1
Grue agrees with green until t
An object is grue if it is examined before time t and green, or not so examined and blue. Every emerald examined before t that was green was therefore grue as well.
Lesson · Science, Induction and Causation · Induction
Grue and the New Riddle
Define “grue” so that every emerald examined so far is both green and grue. The two hypotheses assign the same probability to every observation before the critical time, so their Bayes factor is exactly 1 and the posterior ratio equals the prior ratio. Which predicate is projectible is a fact about the prior.
grue: green if seen before t, else blue
same rows before t: BF = 1 exactly
posterior ratio = prior ratio
100 green emeralds do not separate them
Two hypotheses, one record
Core ideas
What this lesson establishes
Hume asked what justifies projecting the past into the future. Goodman's new riddle asks which pattern in the past to project.
2
A Bayes factor of 1 means the data did not discriminate
If two hypotheses give the observed data the same probability, the data move neither relative to the other, however many observations there are.
3
Projectibility lives in the prior
Treating green as projectible and grue as not is a judgement made before the data, since the data are the same for both.
The material
Defining the rival
The definition first, then the table of what each hypothesis says, then the arithmetic.
Every emerald ever examined has been green. That supports “all emeralds are green”, and by the reasoning of the last lesson it supports “the next emerald will be green”. Now introduce a word.
Definition · Grue
Fix a time t in the future. An object is grue if it is examined before t and is green, or is not examined before t and is blue.
Every emerald examined so far was examined before t and was green, so it was grue. “All emeralds are grue” has exactly as much observed support as “all emeralds are green”, and the two predict opposite colours for the first emerald examined after t.
an emerald examined H-green says H-grue says
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before t green green
after t green blue
The lab takes two outcomes, green and blue, for an emerald as examined. Before t both hypotheses give green probability 1, so their likelihood rows are identical. Put the prior 3/4 on green and 1/4 on grue and enter 100 green emeralds.
Theorem · Identical rows give a Bayes factor of 1
If P(data | H) = P(data | H′) then the Bayes factor of H against H′ is 1, and the posterior odds equal the prior odds. With only these two hypotheses it follows that the posterior of H equals its prior.
The lab prints exactly that: the Bayes factor is 1, the posterior of green is still 3/4, and the tile that classifies the data says they are neutral. One hundred emeralds and one thousand do the same, because each multiplies both likelihoods by 1.
The second preset adds the one difference that matters. Suppose the next emerald is examined after t and is green. Under grue it should have been blue, so grue gives it probability 0 and green gives it 1. The Bayes factor is infinite, the posterior of green is 1, and the lab says the data prove green against this one rival. Until such an observation arrives, nothing in the evidence tells the hypotheses apart.
Goodman's riddle is that no formal feature of the data prefers green to grue. The standard replies place the difference elsewhere. One says grue is gerrymandered, because it mentions a time; but a speaker of a language with grue and bleen as primitives would call green the gerrymandered word. Another says natural kinds are the projectible ones, which relocates the question to what makes a kind natural. Each is a claim about the prior, and the lab can show that it must be.
The lab's limit is the same as in the last lesson: it computes from the prior it is given. It does not say that 3/4 on green is the right number, and a reader who sets the prior on grue at 3/4 gets the mirror result.
Interactive
Exact Bayesian updating
prior times likelihood, normalised; the Bayes factor is the weight of the data
Method
Testing whether data favour one hypothesis
Four steps, applied to any pair of rival hypotheses.
1
Write what each predicts
For each observation you have, record the probability each hypothesis gave it. Do this before computing anything.
2
Compare the rows
If the two rows agree on every outcome that occurred, the Bayes factor is exactly 1 and the data are idle between them.
3
Find the outcome where they differ
Look for an observation, possible in principle, on which the rows disagree. That is the only kind of evidence that can separate them.
4
Ask where the preference came from
If you favoured one hypothesis while the Bayes factor was 1, the preference came from the prior. Say what justified it.
Green, grue and the data
Score 0 / 0Choose the answer you can justify.
A hundred green emeralds
Prior 3/4 on green and 1/4 on grue; both give a green emerald probability 1 before t.
likelihood of the data under green: 1 to the 100th = 1
likelihood of the data under grue: 1 to the 100th = 1
Bayes factor of green against grue: 1/1 = 1
posterior of green: (3/4 · 1) / (3/4 · 1 + 1/4 · 1) = 3/4
prior ratio 3 to 1; posterior ratio 3 to 1
The posterior ratio equals the prior ratio to the digit. One hundred emeralds were examined, and the evidence for green is exactly the evidence for grue.
Common mistakes
1
Thinking more green emeralds favour green over grueEach green emerald examined before t has probability 1 under both hypotheses. The Bayes factor stays at exactly 1 after one emerald and after a hundred, and the posterior ratio is the prior ratio, so the count of green emeralds is no argument at all.
2
Treating “grue” as a trick of language that the data can sweep asideThe hypothesis is as well defined as green and fits as well. It is excluded by a judgement that colours are projectible and time-indexed colours are not, and that judgement is prior to the data.
3
Concluding that induction is hopelessThe lab shows only that induction needs a prior that treats some predicates as projectible. A reader can hold such a prior, defend it, and compute with it; what cannot be done is to derive it from the observations it is applied to.
Check your understanding
✓
Finish when you can show a Bayes factor of 1 and say what follows.
Given two hypotheses and a record of observations, you should be able to write their likelihood rows, show that the Bayes factor is 1, compute the posterior from the prior, and name the kind of observation that would separate them.
Note: The time t is stipulated and never reached by the data in the first preset; the second preset is the one where it is reached. Some authors prefer to make the riddle about properties rather than times, and the arithmetic of identical rows is the same.