Posted on Mar 14th, 2018

Einstein confused his students with a riddle about probability – or was it Einstein himself who was confused?

Albert Einstein disliked the idea that the laws of nature were inherently probabilistic. ‘God does not play dice with the universe,’ he stated famously and repeatedly. Yet, physicists like Niels Bohr strongly advocated the idea –based on the ‘Copenhagen interpretation’ of quantum theory– that chance is an inalienable and inevitable aspect of nature itself.

To Einstein, the recourse to chance must have felt like giving up. A perfect understanding of nature should allow perfect predictions; hence, the statement that chance is inherent to nature is nothing more than a veiled confession of ignorance – something is not being understood, and this lack of understanding is mistaken for ‘chance’. Even though we sympathize with Einstein’s view on general principle, the modern consensus in physics is that Bohr won the debate (Kumar 2009); there exist interpretations of quantum theory that are consistent with a deterministic universe (Earman 1986), but they are currently not mainstream.

In the standard, ‘Copenhagen’ interpretation of quantum mechanics, “physical systems generally do not have definite properties prior to being measured, and quantum mechanics can only predict the probabilities that measurements will produce certain results. The act of measurement affects the system, causing the set of probabilities to reduce to only one of the possible values immediately after the measurement.”

Einstein objected to the Copenhagen interpretation, but there is some debate about which aspect of the interpretation bothered him the most. Max Born, the 1954 Nobel laureate in physics and a lifelong friend of Einstein, believed that Einstein was reluctant to sacrifice the idea that the universe is deterministic and that everything has a cause. This opinion is consistent with Einstein’s statement that ‘God does not play dice with the universe’. In addition, Einstein explicitly stated that the probabilistic basis of the Copenhagen interpretation was unsatisfactory to him: “I am, in fact, firmly convinced that the essential statistical character of contemporary quantum theory is solely to be ascribed to the fact that this [theory] operates with an incomplete description of physical systems.” (Einstein 1949, as cited in Kumar 2009)

On the other hand, Wolfgang Pauli, the 1945 Nobel laureate in physics, believed that Einstein’s primary objection to the Copenhagen interpretation was one of realism, not determinism; in support of this conjecture, Einstein himself wrote that “At the heart of the problem is not so much the question of causality but the question of realism” (Einstein, 1950, as cited in Kumar 2009). In other words, Einstein rejected the Copenhagen interpretation that reality does not exist independent of the observer, and he once asked a colleague “Do you really think the moon isn’t there if you aren’t looking at it?”

The foregoing suggests that *both* aspects of the Copenhagen interpretation Einstein found intolerable. To Einstein, the fact that probability needed to be invoked suggested that the theory was incomplete, and the idea that reality does not exist independently from the observer suggested that the theory was flawed. More details are provided in Kumar (2009).

It is against this background –Einstein arguing that chance is not fundamental to nature, and that the concept only acts to mask our ignorance– that the anecdote below acquires its appeal.

In the excellent article by Holmes (2018), we encounter the following fragment^{1}:

David Mumford suggested in a paper entitled ‘Intelligent design found in the sky with p < 0.001’ that the positioning of the stars in the Orion constellation were so particular that only intelligent design could explain them. However, in his calculations he starts by asking what the probability is that the stars of Orion are so aligned. This argument suffers from the blade-of-grass paradox, as put by Diaconis (New York Times, 1990):

“If you were to stand in a field and reach down to touch a blade of grass, there are millions of grass blades that you might touch. But you will, in fact, touch one of them. The a priori fact that the blade you touch will be any particular one has an extremely tiny probability, but such an occurrence must take place if you are going to touch a blade of grass.”^{*}

^{*} A similar argument by Einstein was related by Wigner in his autobiography: “Life is finite. Time is infinite. The probability that I am alive today is zero. In spite of this, I am now alive. Now how is that?” None of his students had an answer. After a pause, Einstein said, “Well, after the fact, one should not ask for probabilities.”

Let’s focus on the Einstein anecdote described by Wigner, yet another physics Nobel prize laureate. At first sight, Einstein appears to make a valid point – yes, Einstein’s existence is a stroke of extreme luck, and so it is for all of us alive today. Let’s drive this point home. You are you in part because of your genetic makeup, and the probability that your mother’s and your father’s genes combined the way they did –to then produce you– is extremely small.^{2} In this sense, you are the unlikely winner of an genetic lottery. But your luck does not stop there. Your father and your mother *also* had to be unlikely winners of a different genetic lottery, as is the case for all of your ancestors. This means that in order to exist, your ancestors had to win a long and unbroken sequence of lotteries against heavy odds.

