[attention getter] what interpretation of probability for causal modelling in the social science? [need] causal analysis in the social science by means of *probabilistic* models issue of the *interpretation* of probability arises [task] my doctoral research on causal modelling in the social science had to take a stance as to the interpretation [main message] causal models in soc sc. need empirically-based Bayesian probabilities. Particularity of such interpretation = pluralist
Objective here is not to go into the technicalities of Bayesianism (of any sort). I suppose familiarity with that, so, focus on an argument to support a Bayesian interpretation –in part, empirically-based- for probabilities in causal modelling. Give the context where this interpretation has to be preferred State the question more precisely – why the question arises Then offer an argument to support empirically-based Bayesianism for causal modelling Discuss some possible objections
1. Start from the datum that causal models are *probabilistic* Just briefly recall that in these models probabilities are used in an essential manner and that a different interpretation of the error term in the structural equation leads to a different conception (deterministic – indeterministic) of the causal relation thereby described. 2. Causal models are used to make inferences at the population-level as well as the individual-leves. Pop-lev average causal relation that hold virtually for every individual Ind-lev particular causal relation for a specific individual Both cases causal relations are *probabilistically* characterized. Let me give you an example. Epidemiological studies about effects of tobacco consumption on lung cancer. Question 1 (pop-lev) what’s the *average* effect of tobacco consumption on lung cancer. Question 2 (ind-lev) what’s the proba that harry will develop lung cancer, given that he smokes / what’s the probability that smoking caused him cancer. We have to make sense of the following: At pop-lev tobacco consumption causes (meaning, rises proba) lung cancer. So, how can it be the case that 2. a heavy smoker never develop cancer; or 3. some non smokers do develop cancer.
The issue of the interpretation arises bcz if probability has to be applied, it also has to be intepreted. Choice of interpretation of proba is not just matter of academic debate in the phil of proba. It arises straightforwardly once we interpret the structural equation and seems to be particularly relevant when causal inference is involved. Twofold conception of causality put forward by several authors in order to overcome difficulties of probabilistic theories of causality in case of improbable consequences or outlier observation. Twofold causality has been supported by claiming that there are two different causal mechanisms operating at the two levels. *I* try to support twofold causality by resorting to a twofold conception of probability. From an epistemological point of view there is a sensible understanding of the levels *without* different causal mechanisms. Pop-lev causal rel are represented by joint proba distributions Ind-lev causal rel are *realizations* of these joint proba distributions Intuition: two levels of causation – two different interpretations at the two levels. Pop-lev objective intepr, proba are frequencies Ind-lev subjective interpr, proba are degrees of belief
Empirically-based Bayesianism. Basic idea is that degrees of belief at the ind-lev are empirically determined by frequencies that hold at the pop-lev. OSS. Soundness of this interpretation depends on soundness of a dual concept of probability. I’ll argue in favour of this pluralist conception. But first recall very briefly varieties of Bayesianism. Literature typically distinguish 1. subjective, 2. empirically-based, 3. objective B. Different versions would agree on 2 basic principles: 1. scientific reasoning is reasoning in accordance with formal principles of proba calculus, 2. B.is learning from experience. Where Bs disagree is on the constraints to put on prior proba. Subj coherence (= obedience to the axioms) is nec *and* suff condition for prior assignments. Obje we need further constraints, empirical and logical. Empirically-based mid-way position: empirical constraints may be sufficient.
As the roman god janus has two faces, one looking at the past and the other at the future, so probability has two sides, one subjective and the other objective. Duality of proba analyzed at length by Hacking. Since epistolary exchange between Pascal and Fermat, proba meant as degree *and* as tendency of a chance device to produce stable relative frequencies. Hacking’s thesis challenged by Gillies: duality appeared a bit later. Surely started with Laplace, it traces back to Poisson, Cournot … So, janus-faced proba is *historically* tenable.
I want to show now that Janus-faced proba is used in many contemporary *subjectivist* accounts. First, borrow Salmon’s distinction btw F-D of subjective proba and C-D of objective proba. Question to be addressed = given that there are 2 kinds of proba, how do they relate to each other? have to understand relation btw frequencies and degrees of belief. F-D frequencies play major role in determining subjective proba C-D objective assignments are based on belief. Whole point is to make clear, in Salmon’s words, who’s in the driver’s sit.
Different subjectivists e.g. Ramsey, Carnap, Salmon, van Fraassen, Shimony of different sorts e.g decision making theorists, obective Bayesian, (pretended) subjective Bayesian ultimately rely on frequencies in order to shape degrees of belief. I can’t go into the details but the point is that they all use *two* concepts of probability. They take the subjective concept of degree of belief and use frequencies in order to shape personal probabilities.
