Gent March05 Presentation

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  • [Attention getter] why should squirrels trouble probabilistic theories of causality? {Introduction} [Need] probabilistic theories of causality all face the problem of the levels of causation raised by the squirrel example. Although traditionally this problem is solved by distinguishing two levels of causation, I’ll argue that this solution rests on an unnecessary metaphysical assumption [task] so, in this talk I will try to sketch a different understanding of the levels of causation, from an epistemological perspective. [take home message] in a nutshell, we need two levels of causation (population-lev vs. individual-lev), but we do not need any metaph assumption to support the two levels. Also, causal relations at population-level are needed to evaluate individual-level causal hypotheses.
  • State the problem precisely, and state the questions to be addressed and the perspective  epistemology I’ll start by presenting the phil debate. We shall see that phil debate turns around the so-called squirrel example, that is supposed to trouble PT, and the solution is to distinguish 2 levels of causation Then I will discuss Eells’ account of token causality and explain why it gives a metaph answ to an epist question. I shall also present two arguments to show that by updating probabilities the squirrel does not become a positive cause. The solution is fine but the argument unsatisfactory, so Epistemology works to give (1) a better understanding of the levels avoiding any unnecessary metaph assumption. And (2) also gives us a principle to connect the levels.
  • Different causal claims suggest that levels of causation ought to be distinguished. The first is general and refer to the population, the second and third make explicit reference to a specific individual. Although not explicit in these claims, we deal with a probabilistic characterization of causal rel. The first, says that smocking raises the prob of lung cancer. When individuals are concerned, things get more complex. two diff situations might be distinguished: we deal with post hoc proba: 0 or 1, and we want to assign causal effectiveness to a causal factor. Ex: harry did develop cancer  what is the causal effectiveness of smocking? given that an individual belongs to a given population, we wonder how probable is that she will develop a certain desease. Ex: what’s the probability that harry will develop cancer? we are dealing with probabilistic th. of causality  bear in mind: (i) the truth of a general causal claim does not imply the truth of the corresponding token; and (ii) a token causal claim is not necessarily an instantiation of the corresponding general one. This means that there is not always a perfect correspondence between what happens at the population and individual level, and this is why the problem of the levels arises. Two epistemological questions. Are there lev of caus? From epistem pdv it means not whether in nature there are diff causal mechanisms for groups or for individuals. But, if our knowledge of causal rel at the two levels is obtained by different epistemic processes. If there are different levels we have also to connect them. Let’s now have a look at the philosophical debate.
  • Apparently the golf environment fascinated various philosophers since the 70s. A first ex was to show a difficulty for Suppes’ prob th., i.e. when we have improbable concequences. [Rosen’s ex: mediocre golfer, in his approach shot hit tree-limb – spectacular results of a birdie- advance of play low prob, after the shot estimate even lower] modified version  problem of negative causes, and consequently, to raise the problem of the levels. [Squirrel ex: golf ball rolling twds cup-prob high- squirrel pops up kicks it – improbably enough number ricochets-birdie- squirrel caused ball fall in, even though, in gen, squirrels lower prob of birides] Explain the difficulty raised: at the pop lev squirrels’ kick are negative causes, at the individual lev the squirrel’s kick seems to be a positive cause. The two probability inequalities at the 2 lev. do not match … whence the trick. How can be the case that the same event – namely, “the squirrel’s kick” – be a negative and a positive causal factor? It follows, PT is in trouble, if we want to claim that the squirrel’s kick is a negative cause and that the squirrel’s kick actually caused the ball to fall in one.
