish the balance.
In situations like this we have to trade off two errors: falling for a false signal (lack of specificity) or ignoring a true signal (lack of sensitivity). At many levels this pattern re-occurs. Our highest level might be our consciousness, where constantly decisions have to be made which sensations to allow into it. Again thresholds adapt with an aim that could maybe be described as relevance. Making such decisions for which we can express the error-probabilities in numeric terms is the art of statistics.
Let us now go the other way and look at processes on ever smaller scales to discover more. Consider the DNA - the genetic information - the blueprint of life. How does DNA come into being? DNA is composed of two long molecules, each being chains of simple molecules. There is a curious relation between the two strands: both are composed of the same elementary molecules (four species of so called bases) and each base opposes precisely one base on the other strand. Moreover, there is a one-by-one relation between types of bases on the opposing strands. This way, a strand can be reconstructed from the other strand which is the mechanism to copy and "transmit" DNA to a new cell. But how does copying work in detail? Given a single bare strand let us imagine we are the copying machine (DNA polymerase) responsible for constructing the opposing strand. Given that we are positioned opposing a base to be copied, we have to recognize this base and attach the corresponding base to the new, nascent strand according to the fixed rule. The only problem is that there is no way of recognizing the base. The problem is analogous to our limitation to tactually perceive ever smaller objects - limited by the size of our receptors. How can we, the polymerase, solve this problem? Essentially by an itching-type solver. Let us assume that the matching rule is implied by the fact that matching bases can come closer to each other than other combinations (which is the case). We can then just sit and wait for bases bouncing around in the cell to come close to the position to be copied and just check for a match by checking how close a given base comes to the opposing strand and - based on a threshold - accept the base by attaching it to the nascent strand. Here, random molecule movements (diffusion) being checked in small intervals of time take the role of the random trigger and the copying machine integrates these triggers by a threshold. DNA copying is remarkably accurate, only one in 10 to 100 million bases is copied in error and it is now clear to us that this error rate depends on the corresponding threshold and can be adapted as needed. DNA copying, a seemingly deterministic process, crucially depends on randomness happening. By similar implications, we would also be unable to perceive an itch without randomness going on in our cells.
We can follow the randomness to ever finer structures. The molecule movements indispensable for the DNA copying can be traced back to movements of atoms within the molecule which depend on movements of their components. Ultimately, it is believed that we arrive at elementary particles without sub-components (the "atoms" that Democrit originally envisioned). In quantum theory, these particles are believed to behave randomly as well. The source of this randomness remains an unexplained mystery of the theory. Anyway, randomenss seems to be truly everywhere.
Preparing a more philosophical point, we look back at this hierarchy of randomness to make one important distinction. Molecules bouncing off of each other in the cell are different from our scratching-decision or the DNA-copying mechanism in one important aspect: they are decisions and decision cost energy to make. If we look at a place of itching we cannot get back the energy invested in moving our head and the DNA polymerase invested energy in joining molecules which remains irretrievable after the fact. In contrast, molecules merrily bouncing off of each other do not expend energy. They have the property that when we reverse time (looking the movie backwards) we cannot distinguish that scenario from what was actually happening, whereas such a reverse look on our scratching behavior or the DNA would look "strange", which is a way of clarifying that energy was expended. Expressing this again in terms of randomness means that a system goes from a state whose probability was low to a state whose probability has increased - a broken cup is unlikely to spontaneously unbreak - that would be a strange indeed.
What about computers? In this big ocean of randomness, computers seem to be a beacon of determinism. Alas, we can confidently conclude that computers, too, depend on randomness. As we saw, decisions take energy and each computational step a computer makes can be considered a decision, namely to accept the result from one step of computation as input for the next step (like we accept a copy of the DNA as template for the next round of copying). In current computers, these decisions are driven by random movements of electrons, differing concentrations of which trigger currents - net flows of electrons - between points. However, any other mechanism of computation would have a similar random component to it, as energy consumption is required.
We have seen that randomness is truly all-pervasive. We also saw that the perspective into which we take randomness is important. Let us conclude by making two observations on a personal level. Let us first reflect on risks in our lives. From what we discussed it seems natural to ignore probabilities of certain individual risks - say a fatal car crash - and rather consider life time risks taking into account the number of riskful events we might expect. You might already be doing this or might even be more sophisticated by accounting for less events to be expected as life progresses. One advantage of becoming older is to be able to become more reckless! Another more subtle consequence is that we should stop worrying about most risks. Like with the scratching decision (we cannot scratch too often) we have limited capacities to influence risks. Let’s stop trying.
A second reflection is more philosophical. The truths that we believe in, say mathematical truths, also depend on randomness. This follows from the fact that computers are the most faithful instrument to derive these truths (maybe you have to grant this point for the moment) and we saw that computers do have this dependence on randomness, too. This makes outcomes computed by computer random themselves (we are never 100% certain that an itch is for real) and ultimately our (personal) truths as well. We might be convinced that the possible error is minuscule and we could conceive a way to prove a certain error rate by starting from physical models and calculating error probabilities for a computer that would derive mathematical truths from axioms. If we use enough computers that would compare their results it would appear that the error probability could be arbitrarily reduced. However, any such probability would fall apart if we allow for an arbitrary amount of universes to be lived or to exist. To come back to our itching example, it could turn out that our patch of skin is really part of a vastly bigger skin for which everything relevant for us would have to be considered utterly insignificant (the real universe is an elephant not a human). This implies that if we believe in the correctness of our deductions we have at the same time to allow for the possibility of their incorrectness because they imply that they might just be a statistical fluctuation. We have just established the stochastic version of the liar’s paradox ("Epimenides the Cretan says: ‘all the Cretans are liars’").
On a more practical level, mathematical truths expressed by individuals are constantly found to be erroneous and subject to revision. Certainly, the starting points of mathematical derivations - the axioms - are chosen because they seem to be interesting for us. In light of what we considered they become quite a relative - random - choice and another world would have quite likely a different set of mathematical truths. In part this can be seen in "our" mathematics when mutually contradictory systems of mathematics exist based on different sets of axioms. Making a somewhat daring leap, you might share the feeling that behavior of many people around you seem to be based on different rules than those of yourself. As an example, a lack of rational thought seems to give way to emotional reactions with certain individuals. We can consider our emotions and transitions therebetween as a set of rules (a calculus) not following the aim of producing rational thought but our well-being and survival. Also - of course - even the most rational thought is deeply routed in emotions like curiosity and purpose. As an experiment you might accept those other sets of emotional rules as an alternative set of truths and try to indulge in some unfamiliar emotions next time (shouting out loudly, dancing, acting, ...).
You cannot escape statistics, statistics acts everywhere and you are already actively taking part in it. You can also reflect on it and be intrigued by it - I hope you will scratch that itch.