And this, of course, only refers to the genetic part of existence. What we have left out is the *foreplay*, that is, the chances that your ancestors grew up and eventually met one another under circumstances conducive to procreation –under a full moon, with violins playing in the background, and perhaps after having had a few drinks– going back all the way to the mindless mating in the primordial soup.

From such considerations it is evident that your existence, just as that of any other creature alive today, depends on a long series of extremely unlikely occurrences – an unbroken chain of continuous dumb luck that stretches back over a billion years. And when it comes to dumb luck, we have not even touched upon other unlikely events of great importance, such as the probability that a clump of stellar dust would cool down to form our earth, a planet that just happens to be ideally positioned and equipped to sustain life, or the probability that a meteor would crash into earth, just in time to liberate it from dinosaur dominance; the list goes on and on.

It seems therefore that Einstein is right: the probability that any of us should be alive today is essentially zero. And yet, the deterministic viewpoint –championed by Einstein himself– leads us to the exact opposite conclusion. Let’s travel back in time to the classroom where Einstein posed his riddle, and introduce Theodore, an inquisitive and at times obnoxious student:

*Einstein*: “Life is finite. Time is infinite. The probability that I am alive today is zero. In spite of this, I am now alive. Now how is that?”

*Theodore*: “Sorry professor, but you are pacing and lecturing right in front of us. This means that –barring dream or hallucination– the probability that you are alive today is very near one. I must be missing something.”

*Einstein*: “Ah, well, yes. Maybe I should clarify what I mean with the word ‘probability’. Clearly I am alive now. And this is the conundrum, for the advance probability of this happening is basically zero; even when you just consider it from a vantage point as recent as the time of Napoleon.”

*Theodore*: “But professor, this makes it even less sense to me. You have stressed that God does not play dice with the universe; we inhabit only a single universe, and it unfolds according to the laws of physics, in which chance ultimately does not exist. Only one universe was ever possible, and it is the one we currently inhabit; in it, you are currently very much alive. So even if we go back to the time of the dinosaurs, your existence was never improbable; in fact, it was *inevitable*: the probability remains one, not zero.”

*Einstein*: [muttering something under his breath]

*Theodore*: “Professor?”

*Einstein*: “Well, perhaps. But we can agree that the advance probability of my existence is zero *from the perspective of any particular person*: it would have been impossible for, say, Napoleon, to have predicted my existence. We can’t even predict next week’s weather reliably! This just underscores my main point, namely that, after the fact, one should not ask for probabilities.”

*Theodore*: “Professor, I agree that the advance probability of your existence is virtually zero when it is viewed as the degree to which any one person could have predicted it. But I don’t think it follows that we can never ask for probabilities after the fact. All we need to do is exercise mental discipline and put our mind in the same state it was in before the data were obtained. This way we can make predictions even for events that have already occurred.”

*Einstein*: “And do you think this is feasible? Suppose we conduct a séance and I am able to talk to Napoleon. He would have to agree that the probability that I am currently alive is one, or very close to one. But how can we clear his mind of the knowledge he now has? We would have to ask him to forget our encounter. Or perhaps we just conduct the séance without me being present. You can then ask Napoleon’s spirit something like ‘what do you think is the probability that, some hundred years after your death, a physicist called Albert Einstein will be teaching a class in Princeton, on a sunny Wednesday afternoon at 3 pm, while his most obnoxious student, Theodore, accuses him of failing to understand his own material?’ Would Napoleon answer ‘that probability strikes me as virtually zero’, or would he suspect that the act of asking the question provides a hint that the probability is non-negligible?”

*Theodore*: “But professor, such guessing-games are the limitation of the human mind. It is not a principled limitation.^{3} Moreover, for less extreme situations it may work very well. For instance, suppose you were to stand in a field and reach down to touch a blade of grass. There are millions of grass blades that you might touch. But you will, in fact, touch one of them. Still we can answer the question ‘given that you were going to reach down and touch a blade of grass, what is the probability of touching this particular one?’. Of course that probability is again 1, because what has happened is the only possibility; but one might ask ‘at the time that I walked unto the field, what prediction would I have made about touching this particular blade?’ The answer will be approximate but it is nonetheless possible to formulate one. Another counterexample concerns events from the past about which we are unsure. For instance, what is the probability that Shakespeare’s plays were written by Edward de Vere? What is the probability that Julius Caesar’s last utterance was ‘et tu, Brute?’ or words to that affect? What is the probability that, on 31 October 1517, Martin Luther nailed ninety-five theses to the door of the All Saints’ Church in Wittenberg?”^{4}

Here we leave Einstein’s classroom. It should be clear by now that, even for the most brilliant scientists, the concept of probability represents a philosophical minefield, one whose secure crossing requires constant vigilance. As mentioned already by Herschel (1851, p. 219), the doctrine of probabilities is “one of the most difficult and delicate among the applications of mathematics to natural philosophy.”