  • Tell the solution of the squirrel example: 2 levels of causation  epis says OK Argument to justify the distinction  2 different causal mechanisms at the 2 levels  metaph answer to epist question, details later. We are here concerned with epistemology. So, 2 doubts: causation really operates differently at the two levels? )  metaph question I won’t deal with. do we need to postulate 2 different mechanisms?  epist question  intuitively no. We need to start afresh, and to develop a better understanding of the levels which does not postulate different mechanisms. Before to accomplish this task, I shall spend some time on Eells’ account …
  • Eells account of token causation, explain why it is a metaphysical answer (somehow unjustified) The relation to analyse is: ‘x’s being X’ at <tx, sx> caused ‘y’s being Y’ at <ty, sy> , x takes place at time and place <tx, sx> , y takes place at time and place <ty, sy> . x , y = actual instantiations or tokens of the corresponding types X and Y Main point Eells wants to make  probability-increase idea that is appropriate for token causation; Roughly: look at how the probability of the later event y actually evolves around the time of the earlier and later events. depiction of the evolution of the probability value of the effect y , we actually draw the trajectory of its probability. Squirrel ex  the right trajectory is supposed to show that the probability of hole-in-one, after the squirrel’s kick, increases. Hence,squirrel regained qua positive causal factor. [explain trajectory] Eells: despite the fact that the probability that the probability of a birdie will take this trajectory is quite low, in this particular token case the probability of the birdie does take this trajectory. This is, in Eells’ opinion, the most plausible way to enforce the intuition that the squirrel token-caused the birdie. OSS. This trajectory is our best guess to regain the squirrel as positive cause
  • The trajectory might “pictorially” match intuitions about the squirrel BUT we have to calculate prob ans show that there has been an increase. Eells: pob traj can be evaluated in crucial points provided that we have complete and exact knowledge Problem: P(Y) at each t = P(y|K&W)  the causal background context and on other causal factors. OSS: To say that in principle the state of the world would allow an extremely precise calculation of the probability of the birdie does not mean that – de facto – we are able to calculate it! Second problem. to evaluate P(y) at different times along the trajectory, we have to consider *different hypothetical probability distributions* that hold respectively at tx , or at any time t between tx and ty , or at ty . OSS: of course probabilities in the single case are evaluated thanks to knowledge of the corresponding population, but: the very possibility of being able to evaluate at every time the corresponding hypothetical distributions and, the fact that if at different times we evaluate probabilities in terms of different hypothetical distributions, we miss the connection to the actual population the token under analysis belongs to Anyway, I take up eells challenge and try to see whether the squirrel’s kick raises the probability of the birdie. To do that I shall first formulate the squirrel example more precisely and than apply two different strategies to update probabilities. I will show that in neither of them the squirrel’s kick is a positive cause.
  • The story goes as usual. A team of experienced golf players is training, and from time to time squirrels pop up. After a large number of shots we can allocate priors as follows … Suppose then we get at the following results … P(birdie) = .9 P(kick) = .15 Since the cond prob is less than the uncond prob, according to probabilistic theories, the squirrel’s kick is a negative cause. BUT, as I told, in this particular situation we feel that the squirrel did play a causal role in the story, consequently we wish to update probability accordingly.
  • B denotes Birdie, in K n is encoded the sequence of data observed in the past concerning the squirrel’s kick. K is the new fact we wish to update upon.  how we should append the new datum K to the past data K n , and perform a global computation of the impact of the entire set K n+1 on B.  rule of recursive updating. Compare to Bayes th.: prior belief P(B) x likelihood P(B|K). In this rule, P(B|K n ) plays role of prior belief, it summarise pas experience. To update, multiply this old belief by likelihood function P(K|K n ,B) measuring the probability of the new datum K given B and past experience. Bayes th expressed in terms of odds and likelihood: O(B|K)= O(B) L(B|A). Prior odds measures predictive support, likelihood represent retrospective support given to birdie by past experience of kicks. Rule recursive updating can be expressed in terms of odds and likelihood as well. NB: new posterior odds is still lower than prior odds  the squirrel’s kick is not a positive cause. BUT this updating strategy is not of much help – we are not allowed to express that contrary to our prior beliefs the squirrel IS positive cause and should be assigned high probability value accordingly.  second strategy