Lewis’ PP also intends to tie the objective and subjective side of proba. However, it goes the opposite direction. Briefly and inormally, PP says that the chance of an event A equals the degree to which the agent believes A. in other words, credence determines chance. Difference btw F-D and C-D accounts is subtle but fundamental. In F-D degrees of belief are take as fundamental, but they’re shaped by resorting to frequencies. In Lewis’ proposal chance is entirely and uniquely determined once our *credence* in the truth of the corresponding proposition is fixed, and evidence does not contradict this credence. The difference btw F-D and C-D is exactly what vindicates the choice for empirically-based B. According to PP different agents with different initial credence functions may ultimately assign *different* objective probability values. But this opens the door to arbitrariness. An empirically-based or objective B. does not run this risks exactly because agents have constraints to put on the priors. Refer again to PP to underline the substantial difference btw the use of obj and subj proba in F-D and in C-D. In-lev goal *not* to claim credence about chance i.e. credence about the strength of smoking of producing cancer in Harry. But to establish a reasonable degree of belief about the hyp the smoking caused Harry’s cancer, *based* on empirical evidence = frequencies that hold at the pop-lev. In other words, degrees of belief are frequency-driven.
Now, discuss a couple of possible objections. First come from the staunch objectivist and the persuaded realist. Despite I eventually favour an empirically-based B, as a matter of fact, emp-based B deals with degrees of belief. Degrees of beliefs are features of an agent’s mental state, hence are in sharp contraposition to objective probabilities. Objection: does the adoption of a subjectivist perspective lead to drop any ambition to acquire knowledge about the world? (and particularly about causal relations?) The answer, in a nutshell, is no, as long as degrees of belief are based on the available evidence i.e. what we know about the world. This ensures that subjective proba are not devoid of empirical content.
Second objection from the staunch subjectivist. In fully personalistic approaches coherence is nec and suff for assignments of prior proba. So, the objection: aren’t frequencies just a pedagogical need? Carnap gives an interesting answer (Philosophical Foundations of Probability, §50-51, 41C). 1. we can do without frequencies if inductive logic is accepted. Reason 1: prob1 (subj) can be explicated as estimate of proba2 (obj). So, if proba2 is know, then proba1 just equals this value. Reason 2: even if proba2 is unknown, we can still compute proba1 as estimate of the unknown proba2 from *frequencies* in the sample
But the subjectivist can still play a last card: the exchangeability argument. Briefly and informally, De Finetti shows that different agents may start with different prior probabilities, but, as evidence accumulates, their posterior proba will tend to converge, thus giving the illusory impression that objective probability exists. It seems to me, this is not a decisive argument against an objective interp, specially if the inter defended is frequentism. Bcz empirically-based B does not reify proba in metaph propensities, nothing metaphysical in using frequencies to shape degrees of belief.
In the end, true, subj B is not ruled out in principle, BUT it makes sense of “learning-from-experience” only after priors have been allocated. On the contrary, especially in proba assignments for individual cases, previous knoweldge *does* play a role, and this is why emp based B, since it imposes empirical constraints on priors, is preferable.
To sum up. In the soc sc, causal analysis is performed by means of probabilistic models. Consequently, we have to take a stance concerning the interpretation of proba. The choice is even more difficult since causal inference are performed both at the level of pop and of ind. So, have to choose an int that fits both levels equally. If no other interpretation does the job, empirically-based B seems to. Emp-based B allows a frequentist int at the pop-lev and a subj inter at the ind-lev. However, those degrees of belief turn out to be frequency-driven. This ensures that degrees of belief are not arbitrary nor devoid of empirical content. A pluralist conception of proab is historically tenable, and it can be shown that many recent subjectivist account actually resort to frequencies to shape degrees of belief. For all these reasons, empirically-based B seems to be the int that better fits causal modelling.
Want now to draw some general conclusions. The issue of the interpretation has relevant consequences and implications on germane issues in philosophy of science. E.g. causality, scientific realism, confirmation … The choice can depend on the context. And even in the same context different interpretation may be equally plausible. That is, we should favour a pluralism in philosophy of probability. The question is not what’s the *right* interpretation, rather what is the interpretation that *better* fits a given context. We have to provide good *philosophical* arguments for or against a given interpretation in a given context.