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^{1} paraphrased for the reader’s benefit.

^{2} We can be more specific. Humans have 23 pairs of chromosomes; of each pair, one is inherited from the father and one from the mother. For a single specific chromosome pair, the a priori probability that it would combine to yours is 0.5 (50% chance of getting the matching one from dad) times 0.5 (50% chance of getting the matching one from mom), equaling 0.25. But this needs to happen for all 23 pairs, yielding an overall probability of 0.25^{23} . This is roughly 1 in 100,000,000,000,000. And this computation ignores biological processes such as genetic recombination through chromosomal crossover, which will lower this probability still further.

^{3} Had Theodore been alive today, he may have exemplified his argument by mentioning artificial intelligence systems. In such systems, mental time travel is possible provided that the system has access to its previous state.

^{4} NB. None of these events are supported by compelling historical evidence.

Holmes, S. (2018). Statistical proof? The problem of irreproducibility. *Bulletin of the American Mathematical Society, 55*, 31-55.

Kumar, M. (2009). *Quantum: Einstein, Bohr and the Great Debate about the Nature of Reality*. London: Icon Books.

Eric-Jan (EJ) Wagenmakers is professor at the Psychological Methods Group at the University of Amsterdam.

Posted on Mar 9th, 2018

*The promised post on Einstein will follow next week.*

More and more psychologists are registering their hypotheses, predictions, and analysis plans prior to data collection. Will such preregistration be the death knell for creativity and serendipity? Gilles Dutilh, Alexandra Sarafoglou, and I recently wrote an article for *Perspectives on Psychological Science* that provides a historical perspective on this question. In the article, we describe the origin and development of “the empirical cycle”, that is, the modern perspective on how scientists can learn from data. In the course of our historical investigations, we came across several interesting anecdotes that lack of space prevented us from including. But we can include them in this series of blog posts. Here is the first story, courtesy of Cornelis Menke.

Francis Bacon is often heralded as the scientific Moses that guided academia out of the dark ages of ignorance and prejudice. As discussed in the article, this idea is probably false. For a counterargument one can do no better than read the 1863 violent take-down by Baron Justus von Liebig, a famous German chemist. In the article we mention von Liebig’s final conclusion about Francis Bacon:

“When a boy, he studied jugglery; and his cleverest trick of all, that of deceiving the world, was quite successful. Nature, that had endowed him so richly with her best gifts, had denied him all sense for Truth.” (von Liebig, 1863b, p. 261)

This is pretty harsh, but not as harsh as some of the other fragments in von Liebig’s piece. Based on the following paragraph, the current guardians of “tone” in scientific discussions (sometimes called “tone police”) would make Baron Justus public enemy number one:

“Whoever studies his ‘Novum Organum,’ or any other of his works, diligently and in good faith, and, with patience and perseverance, follows up a single thought of his through all its turnings, and into all its corners, will find that it resembles, in its origin, a merry spring bursting forth out of the ground. It gives us reason to hope that we shall be led by it, through green and flowery meadows, to cool shady woods, to a brook with mills beside it, and, at last, to a stream bearing ships upon its waters; but, instead of this, the wanderer is brought into a desert, where there is no life, and where the rivulet vanishes amid the barren sand. At first, we take this for a chance occurrence, and think that, in a second or third attempt, we shall be led in other directions more satisfactory; but at last the conviction forces itself upon us, that the whole was but a painted decoration. Eventually we discover the intention, and are ashamed of ourselves for having been deluded by so coarse a deception.” (von Liebig, 1863a, p. 242)

Von Liebig’s poetic assessment can be applied to other frameworks that are misleading yet highly influential. For instance, some people might feel it is apt to replace “Novum Organum” with works such as “Statistical Methods for Research Workers” or with “Frequentist Probability and Frequentist Statistics”.^{1} The main problem with this analogy, however, is that the frequentist framework, in its origin, resembles a smelly tar pit rather than a merry spring. But this is the topic of another day. 🙂

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^{1} In my opinion this is going too far, but only because the frequentists succeeded in misleading not only their readership, but also themselves. I certainly would not want to argue that there was an explicit attempt at deception, the way von Liebig argues is the case with Francis Bacon.