  • We think that the squirrel does have positive causal effectiveness, although we allow for other factors (e.g. expertise of the player, wind, etc) to play a causal role.  new prob value of K is not 1 After assignment of new value for K, we wish to ascertain how it influences other beliefs. In particular, we expect P n (K) to raise old belief P(B).  Jeffrey’s Rule = principle of probability kinematics.  we can compute new P(B) once we know new P(K) and nothing else. NB the high prob value assigned to K is not enough to raise the new probability value of B. OSS. Plausibility of this updating strategy rests on adoption of empirically-based Bayesianism.  probabilities can be interpreted differently at population and individual level. Pop lev  rel frequencies or proportions in the pop at hand  how often golf ball falls into the cup, Ind lev  subjective prob shaped on this empirical knowledge Subjective prob are empirically-based At ind lev prob are not frequencies – best understanding of single-case prob is subjective. NB this is not counterintuitive: background knowledge  squirrels are negative causes. A single observation cannot be so powerful to reverse our priors. Hundreds of squirrels would be needed to raise new prob value of B, updating after updating. In the end, I am delighted to announce you that squirrels are not positive causes. Nonetheless …
  • P(birdie&kick) is small but it definetively exists. Of course it is more probable that the ball will fall in one if the squirrel does not kick it, rather than if it does, but this possibility exists. This remark leads us to the following understanding of the levels … So, to the first question, epist reply: Yes, there are levels Causal conclusions drawn from stat analyses concern the pop Individual causal relations take place between individuals within the population Pop-lev  causal rel *represented* by joint prob distributions Ind-lev causal rel are *realizations* of joint probability distributions Info encoded in prob distr at pop-lev already includes the possibility of joint realization of birdie&kick Since this poss exists, I don’t see any real trouble for PT, PT developed in order to avoid nec&suff causes This provides sensible understanding of non instantiated cases or outlier observations. Ind. causal relations as realizations of joint prob functions epistemologically makes sense of the golf ball falls into the cup, despite the squirrel’s kick. (More importantly) Harry didn’t develop lung cancer, in spite of his heavy smoking. OSS:I’m not reducing causes to probabilities, I’m suggesting that to account for the levels epistemology no need to postulate different mechanism. OSS2:why is this a stat understanding? Bcz knowledge of causal rel through *correlational* data Next question: how to relate the levels?
  • Stated in Sober’s 1986, CP entirely neglected in the philosophical literature. Although stat analyses allow to conclude that smocking causes lung cancer, we might be interested in the chance of Harry’s getting lung cancer, or how likely is that it was his smoking to cause him to get cancer. OSS: These are two different issues. concerns the way priors are assigned in particular cases, while is concerned with the likelihood of hypotheses. CP deals directly with the (ii)
  • OSS: while the rough idea seems correct, but sober’s formulation seems imprecise, compared to what is stated before. Rough idea: S(token hypothesis) is *proportional* to the strength of the causal relation at the population level, the support of C(t1)  E(t2) is proportional to the strength of C  E . Formulation: if *strength* of C  E is m , then the *support* of C(t1)  E(t2) , given that C(t1) and E(t2) occurred, will be * m* as well. My doubt: not sure that Sober is interested in the support of the individual hypothesis given that the relata occurred, as the formulation states. Rather, Sober seems interested in the support of the individual hypothesis, given that at the population level the corresponding causal relation holds. So, I tried to reformulate CP in order to make it consistent with the rough idea
  • OSS: new notation c , e  causes and effects at the population level. m  magnitude or strength of the population level causal relation. Consider S(H|E): E  evidence, or prior knowledge when interested in the assessment of *support of individual hypotheses*. Hence E denotes the population-level causal relation, having a certain strength m . H  ind-level causal relation; in particular, H refers to the ind hypothesis under analysis. CP reformulated, says that if we know that a certain population-level causal relation has strength m , then the support of the corresponding token hypothesis, given this evidence, is *proportional* to m . The stronger the pop-lev causal rel, the higher the supp of ind-lev causal rel
  • Strength m The choice of the strategy to assess the causal strength not of secondary importance BUT not main worry: other possible measures of correlation as long as different strategies are coherent (= detect causal strength in same direction positive or negative) it does not matter which one is preferred. Replacement of equality better mirrors the intuition expressed in the rough idea: the higher the strength of the population level relation, the greater the support of the ind relation. OSS: different measures of correlations will give different values, so it seems odd to say that the value of S *equals* m