Empirically-based Bayesian probabilities in the social science Federica Russo Institut Supérieur de Philosophie, Université Catholique de Louvain
Overview <ul><li>The context </li></ul><ul><ul><li>Causal modelling in the social science </li></ul></ul><ul><li>The question </li></ul><ul><ul><li>What interpretation of probability </li></ul></ul><ul><ul><li>for causal models? </li></ul></ul><ul><li>The answer </li></ul><ul><ul><li>Empirically-based Bayesianism </li></ul></ul><ul><li>Objections </li></ul><ul><ul><li>From the objectivist and the realist </li></ul></ul><ul><ul><li>From the subjectivist </li></ul></ul>
Causal modelling in the social science <ul><li>Probabilistic causal models </li></ul><ul><li>Causal inference: </li></ul><ul><ul><li>Population-level and individual-level </li></ul></ul> The context The question The answer Objections
What interpretation of probability? <ul><li>If probability has to be applied, </li></ul><ul><li>it has to be interpreted </li></ul><ul><li>Two levels of causation, </li></ul><ul><li>two interpretations of probability </li></ul>The context The question The answer Objections
Empirically-based Bayesianism <ul><li>Varieties of Bayesianisms </li></ul><ul><ul><li>Subjective </li></ul></ul><ul><ul><li>Empirically-based </li></ul></ul><ul><ul><li>Objective </li></ul></ul><ul><li>Same motivations, </li></ul><ul><li>different constraints on priors </li></ul>The context The question The answer Objections
Janus-faced probability <ul><li>Historical claims: </li></ul><ul><ul><li>Hacking and Gillies </li></ul></ul><ul><li>A pluralist conception of </li></ul><ul><li>probability since Laplace </li></ul>The context The question The answer Objections
Janus-faced probability <ul><li>Who’s in the driver’s sit? </li></ul><ul><li>Frequency-driven accounts of </li></ul><ul><li>subjective probability </li></ul><ul><li>vs. </li></ul><ul><li>Credence-driven accounts of </li></ul><ul><li>objective probability </li></ul>The context The question The answer Objections
Frequencies in subjectivist accounts <ul><li>Ramsey (1926) </li></ul><ul><li>Carnap (1950) </li></ul><ul><li>Salmon (1967) </li></ul><ul><li>Van Fraassen (1983) </li></ul><ul><li>Shimony (1988) </li></ul>The context The question The answer Objections
Credence about chance <ul><li>Lewis’ Principal Principle </li></ul><ul><li>The difference between </li></ul><ul><li>F-D and C-D accounts </li></ul><ul><li>is subtle but fundamental </li></ul>The context The question The answer Objections
The staunch objectivist and the persuaded realist rebut … <ul><li>Do degrees of belief lead to drop </li></ul><ul><li>any ambition to acquire knowledge </li></ul><ul><li>about the world? </li></ul>The context The question The answer Objections
The staunch subjectivist rebuts… <ul><li>Aren’t frequencies just </li></ul><ul><li>a pedagogical need? </li></ul>The context The question The answer Objections
The staunch subjectivist plays her last card … <ul><li>De Finetti’s exchangeability argument: </li></ul><ul><li>objective probabilities are illusory </li></ul>The context The question The answer Objections
Bayesian quarrels <ul><li>Subjective Bayesianism is not ruled out </li></ul><ul><li>in principle, but makes sense of </li></ul><ul><li>learning from experience too late </li></ul>The context The question The answer Objections
To sum up <ul><li>I gave a context: </li></ul><ul><li>causal modelling in the social science </li></ul><ul><li>I claimed: </li></ul><ul><li>empirically-based Bayesianism </li></ul><ul><li>is a sensible answer </li></ul><ul><li>I offered an argument: </li></ul><ul><li>a pluralist interpretation fits </li></ul><ul><li>causal modelling </li></ul>
To draw some conclusions <ul><li>Interpreting probability has </li></ul><ul><li>consequences on other issues </li></ul><ul><li>The choice of the interpretation </li></ul><ul><li>is highly contextual </li></ul><ul><li>To argue pro or contra an interpretation </li></ul><ul><li>is the philosopher’s job </li></ul>
Selected references <ul><li>Carnap R. (1950), The Logical Foundations of Probability . </li></ul><ul><li>De finetti B. (1937), “Foresight. Its logical laws, its subjective sources”. </li></ul><ul><li>Gillies D. (2000), Philosophical Theories of Probability . </li></ul><ul><li>Hacking I. (1975), The Emergence of Probability . </li></ul><ul><li>Lewis D. (1971), “A Subjectivist Guide to Objective Chance”. </li></ul><ul><li>Ramsey F. P. (1926), “Truth and Probability”. </li></ul><ul><li>Salmon W.C. (1967), Foundations of Scientific Inference . </li></ul><ul><li>- (1988) “Dynamic Rationality”. </li></ul><ul><li>Shimony A. (1988), “An Adamite Derivation of the Principles of the Calculus of Probability”. </li></ul><ul><li>Van Fraassen B.C. (1983), “Calibration: a Frequency Justification for Personal Probabilities”. </li></ul><ul><li>Williamson J. (2005), Bayesian Nets and Causality. </li></ul>
<ul><li>Comments? </li></ul><ul><li>Mailto: </li></ul><ul><li>[email_address] </li></ul><ul><li>Interested in further material? </li></ul><ul><li>Check out: </li></ul><ul><li>http://www.geocities.com/russo_fede </li></ul>Thanks to Jon Williamson, Andrea l’Episcopo and two referees for useful comments
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