Scientific Inference: Do We Really Need Induction?

 

Part 2. “We don’t need no stinkin’ induction” — Sir Karl Popper

The traditional scientific method supposedly employs induction both in the course of forming hypotheses and empirically confirming them. The examples of inductive inference from the previous note—whether the sun will rise tomorrow, or whether the sequence 1,3,5,7,9,… anticipates all odd numbers—were discussed in the context of hypothesis formation. They were simplified illustrations, but there are real-world examples. Take Charles Darwin: it was through his careful study and direct observation of individual animal specimens that he came to discern the hints of a much grander and extensive law of nature. By examining the beak shapes of particular finches in the Galapagos, Darwin was able to induce a wide-ranging principle that applied not only to other species but across eons of time. Incidentally, this is an example of strong induction, as evidenced by the wild success of the theory of evolution by natural selection. There are other great hypotheses, however, that do not share this inductive genealogy: Schrödinger’s non-relativistic wave equation comes to mind, as does Einstein’s relativity theory. In Schrödinger’s case, the equation was the result of a creative, non-logical affair incorporating theoretical aesthetic, bold conjecture, and apparently a skiing holiday in the Swiss Alps; in Einstein’s case, it was a thought experiment that got terribly out of hand.

Surely these ideas came from somewhere; after all, not since Locke introduced the world to empiricism have people actually believed that facts about nature can be known a priori¹. Indeed, they come from observations, but not systematically and not without man’s psychological propensity for pattern-seeking or his inborn sense of causal relation. Philosopher of science, Karl Popper, rejected hypothesis-by-induction outright, arguing instead that science progresses not through the steady accumulation of evidence to universals, but instead as a series of conjectures and refutations, of “inventions—conjectures boldly put forward for trial, to be eliminated if they clashed with observations…” that arise “psychologically or genetically a priori, prior to all observational experience.” [1] It’s easy to misinterpret Popper’s statement: he does not literally mean all observation experience, but that precursor experience (“genetically a priori“) from which inductive generalizations are reached. We’ll grant that induction has nothing necessarily to do with the formation of hypotheses and examine instead the main theme of this series: its role in justifying hypotheses.

If we consult again the examples from the last note, they are just as many acts of hypothesis justification as they are examples of hypothesis formation. For example, the observation of a few black jellybeans motivated the hypothesis (if unwisely) that all jellybeans in the jar were black. Equivalently, the observation of a few black jellybeans also constitutes a certain amount of evidence in favor of the proposed hypothesis that all jellybeans in the jar are black. And of course, the comparison of the data with the hypothesis fundamentally involves induction. Schematically, the process of hypothesis testing goes as follows: from hypothesis, $H$, an observable prediction, $O$, is deduced: $H\rightarrow O$. This prediction will be about a track in a cloud chamber or a fossil in the Cambrian. We then set up an experiment or organize an expedition to test the prediction. To confirm the hypothesis, we must use inductive inference: we generalize the observation and compare it against the hypothesis: $O\rightarrow H$. Now, if $O$ agrees with the prediction, have we confirmed $H$? Sadly, no, because $O\rightarrow H$ is not a deductively valid move. It is an example of the logical fallacy of affirming the consequent: the assertion $O\rightarrow H$ does not follow from $H\rightarrow O$ because it is in general possible to find some other hypothesis, say, $H'$, that is also compatible with the observation, $H’\rightarrow O$. This is the problem of projectability that we just discussed last time: for example, the observation of the numbers $1,3,5,7,9$ agrees with both the hypothesis $x_n=2n+1$ and $x_n=(1−2n-1)(3−2n-1)(5−2n-1)(7−2n-1)(9−2n-1)+2n+1$.

Sometimes we can legally get away with affirming the consequent if we know that a particular hypothesis describes all the necessary and sufficient conditions for the observation. The only way to know is to do comprehensive experiments that account for all the relevant conditions, a feat that is seldom possible in practice: as Figure 1 shows, the data might fail to reveal the causal relationships between the experimental conditions and the observations, leaving multiple hypotheses in the running². In general, though, certain and exact confirmation of unique hypotheses is not possible.

Fig 1. The danger of working from observation, $O$, to hypothesis, $H$. Given the observation (circled in blue) that the past week as been both cloudy and rainy, the causal relationship between rain and clouds is unclear—it is not known whether clouds are a necessary or sufficient condition for rain. The observation is therefore compatible with at least two hypotheses: $H$, which asserts that clouds are a sufficient condition for rain, and $H'$, which asserts that clouds are a necessary condition.

