1
A forbidden outcome is a likelihood of zero
If P(o | H) = 0 then observing o makes P(H | o) = 0, whatever P(H) was. No prior survives it except a prior of exactly 1 that was never really in doubt.
Lesson · Science, Induction and Causation · Confirmation
Falsification and What a Theory Forbids
A hypothesis forbids the outcomes it gives probability 0. One such observation drives its posterior to 0 whatever the prior, while any number of allowed observations leaves it short of 1. A hypothesis with no zero in its row cannot be refuted, and one whose row matches its rival's cannot be moved at all; the more a theory forbids, the more it gains when it survives.
forbids o ⟺ P(o | H) = 0
o observed ⟹ P(H | data) = 0, any prior
no zero in the row ⟹ never refuted
same row as the rival ⟹ never moved
survival is worth more when more is forbidden
What a theory forbids
Core ideas
What this lesson establishes
Popper's thought was that a theory is scientific to the degree that it says what cannot happen. In probability terms that is a likelihood of zero.
2
Confirmation never reaches proof
Evidence that raises P(H) leaves it below 1 as long as some rival gave the data positive probability. A refutation is final; a confirmation is not.
3
A theory that forbids nothing risks nothing
A hypothesis with no zero in its row cannot be refuted by any single observation. If its row also matches its rival's, nothing can move it at all, and survival is worth nothing. Between the two lies every hypothesis whose row differs from its rival's without a zero: those are moved, by the Bayes factor of the last lesson, and never to 0.
The material
The asymmetry in the numbers
One forbidden outcome, one theory that forbids nothing, and one that forbids a great deal.
Take a hypothesis H and three possible outcomes, o1, o2 and o3. Suppose H gives probability 1/2 to each of the first two and 0 to the third: it forbids o3. A rival, which is just not-H, gives each outcome 1/3. Give H a prior of 99/100.
Definition · Forbidden outcome
An outcome is forbidden by H when P(o | H) = 0. A hypothesis is falsifiable to the extent that it forbids some outcome that can be observed.
Now observe o3. The likelihood of the data under H is 0, so the joint probability of H and the data is 0 and the posterior is 0. The lab calls that refuted, and the Bayes factor is 0. Nothing about the prior of 99/100 mattered; multiplying anything by 0 gives 0.
Theorem · The asymmetry of refutation
If P(o | H) = 0 and o is observed, then P(H | o) = 0 for every prior. But if P(o | H) > 0 for every observed o, and a rival gives some of them positive probability, then P(H | data) < 1 however many there are.
The second half is the one the misconception ignores. Feed the same hypothesis ten observations of o1, which it allows, and its posterior rises from 99/100 to 5845851/5846875, about 0.9998, and does not reach 1, because the rival never gave o1 probability 0. The lab reports the verdict as “confirms”, not “proves”. One refuting observation does what no number of confirming ones can.
The preset for a hypothesis that forbids nothing takes the opposite case: no zero in its row, and the same row as its rival, 1/3 on each outcome. Whatever is observed, the two likelihoods are equal at every step, so the posterior is the prior and the lab calls the data neutral. Two things are true of this hypothesis, and they should be kept apart. Because it has no zero, no single observation can refute it. Because its row matches its rival's, nothing can confirm it either. Edit its row to 1/2, 1/4, 1/4 and the first stays true while the second fails: the hypothesis still forbids nothing, and the same four observations now confirm it, by a factor of 81/64. The slogan that a theory which forbids nothing risks nothing is exact about refutation and a limiting case about the rest. The general measure of what a theory risks is how far its row departs from its rival's, and a zero is the furthest it can go.
Between the two lies the last preset. H now forbids two of the three outcomes and gives all its probability to o1. Observing o1 is what H said would happen, and the rival, which spread its probability, gave it only 1/3. The factor is 3, and H goes from 1/2 to 3/4. The more a hypothesis forbids, the more its survival counts.
Popper himself refused the reading given here. On his account a theory that survives a severe test is corroborated, not made more probable; he held that every universal law has probability 0 on any evidence, and that the search for probable theories is the wrong aim, since the most probable hypothesis is the one that says least. The lesson borrows his asymmetry and puts it into the frame he rejected. A reader who follows him keeps the first tile of the first preset, the 0, and declines to read the posteriors of the others as anything but bookkeeping; what that reader then owes is an account of why a corroborated theory should be relied on, which Popper agreed he could not give.
None of this makes falsification a procedure. A theory is rarely tested alone, and “The Duhem–Quine Problem” asks what a forbidden outcome refutes when the prediction depends on more than the theory. What the lab shows is only the arithmetic: a likelihood of zero is the strongest thing a hypothesis can stake.
Interactive
Exact Bayesian updating
prior times likelihood, normalised; the Bayes factor is the weight of the data
Method
Reading what a hypothesis forbids
Four steps, applied to any hypothesis with a row of likelihoods.
1
List the outcomes with likelihood 0
Scan the hypothesis's row for zeros. Those are the outcomes it forbids, and they are what makes it falsifiable.
2
Check the data against the zeros
If any observed outcome sits under a zero, the posterior is 0 and the work is over; no prior or further data can restore it.
3
Otherwise compare with the rivals
If nothing forbidden occurred, the Bayes factor against the rivals gives the confirmation. A row identical to a rival's gives a factor of 1.
4
Note how much was staked
The more outcomes forbidden, the larger the factor when the data fall in the allowed region. Compare the surviving factor with the share of outcomes that were allowed.
Forbidding and surviving
Score 0 / 0Choose the answer you can justify.
A prior of 99/100 and one forbidden outcome
The hypothesis H forbids o3; its rival gives every outcome 1/3.
P(o3 | H) = 0, P(o3 | not-H) = 1/3
joint probability of H and o3: 99/100 · 0 = 0
joint probability of not-H and o3: 1/100 · 1/3 = 1/300
P(H | o3) = 0 / (0 + 1/300) = 0
Bayes factor = 0
One observation took 99/100 to 0. Had the prior been 999999/1000000 the result would have been the same, because the joint probability would still be 0.
Common mistakes
1
Thinking a theory confirmed many times is provenTen observations of o1 take H from 99/100 to a figure close to 1 and not equal to it, because the rival gave o1 probability 1/3. The lab labels this “confirms”, and a single o3 afterwards would still bring it to 0.
2
Thinking an unfalsifiable theory is the best-supported oneA theory that forbids nothing is never contradicted, but with likelihoods equal to its rival's it is never confirmed either. The posterior equals the prior at every step.
3
Reading a likelihood of zero as a prior of zeroA prior is what you believed before the data. A likelihood of zero is what the hypothesis says about an outcome. The first can be moved by evidence and the second is the reason one observation moves it all the way.
Check your understanding
✓
Finish when you can identify what a hypothesis forbids and compute the effect.
Given a row of likelihoods you should be able to list the forbidden outcomes, say what happens to the posterior if one occurs, and say what happens if only allowed outcomes occur.
Note: In the lab the rival is simply not-H, and its row is a stated average of the other ways H could fail. A real theory has a particular rival or several, and the Bayes factor changes with which.