Transcript Subjective probability - School of Life Sciences

PHILOSOPHY OF SCIENCE:
Bayesian inference
Thomas Bayes
1702-1761
Zoltán Dienes, Philosophy of Psychology
Subjective probability:
Personal conviction in an opinion – to which a number is assigned
that obeys the axioms of probability.
Probabilities reside in the mind of the individual not the external
world. There are no true or objective probabilities.
You can’t be criticized for your subjective probability regarding any
uncertain proposition – but you must revise your probability in the
light of data in ways consistent with the axioms of probability.
Personal probabilities:
Which of the following alternatives would you prefer? If your
choice turns out to be true I will pay you 10 pounds:
1. It will rain tomorrow in Brighton
2. I have a bag with one blue chip and one red chip. I
randomly pull out one chip and it is red.
Personal probabilities:
Which of the following alternatives would you prefer? If your
choice turns out to be true I will pay you 10 pounds:
1. It will rain tomorrow in Brighton
2. I have a bag with one blue chip and one red chip. I
randomly pull out one and it is red.
If you chose 2, your p(‘it will rain’) is less than .5
If you chose 1, your p(‘it will rain’) is greater than .5
Assume you think it more likely than not that it will rain.
Now which do you chose? I will give you 10 pounds if your chosen
statement turns out correct.
1. It will rain tomorrow in Brighton.
2. I have a bag with 3 red chips and 1 blue. I randomly pick a chip
and it is red.
Assume you think it more likely than not that it will rain.
Now which do you chose? I will give you 10 pounds if your chosen
statement turns out correct.
1. It will rain tomorrow in Brighton.
2. I have a bag with 3 red chips and 1 blue. I randomly pick a chip
and it is red.
If you chose 2, your p(‘it will rain’) is less than .75 (but more than .5)
If you chose 1, your p(‘it will rain’) is more than .75.
By imagining a bag with differing numbers of red and blue
chips, you can make your personal probability as precise as
you like.
e.g. if you prefer, in order to get 10 pounds, gambling on
‘It will rain tomorrow in brighton’
Rather than on
‘Selecting one red chip out of a bag composed of 6 reds and 4
blues’
Then your personal probability that it will rain is greater than
0.6.
This is a notion of probability that applies to the truth of theories
(Remember objective probability does not apply to theories)
So that means we can answer questions about p(H) – the probability
of a hypothesis being true – and also p(H|D) – the probability of a
hypothesis given data (which we cannot do on the Neyman-Pearson
approach).
Consider a theory you might be testing in your project.
What is your personal probability that the theory is true?
ODDS in favour of a theory = P(theory is true)/P(theory is false)
For example, for the theory you may be testing ‘extroverts have
high cortical arousal’
Then you might think (it’s completely up to you)
P(theory is true) = 0.5
It follows you think that
P(theory is false) = 1 – P(theory is true) = 0.5
So odds in favour of theory = 0.5/0.5 = 1
Even odds
If you think
P(theory is true) = 0.8
Then you must think
P(theory is false) = 0.2
So your odds in favour of the theory are 0.8/0.2 = 4 (4 to 1)
What are your odds in favour of the theory you are testing in your
project?
Odds before you have collected your data are called your prior odds
Experimental results tell you by how much to increase your odds (the
Bayes Factor, B)
Odds after collecting your data are called your posterior odds
Posterior odds = B * Prior odds
For example, a Bayesian analysis might lead to a B of 5.
What would your posterior odds for your project hypothesis be?
You can convert back to probabilities with the formula:
P (Theory is true) = odds/(odds + 1)
So if your prior odds had been 1 and the Bayes factor 5
Then posterior odds = 5
Your posterior probability of your theory being true = 5/6 = .83
Not a black and white decision like significance testing (conclude
one thing if p = .048 and another if p = .052)
If B is greater than 1 then the data supported your experimental
hypothesis over the null
If B is less than 1, then the data supported the null hypothesis over
the experimental one
If B = about 1, experiment was not sensitive.
(Automatically get a notion of sensitivity;
contrast: just relying on p values in significance testing.)
For each experiment you run, just keep updating your
odds/personal probability by multiplying by each new Bayes
factor
No need for p values at all!
No need for power calculations!
No need for critical values of t-tests!
And as we will see:
No need for post hoc tests!
No need to plan number of subjects in advance!
EXAMPLE WITH REAL DATA:
Sheldrake’s (1981) theory of morphic resonance
EXAMPLE WITH REAL DATA:
Sheldrake’s (1981) theory of morphic resonance
- Any system by virtue of assuming a particular form, becomes
associated with a “morphic field”
- The morphic field then plays a causal role in the development and
maintenance of future systems, acting perhaps instantaeously through
space and without decay through time
- The field guides future systems to take similar forms
- The effect is stronger the more similar the future system is to the
system that generated the field
- The effect is stronger the more times a form has been assumed by
previous similar systems
- The effect occurs at all levels of organization
Nature editorial by John Maddox 1981:
The “book is the best candidate for burning there has been in many years
. . . Sheldrake’s argument is pseudo-science . . . Hypotheses can be
dignified as theories only if all aspects of them can be tested.”