In the 1930s, Karl Popper was moved to abandon confirmation as an instrument of scientific discovery because he did not accept the inductive scaffolding on which it is built. He recognized an important asymmetry between the acts of confirmation and refutation: while it is impossible to verify a universal statement by observing singular instances, universal statements can be contradicted by individual observations. Take the prototypical sample hypothesis that all swans are white—though the number of swans is arguably finite, we cannot search the whole world over in an attempt to verify this hypothesis (and if we could, it would not result in a predictive theory). We can, however, falsify it, swiftly and assuredly by observing just a single black swan. Unlike confirmation, the falsification of hypotheses is a deductively-sound procedure employing the classical rule of modus tollens: if $H\rightarrow O$, then $\overline{O}\rightarrow \overline{H}$ (where $\overline{O}$ is read “not $O$”). While in practice statistical limitations might require several disconfirming trials before a hypothesis or theory is discarded, sometimes the evidence is resounding: for example, the discovery of the cosmic microwave background in 1965 decapitated Hoyle’s steady-state universe model
in one fell swoop. By proposing falsification as the means of testing hypotheses, Popper sought to establish a logically valid, induction-free approach to the justification of all scientific proposals based on the simple maxim: “it must be possible for an empirical scientific system to be refuted by experience” [2].

Attempts at falsification: The p-value catastrophe

Falsification is the presumed mode of inference employed by much of contemporary science based on frequentist, or classical, statistics. The weapon of choice is the most well-worn instrument of statistical terror ever to be foisted upon the scientific process—the p-value.  Suppose we wish to test some hypothesis, $H_0$.  The subscript ‘0’ indicates that this hypothesis generally refers to some fiducial model, perhaps one for which the effect we’re seeking is absent.  In this case, it is appropriately called the null hypothesis, and the purpose of the p-value is to determine the agreement between the observed data, $O$, and the null hypothesis, $H_0$, when there are other possible explanatory hypotheses. Specifically, it provides the probability that we would observe the data $O$ under the assumption that $H_0$ is correct, $p(O|H_0)$³. A small p-value, say less than 0.05, means that there is a smaller than 5% chance that we would have measured the data $O$ given that $H_0$ is true.  In other words, $H_0$ is correct, and the tension with the data arises instead due to chance (whether it’s from noise in the measuring device or some fundamental uncertainty of the physical process being measured.) Often, however, in the statistical parlance one reads that the p-value as the probability of “falsely rejecting the null hypothesis”.  This agrees with our understanding of the p-value as the chance of a statistical fluke, but we are actually in no way licensed to reject, or falsify, the null hypothesis if the p-value happens to be small.

Let’s see what goes wrong when we use p-values to make inferences. Upon making observation $O$, we compute $p(O|H_0)$, the probability that we measure $O$ given the truth of $H_0$. Suppose the p-value is found to be $<0.05$. Next, we equate $p(O|H_0)$ with $p(H_0|O)$, the probability of the truth of $H_0$ given the observation, $O$. Then, we conclude $p(\overline{H}_0|O) = 1 – p(H_0|O)$, which asserts that some alternative hypothesis (or group of hypotheses), $\overline{H}_0$, which is typically a model (or group of models) that supports the observed effect, is correct at $95\%$ confidence. Can you spot the error? In the very first step we took $p(O|H_0) = p(H_0|O)$. This is wrong—it’s called the fallacy of the transposed conditional. It’s not as bad as affirming the consequent (for which we’d always have $p(O|H_0)=p(H_0|O) = 1$), but it can really get us into trouble. An especially striking example of this confusion is discouragingly called the prosecutor’s fallacy because it has been used by the prosecution in criminal court to argue for the guilt of the defendant: the probability of getting certain evidence, say DNA evidence ($E$), given the innocence ($I$) of the defendant, $P(E|I)$ (which is typically low) is equated with the probability of innocence given the evidence, $p(I|E)$ (which is then likewise low). What is neglected here is the chance that the defendant is innocent in the first place before the evidence is brought. Catastrophic when misused in the court of law, it is also tragic that it provides the statistical foundation of many important scientific results, from drug efficacy to climate change.

In practice, we see that the p-value is not really about falsification: it’s used ostensibly to confirm a hypothesis alternative to the null hypothesis, which it must falsify in the course of doing business. And we must commit a logical fallacy in order to accomplish this inference. Popper would not be pleased, nor would anyone seeking a coherent scientific inference. P-value catastrophe aside, Popper’s model of falsification still has its dissenters. After all, what good is the scientific method if it can only tell us when we get stuff wrong?