Fisher, R. A. (1925). *Statistical methods for research workers*. Edinburgh: Oliver and Boyd.

Neyman, J. (1977). Frequentist probability and frequentist statistics. *Synthese, 36*, 97-131.

von Liebig, J. (1863a). Lord Bacon as natural philosopher (Part I). *Macmillan’s Magazine, 8*, 237-249.

von Liebig, J. (1863b). Lord Bacon as natural philosopher (Part II). *Macmillan’s Magazine, 8*, 257-267.

Wagenmakers, E.-J., Dutilh, G., & Sarafoglou, A. (in press). The creativity-verification cycle in psychological science: New methods to combat old idols. *Perspectives on Psychological Science*. Preprint at https://psyarxiv.com/37ntp.

Eric-Jan (EJ) Wagenmakers is professor at the Psychological Methods Group at the University of Amsterdam.

Posted on Feb 21st, 2018

Cicero eloquently summarized the philosophical position that the universe is deterministic – all events are preordained, either by nature or by divinity. Although “ignorance of causes” may create the illusion of Fortune, in reality there is only Necessity.

The male academic who cites Cicero generally lacks the insight that, instead of imbuing his writing with gravitas, he inevitably conveys the impression of being a pompous dickhead (‘glans inflatus’). Particularly damaging to a writer’s reputation are Cicero quotations that occur at the start of an article; for, as Horace reminds us, “parturiunt montes, nascetur ridiculus mus”. Indeed, the only academics who seem to get away with citing Cicero are those who study Cicero’s work professionally.

Posted on Feb 16th, 2018

**Summary:** When Bayesians speak of probability, they mean plausibility.

The famous Matrix trilogy is set in a dystopic future where most of mankind has been enslaved by a computer network, and the few rebels that remain find themselves on the brink of extinction. Just when the situation seems beyond salvation, a messiah –called Neo– is awakened and proceeds to free humanity from its silicon overlord. Rather than turn the other cheek, Neo’s main purpose seems to be the physical demolition of his digital foes (‘agents’), a task that he engages in with increasing gusto and efficiency. Aside from the jaw-dropping fight scenes, the Matrix movies also contain numerous references to religious themes and philosophical dilemma’s. One particularly prominent theme is the concept of free will and the nature of probability.

Posted on Feb 9th, 2018

In an earlier blog post we discussed a response (co-authored by 88 researchers) to the paper “Redefine Statistical Significance” (RSS; co-authored by 72 researchers). Recall that RSS argued that *p*-values near .05 should be interpreted with caution, and proposed that a threshold of .005 is more in line with the kind of evidence that warrants strong claims such as “reject the null hypothesis”. The response (“bring your own alpha”, BYOA) argued that researchers should pick their own alpha, informed by the context at hand. Recently, the BYOA response was covered in *Science*, and this prompted us to read the revised, final version (hat tip to Brian Nosek, who attended us to the change in content; for another critique of the BYOA paper see this preprint by JP de Ruiter).

Posted on Feb 1st, 2018

*This is a guest post by Scott Glover.*

In a recent blog post, Eric-Jan and Quentin helped themselves to some more barbecued chicken.

The paper in question reported a *p*-value of 0.028 as “clear evidence” for an effect of ego depletion on attention control. Using Bayesian analyses, Eric-Jan and Quentin showed how weak such evidence actually is. In none of the scenarios they examined did the Bayes Factor exceed 3.5:1 in favour of the effect. An analysis of this data using my own preferred method of likelihood ratios (Dixon, 2003; Glover & Dixon, 2004; Goodman & Royall, 1988) gives a similar answer – an AIC-adjusted (Akaike, 1973) value of λadj = 4.1 (calculation provided here) – meaning the data are only about four times as likely given the effect exists than given no effect. This is consistent with the Bayesian conclusion that such data hardly deserve the description “clear evidence.” Rather, these demonstrations serve to highlight the greatest single problem with the *p*-value – *it is simply not a transparent index of the strength of the evidence*.

Beyond this issue, however, is another equally troublesome problem, one inherent to null hypothesis significance testing (NHST): *any-sized effect can be coaxed into being statistically significant by increasing the sample size* (Cohen, 1994; Greenland et al., 2016; Rozeboom, 1960). In the ego depletion case, a tiny effect of 0.7% is found to be significant thanks to a sample size in the hundreds.