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Bayésianisme objectifet modélisation causale
en sciences socialesFederica Russo
Philosophie, Louvain & Kent
IHPST – 5 décembre 2008
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OverviewProbabilistic causal claims in CM
Generic / Single-case
Interpreting probability:a rush course
Interpreting probability in CM:Frequency-driven epistemic probabilitiesObjective Bayesian probabilities
Disclaimer: cet exposé sera en “franglais”
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Probabilistic causal claims
Causal claims:Tendency of an event to cause anotherCausal effectiveness of an eventFrequency of occurrence of a causal relation…
Causal claims are probabilisticallyprobabilistically modelled
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Probabilistic causal claimsGeneric
About the population as a whole;Describe an average causal relation;Concern frequency of occurrence.
Single-caseAbout a particular ‘individual’;Occur at particular time and space;Express a rational belief about what will or did happen
What interpretation fits generic andand single-case?
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Interpreting probability:a rush courseKolmogorov axiomatisation
1. Probabilities are non-negative numbers2. Every tautology is assigned value 13. The sum of 2 mutually inconsistent sentences
is equal to the probability of their disjunction
Conditional probability:
Bayes’ theorem then follows:
0)(,)()&()|( BP
BPBAPBAP
def
0)(,)(
)()|()|(
APAP
BPBAPABP
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Interpreting probability:a rush course
Classical and logicalP = ratio # of favourable cases / # of all equipossible cases
Physical: frequency and propensityP = limiting relative frequency of an attribute in a reference classP = tendency of a type of physical situation to yield an outcome
Bayesian interpretationSubjectiveEmpirically-basedObjective
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Interpreting probability:a rush critique
Classical and logicalWell suited to games of chance,not quite to express generic causal
knowledgenor individual hypotheses
PhysicalDoes not make sense in the single case,it is of scarce applicability to evaluateindividual hypotheses
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Bayesian interpretationsAn epistemological stanceabout scientific reasoning
We reason according to the formal principle of probability theory
Bayesianism provides an account of how we can/should learn from experience
Probability expresses rational degree of belief
Bayesians disagree as to howdegrees of belief are shaped
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Bayesian interpretationsSubjective
Choose any probability you wish,just preserve coherence
Empirically-basedChoose any probability you wish, preserve coherence,andand incorporate empirical constraints
ObjectiveChoose any probability you wish, preserve coherence,incorporate empirical constraints andand logical constraints
BayesianismBayesianismempirically-based or objective
is the interpretation that best fit CM
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Janus-faced probabilityHistorically tenable
Who’s in the driver’s sit?
Frequency-driven epistemic probabilitiesDegrees of belief are shapedupon knowledge of frequencies
Credence-driven physical probabilitiesCredence in the truth of a propositionfixes the chance of the event(as long as evidence does not contradict)
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Frequency-drivenepistemic probabilities
Account for different types ofprobabilistic causal claims
because they are Janus-faced
Make sense of learning from experience because they incorporate empirical
constraints
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Credence-drivenobjective probabilities
Lewis’ Principal PrincipleLet C be any reasonable initial credence function. Let t be any time. Let x be any real number in the unit interval. Let X be the proposition that the chance, at time t, of A's holding equals x. Let E be any proposition compatible with X that is admissible at time t. Then, C(A | XE) = x.
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The Janus you choosemakes the whole differenceLewis 1986:
Carnap did well to distinguish two concepts of probability, insisting that both were legitimate and useful and that neither was at fault because it was not the other. I do not think Carnap chose quite the right two concepts, however. In place of his ‘degree of confirmation’ I would put credence or degree of belief; in place of his ‘relative frequency in the long run’ I would put chance or propensity, understood as making sense in the single case. The division of labor between the two The division of labor between the two concepts will be little changed by these replacements.concepts will be little changed by these replacements. Credence is well suited to play the role of Carnap's probability1, and chance to play the role of probability2 .
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In fact:C-D accounts leave room to arbitrariness
Different agents with different initial credence functions
will assign different chances to the same event
Additionally:In the single-case the goal is notto claim credence about chance but to express a rational degree of beliefin an individual hypothesis
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The case against?Bayesian probabilities are degrees of belief.So is causal knowledge given up?
Not quite: empirical constraints ensure thatprobabilities are not devoid of empirical content
Rational decision making does not usesfrequencies at all …
Perhaps, but still, experience informs ourdegrees of beliefs in many ways
Exchangeable sequences show that‘probability does not exist’
Well, there’s nothing ‘metaphysical’in making an epistemic use of frequencies
The full-blown advantage of objective Bayesianismobjective Bayesianism
In the design and interpretation of testsIn guiding action
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Hypothesis testingBasic idea
To compare hypotheses with dataElements
Null hypothesis: observed variation is chancyAlternative hypothesis: observed variation is
realTest statistic
Null hypothesis is accepted/rejecteddepending on the chosen p-value
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Probability in hypothesis testing
From frequentist viewpoint:Evaluate the probability to obtain the sampleif the hypothesis is true;
‘The probability of a hypothesis’has no meaning because it is single-case
But Bayesians cancan evaluatethe probability of a hypothesis
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Consider:‘The unknown parameter lies in (1, 2)with confidence level 95%’
This means:If we draw many samples of the same size andbuild the same interval around ,then we can expect that 95% of confidence intervalswill contain
This is not the probability that will lie in (1, 2)
Freedman et al: “Chances are in the sampling procedure,not in the parameter.”
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What’s the probabilityof a hypothesis?
Frequentists cannotcannot answer this
But Bayesians cancan:A long-lasting project to rephrase(frequentist) statistical problems in Bayesian
termsJaynes, Florens & Mouchart, Drèze & Mouchart,Berger, Bernardo, …
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… the probability of which hypothesis?
Hypothesis testing teststhe Null H against the Alternative H
Acceptance/rejectiondirectly concerns Null H, not Alternative H
Objective Bayesianism treat both Hs on a par,unless evidence suggests doing otherwise
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A problem of error?Type I error
Reject Null H when it is in fact trueType II error
Accept Null H when it is in fact false
Type I is weightier than type II:Be more cautious to accept the H thatthe observed variation is true rather than chancyBe more cautious in accepting ‘causal’ variations
Solution: restrict rejection region
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But what about the probability of the AlternativeAlternative hypothesis?
Type I error has probability , aka p-valueType II error has probability
depends but is not determined by
The frequentist does not treat them on a par
The BayesianAssigns different and only based on evidenceChooses Null or Alternative H on the basis of the
posteriorposterior
Not a problem of error anymore,a problem of evidenceevidence
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Probability isthe very guide of life
That is, probability guides decisions
DecisionsTo accept/reject a hypothesisTo take action
Policy-makingAbout individuals
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Individual decisions
Concern the single case,e.g. a medical patient
Bayesian probabilities are applicable in the single case
† Frequencies aren’t
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Decisions in policy makingFrom the Policy Hub of UK Civil Service
Policy making is: 'the process by which governments translate their political vision into programmes and actions to deliver 'outcomes' - desired changes in the real world'. This concern with achieving real changes in people's lives is reflected in the Government's overall strategy for improving public services published in March 2002 Promoting good practice in policy making is fundamental to the delivery of quality outcomes for citizens and to the realisation of public sector reform. Policy makers should have available to them the widest and latest information on research and best practice and all decisions should be demonstrably rooted in this knowledge.
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To sum up and concludeCausal claims in CM
Generic / Single-case
Both are probabilistic
What interpretation of probability?
Not just a Bayesian,but an objective Bayesianobjective Bayesian