Corroboration without induction?

Popular objections to Popper’s program tend to center around the perceived negativity associated with the act of falsification; call it a psychological aversion to failure. More practically, the concern is that science should not be about what doesn’t work—how can we hope to increase our knowledge of the world if we cannot confirm, or verify, scientific hypotheses? Shouldn’t science be a constructive, rather than destructive, pursuit? According to Popper, the only tenable position is that all of our theories are basically wrong, awaiting falsification. Importantly, though, Popper views the cycle of conjecture and refutation as generative, as one that puts scientific hypotheses through a kind of optimization process in order to develop theories that, though ultimately incorrect, are the very best possible prototype of the truth: “Theories are nets cast to catch what we call ‘the world’: to rationalize, to explain, to master it. We endeavor to make the mesh ever finer and finer.” [2] This optimization process must be a relentless, vigorous assault on all hypotheses and scientific proposals bent on striking down those that miss the mark, weeding out the “unfit” hypotheses by “exposing them all to the fiercest struggle for survival.”

While Popper advocates for falsification, those hypotheses that escape it are not all considered equal. Those that pass the most strenuous attempts at falsification are in some sense preferred and were said by Popper to be corroborated. Corroboration is not an absolute condition: a theory is either more corroborated than a competitor, or new data can further corroborate an existing theory. Corroboration is essentially a measure of the testability, or falsifiability, of a hypothesis relative to alternatives. And testability, Popper argues, is logically related to the empirical content of the theory: “the more a theory forbids, the more it says about the world of experience.” What this means is that theories with more universal or precise statements have greater testability because there are more opportunities for a misstep, more places for them to go wrong.

But when all is said and done and we end up with our most stream-lined, best-fitting model, there is a strong sense that it has been given positive support by the data. Is this just an illusion? Has Popper really, truly done away with confirmation? I don’t think so. Let’s revisit the “experiment” conducted in Figure 1, but suppose we’ve collected more data. Furthermore, our experiment has been improved so that we can also measure the existence of lightning,

Fig 2. An illustration of corroboration. Both $H'$ and $H”$ are consistent with the observation, but $H”$ is more prone to falsification than $H'$ because its “predictive range” is smaller. Hypothesis $H”$ is more corroborated according to Popper.

Our dataset is such that we’ve observed some cloudy days without rain, falsifying hypothesis $H$, and it just so happens that every rainy day also had lightning. This latter observation supports a new hypothesis $H”$, that whenever it rains, there is lightning. On the question of causality, it is an alternative to $H'$: that whenever it rains, there are clouds. Now, while both $H'$ and $H”$ are consistent with the data, we see $H”$ is more easy to falsify because its “predictive range” is smaller, in fact contained within, that of $H'$. In other words, fewer observations are needed to falsify $H”$ than $H'$. $H”$ is therefore more corroborated than $H'$ and by this we must mean that it gets more things right than $H'$. Popper called this tendency towards truth verisimilitude [1] and, while it is meant to apply to ultimately false hypotheses, it is an accounting of correct statements made by the hypothesis, i.e. it is an accounting of the degree of support that the data lends to the hypothesis. This suggests a tempered view of confirmation, one that surrenders certainty for degrees of confidence instead, in line with an inductive mode of justification. While Popper might wish to view the hypothesis that passes experimental muster as ultimately false and merely awaiting rejection, until that time we consider it the most successful generalization of observed phenomena. Popper discounts the validity of the latter view, but what else is corroboration if not tentative acceptance? Both refer to the same logical relation between evidence and hypothesis. My impression is that, despite Popper’s attempts to exorcise him, Hume’s specter still lurks in the idea of corroboration. Next time, we’ll see about developing a theory of confirmation.

Read on to Part 3: Balancing Predictive Success with Falsifiability

 

References and Footnotes

[1] Conjectures and Refutations: The Growth of Scientific Knowledge, Karl Popper, Routledge (1963).
[2] The Logic of Scientific Discovery, Karl Popper, Routledge (1934).

¹Think babies born with electrodynamics already in their heads. back
²Charlatans and conspiracy theorists alike adore this kind of situation because they believe it gives faith healing and UFO abductions equal room on the stage of potential explanations for a given piece of evidence. It doesn’t, and we’ll see why shortly. back
³The notation $p(x|y)$ should be read “the probability of $x$ given that $y$ is true.” back