  • CP relates to the concept of LIKELIHOOD, in particular, Fisher (1922)  likelihood = predicate of hypotheses in the light of data. So, the likelihood of a hypothesis relative to an observation indicates how well the observations support the hypothesis. Rephrased L(H|E) = likelihood of the hypothesis that c token-caused e , given that the corresponding population-level causal relation holds and has a certain strength m . Edwards (1972)  S(H|E) is def as the ln of the likelihood ratio. CP  *S*(H|E) is *proportional* to the causal *strength*, namely the stronger the population-level causal relation, the more likely the hypothesis about ind-level causal relation. OSS: likelihoods allow comparison among different sets of data or of different hypotheses. Ex: support of the hypothesis that Harry got lung cancer because of his heavy smoking, given that in the population smoking is a considerable risk factor. Or, support that asbestos caused him lung cancer, given the same results at the population-level. [rapp likelihood – probability] Edwards (1972)  likelihood L(H|E) proportional to P(E|H) ; the likelihood ratio of two hypotheses on some data is the ratio of their likelihoods on the same evidence
  • IN SUM: Sober interprets the support S(H|E) as ln of likelihood ratio of the hypothesis relative to the available evidence. strength of pop-lev causal rel assessed by measure of correlation. Then, knowledge of pop-lev causal rel used to allocate priors in the ind case and to establish the support of the corresponding token hypothesis.  CP* allows to relate epistemologically the two levels of causation: from prior knowledge of pop we are able to assess the support of the corresponding ind-lev hypothesis.  CP* epistemologically relates pop-lev causal claims and ind-lec causal claims. OSS1: CP* not to assign a strength to the ind-lev causal relation; rather, CP* tells us what is to be expected at the ind-lev. OSS2: this understanding of the levels is metaph parsimonious  it abstains from postulating different causal mechanisms at the two lev. OSS3: distinction btw epist - metaph of causality is fundamental  ind -lev causation clearly ontologically primitive: pop are made of individuals among which token causes and not type causes take place. BUT pop-lev causation is epistemologically primitive. –bcz- causal conclusions drawn from statistical analyses concern pop, not directly indiv in the population, to assign prob values to individuals we refer to probability values pertaining to the corresponding pop
  • Squirrels show: that we definitively need two levels of causation, but we do not need to postulate that causation operates differently at the two levels. Of course this does not hold only for squirrels … If we take up the epistemological perspective, we obtain a meaningful understanding of the levels: (i) metaphysically parsimonious, (ii) accounts for discrepancies at the individual and population level. CP says how we should conceive of the relation of the two levels. (i) ruled out the possibility that from knowledge of the population we can know the strength of the individual causal relation, (ii) but from this knowledge we can assess the likelihood of individual hypotheses.
  • Gent March05 Presentation

    1. 1. Levels of Causation Federica Russo http://www.geocities.com/russo_fede Institut Supérieur de Philosophie, UCL, Belgium Centre for Philosophy of Natural and Social Science, LSE, UK
    2. 2. Overview <ul><li>The problem, the questions, and the perspective </li></ul><ul><li>The philosophical debate: </li></ul><ul><ul><li>probabilistic squirrels </li></ul></ul><ul><ul><li>twofold causality </li></ul></ul><ul><li>Metaphysical answers to </li></ul><ul><li>Epistemological questions </li></ul><ul><ul><li>Eells’ account of token causality </li></ul></ul><ul><ul><li>Bayesian squirrels </li></ul></ul><ul><li>The Epistemology of Causality </li></ul><ul><ul><li>a statistical understanding of </li></ul></ul><ul><ul><li>the levels of causation </li></ul></ul><ul><ul><li>the Connecting Principle </li></ul></ul>
    3. 3. The Problem <ul><li>Compare: </li></ul><ul><li>- Smocking causes lung cancer. </li></ul><ul><li>- Harry’s smoking caused him cancer. </li></ul><ul><li>- In spite of his heavy smocking, Harry </li></ul><ul><li>didn’t develop cancer. </li></ul>The Questions Are there levels of causation? If so, how they relate to each other?
    4. 4. Probabilistic Squirrels <ul><li>Population-level Causation </li></ul><ul><li>P(B|K) < P(B) Squirrels’ K icks are </li></ul><ul><li>negative causes </li></ul><ul><li>for B irdies </li></ul>Individual-level Causation P(b|s) > P(b) The squirrel’s k ick is a positive cause for the b irdie
    5. 5. Twofold Causality solves the problem: <ul><li>Two Levels of causation: </li></ul><ul><li>population -- individual </li></ul><ul><li>Two Different Causal Mechanisms </li></ul><ul><li>at the two levels </li></ul>
    6. 6. Metaphysical answers to Epistemological questions
    7. 7. Token probability trajectories face two problems: <ul><li>The exact specification of the causal context and of all the factors involved </li></ul><ul><li>The reference to several hypothetical populations </li></ul>
    8. 8. Bayesian Squirrels P(birdie) = .9; P(kick) = .15; P(birdie&kick) = .05 A team of experienced golf players is training … squirrels pop up… eventually, we allocate priors as follows: squirrels are negative causes P(birdie|kick) = .333 < P(birdie)
    9. 9. Recursive Bayesian updating The new posterior odds is lower than the prior odds The squirrel’s kick is not a positive cause !