Wolpert, 1984:
“ . . . It is possible to hold absurd theories which are testable, but that
does not make them science. Consider the hypothesis that the poetic
Muse resides in tiny particles contained in meat. This could be tested by
seeing if eating more hamburgers improved one’s poetry”
Repetition priming
Subjects identify a stimulus more quickly or accurately with repeated
presentation of the stimulus
Lexical decision
Subjects decide whether a presented letter string makes a meaningful
English word or not (in the order actually presented).
Two aspects of repetition priming are consistent with an explanation
that involves morphic resonance:
Durability, stimulus specificity
Unique prediction of morphic resonance:
Should get repetition priming between separate subjects! (ESP)
Design:
Stimuli:
shared+unique
shared
shared+unique . . .
Subject no:
1
2..9
10,
...
boosters
resonator
...
Subject type: resonator
Design:
Stimuli:
shared+unique
shared
shared+unique . . .
Subject no:
1
2..9
10,
...
boosters
resonator
...
Subject type: resonator
- There were 10 resonators in total with nine boosters between each.
Resonators were assigned randomly in advance to their position in the
sequence.
- The shared stimuli received morphic resonance at ten times the rate
as the unique stimuli
- There was a distinctive experimental context (white noise, essential
oil of ylang ylang, stimuli seen through a chequerboard pattern)
Design:
Stimuli:
shared+unique
shared
shared+unique . . .
Subject no:
1
2..9
10,
...
boosters
resonator
...
Subject type: resonator
Prediction of theory of morphic resonance:
The resonators should become progressively faster on the shared as
compared to the unique stimuli
80
60
40
20
0
-20
-40
-60
-80
1
2
3
4
5
Resonator number
Data for words.
slope (ms/resonator) = - 5.0 , SE = 1.5
Neyman-Pearson: p = 0.009 significant
6
7
8
9
10
With some plausible assumptions, Bayes factor = 12
i.e. whatever one’s prior odds in favour of morphic resonance you
should multiply them by 12 in the light of the data.
Contrast Neyman-Pearson: Result was significant so should
categorically reject null hypothesis
With Bayesian approach, if before you had a very low odds in favour of
morphic resonance, they can still be very low afterwards.
NB:
I ran further three further studies with same paradigm
Including one in which boosters were run at Sussex and resonators
run in Goetingen – could they show which word set was being
boosted in Sussex??
All results flat as a pancake.
Combined Bayes factor for non-word data = about 1
Combining with word data = about ½
Does not rule out morphic resonance – just changes our odds
Summary:
A Bayes factor tells you how much to multiply your prior odds in the
light of data.
Advantages:
Low sensitive experiments show up as having Bayes’ factors near 1.
You are not tempted to accept the null hypothesis just because the
experiment was insensitive.
Contrasts with Neyman-Pearson:
1. A significant result (i.e. accepting the experimental hypothesis
on Neyman Pearson) can give rise to a very small Bayes factor!!
See handout for example. (It arises for vague theories – vague
theories are punished by Bayes)
2. On Neyman Pearson, it matters whether you formulated your
hypothesis before or after looking at the data.
Post hoc vs planned comparisons
Predictions made in advance of rather than before looking at the
data are treated differently
Bayesian inference: It does not matter what day of the week you
thought of your theory
The evidence for your theory is just as strong regardless of its timing
3. On Neyman Pearson standardly you should plan in advance how
many subjects you will run.
If you just miss out on a significant result you can’t just run 10 more
subjects and test again.
You cannot run until you get a significant result.
Bayes: It does not matter when you decide to stop running subjects.
You can always run more subjects if you think it will help.
4. On Neyman Pearson you must correct for how many tests you
conduct in total.
For example, if you ran 100 correlations and 4 were just significant,
researchers would not try to interpret those significant results.
On Bayes, it does not matter how many other statistical hypotheses you
investigated. All that matters is the data relevant to each hypothesis
under investigation.
The strengths of Bayesian analyses are also its weaknesses:
1. Are our subjective convictions really susceptible to the
assignment of precise numbers and are they really the sorts of
things that do or should follow the axioms of probability?
Should papers worry about the strength of our convictions in
their result sections, or just the objective reasons for why
someone might change their opinions?
BUT can just report Bayes factor
2. Bayesian procedures – because they are not concerned with
long term frequencies - are not guaranteed to control error
probabilities (Type I, type II).
Which is more important to you –to use a procedure with known
long term error rates or to know the amount by which you
should change your conviction in a hypothesis?