Objective Bayesianismand causal modellingin the social sciences

Federica RussoPhilosophy, Louvain & Kent

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Overview

Probabilistic causal claims in CMGeneric / Single-case

Interpreting probability:a rush course

Interpreting probability in CM:Frequency-driven epistemic probabilitiesObjective Bayesian probabilities

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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;Expresse 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