    10. 10. Bayesian updating by Jeffrey’s Rule P n (K) = .95 P n (B) = .333 .95 + .944 .05 = .363 < P(B) The new value of B is lower than the old value The squirrel’s kick is not a positive cause !
    11. 11. A statistical understanding of the levels of causation <ul><li>At the population-level , </li></ul><ul><li>causal relations are represented by </li></ul><ul><li>joint probability distributions </li></ul><ul><li>At the individual-level, </li></ul><ul><li>causal relations are realizations </li></ul><ul><li>of joint probability distributions </li></ul>P(birdie & kick) is tiny, but it exists
    12. 12. The Connecting Principle <ul><li>The rough idea: </li></ul><ul><li>If a token event of type C is followed by a token event of type E , then </li></ul><ul><li>the support of the hypothesis that the </li></ul><ul><li>first event token-caused the second increases </li></ul><ul><li>as the strength of the property causal relation of C to E does. </li></ul>
    13. 13. The Connecting Principle <ul><li>The formulation: </li></ul><ul><li>If C is a causal factor of magnitude m </li></ul><ul><li>for producing E in a population P , then </li></ul><ul><li>S { C t1 token caused E t2 | C t1 and E t2 occurred </li></ul><ul><li>in the population P } = m </li></ul>
    14. 14. The Connecting Principle reformulated <ul><li>If c is a causal factor of magnitude m </li></ul><ul><li>for producing e in a population P , then </li></ul><ul><li>S(H| E) is proportional to m . </li></ul><ul><li>Notation: </li></ul><ul><li>c, e : causes and effects at the population-level </li></ul><ul><li>E : evidence  population-level causal relation with strength m </li></ul><ul><li>H : hypothesis  individual-level </li></ul><ul><li>causal relation </li></ul>
    15. 15. Connecting the Levels, let’s discuss about: <ul><li>the strength m </li></ul><ul><li>the replacement of equality by proportionality </li></ul><ul><li>the measure of support S(H|E) </li></ul>
    16. 16. Likelihood and Support <ul><li>The Fisherian concept of Likelihood: </li></ul><ul><li>A predicate of hypotheses </li></ul><ul><li>in the light of data </li></ul><ul><li>Edwards’ definition of support : </li></ul>
    17. 17. Epistemological Morals <ul><li>The Connecting Principle allows to compute </li></ul><ul><li>the likelihood of individual hypotheses, </li></ul><ul><li>not their strength – this is metaphysics </li></ul><ul><li>Population-level causation is </li></ul><ul><li>epistemologically primitive </li></ul><ul><li>Individual-level causation is </li></ul><ul><li>ontologically primitive </li></ul>
    18. 18. To sum up … <ul><li>We need two levels of causation, </li></ul><ul><li>but we do not need two causal mechanisms </li></ul><ul><li>The Epistemology of causality: </li></ul><ul><ul><li>Gives us a meaningful understanding of the levels </li></ul></ul><ul><ul><li>Tells us how to relate the levels </li></ul></ul>
    19. 19. <ul><li>References </li></ul><ul><ul><li>Edwards A.W.F. (1972), Likelihood , Cambridge University Press. </li></ul></ul><ul><ul><li>Eells E. (1991), Probabilistic Causality , Cambridge University Press. </li></ul></ul><ul><ul><li>Fisher R.A. (1922), “On the mathematical foundations of theoretical statistics”, Philosophical Transactions of the Royal Society. </li></ul></ul><ul><ul><li>Hacking I. (1965), Logic of Statistical Inference , Cambridge University Press. </li></ul></ul><ul><ul><li>Sober. E. (1986), &quot;Causal Factors, Causal Influence, Causal Explanation&quot;, Proceedings and Addresses of Aristotelian Society, 60, pp. 97-136. </li></ul></ul>Comments? Mail to: [email_address]

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