Abstract
Understanding probabilities as something other than point values (e.g., as intervals) has often been motivated by the need to find more realistic models for degree of belief, and in particular the idea that degree of belief should have an objective basis in “statistical knowledge of the world.” I offer here another motivation growing out of efforts to understand how chance evolves as a function of time. If the world is “chancy” in that there are non-trivial, objective, physical probabilities at the macro-level, then the chance of an event e that happens at a given time is \(<1\) until it happens. But whether the chance of e goes to one continuously or not is left open. Discontinuities in such chance trajectories can have surprising and troubling consequences for probabilistic analyses of causation and accounts of how events occur in time. This, coupled with the compelling evidence for quantum discontinuities in chance’s evolution, gives rise to a “(dis)continuity bind” with respect to chance probability trajectories. I argue that a viable option for circumventing the (dis)continuity bind is to understand the probabilities “imprecisely,” that is, as intervals rather than point values. I then develop and motivate an alternative kind of continuity appropriate for interval-valued chance probability trajectories.
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Notes
I will assume that the basic form of these chance probabilities is unconditional; this is in contrast to general probability, which applies to classes of event and whose basic forms is conditional. I assume this for clarity and convenience only: the continuity issues I deal with here are not sensitive to whether the physical probabilities of chance are analyzed in the standard Kolmogorovian way or some other way, with a different conditionalization and/or with conditional probabilities as the basic form; see for example Hájek (2003).
As an anonymous reviewer points out, there are other equally (perhaps more) plausible ways of understanding the token squirrel kick’s effect on the probability trajectory, e.g., it might be understood as “immediately” raising the probability if focusing on how it “immediately” improves the balls trajectory, or understood as smoothly lowering it if focusing on the chance of the squirrel collision becoming more and more likely. But nothing here turns on these particulars—as long as some sort of discontinuity is plausible in some setting, which defensible understandings of some quantum examples provide. The intent of the example here is only to illustrate clearly a chance discontinuity. The point drop rendering above (following Eells) is particularly helpful (though not essential) for my purposes because it exhibits two different discontinuities. I note too that since all most all (excepting Eells) probabilistic analyses of causation are explicitly neutral with respect to the continuity question, the mere possibility of discontinuities needs to be explicitly accommodated or ruled out, since the possibility itself undercuts such analyses. Both of these points will be taken up at length below.
A jump discontinuity is one in which the left- and right-hand limits exist, but are not equal. The other two possibilities, that the left and right hand limits exist and are equal, or that one (or both) fail to exist are called removable and essential discontinuities, respectively. The essential discontinuity case will come up again below.
Eells, for one, does recognize that chance could also be represented in a continuous fashion, with the probability continuously approaching one from below. But he writes that his analysis does not “pay attention” to whether the trajectory is continuous at the time the event occurs (Eells 1991, 294, note 6). See Peressini (forthcoming) for an argument to the contrary.
A possible exception to this might be an irreducibly probabilistic (point) event, e.g., whether element U-238 will emit an electron by time t. According to prevalent interpretations of quantum physics this probability will be bound away from one right up to the instant it happens, at which point it will “jump” to one. I discuss this case below.
I will not distinguish in what follows between causally and probabilistically relevance. The questions of if and how these notions coincide is of course at the center of the debate about whether causation can analyzed probabilistically. For the purposes of this paper, probabilistic relevance is sufficient, since the concern here is with probabilistic analyses of causation.
It is important to stress that since the argument requires the construction of a series of events upon which Y probabilistically depends, it most obviously succeeds when there is a space-time process leading up to or constituting the event Y, as there is in Example 1. And as a consequence, the argument does not necessarily apply to certain classes of quantum events, which (under certain interpretations) fail to have such probabilistically relevant antecedent events; this is as it should be as there is nothing incoherent about such quantum-level examples. While there is debate about whether all macro-level examples of causation need to have such an intermediate process, even accepting a pluralistic view, e.g., Hall (2004), it is sufficient for my argument here that it work for the large class of macro-level cases (like Example 1) in which there is such a mediating process. I owe thanks to an anonymous reviewer for help with this point.
I note that CDJP does not give rise to any novel problems from those that follow from DJP, and in fact may be seen as following from DJP, since DJP entails that there be discontinuous “jumps” in an event’s trajectory at each moment its “causes” occur. But while intuitive, the actual argument to establish this entailment is far from trivial; see Peressini (forthcoming). Furthermore, it is important to distinguish between the two principles because the rationales for introducing each are different, namely causal concerns versus more general ontological concerns regarding determinism, chance and event ontology.
Menzies’ account, like Kvart’s, while not explicitly addressing continuity, does implicitly constrain discontinuities. He builds on Lewis’ (1986) counter-factual analysis in terms of unconditional probabilities. Menzies requires that causally related events c and e be probabilistic dependent—which amounts to there being intermediate events corresponding to any finite set of intervening times between the times of c and e such that the actual probability of each of the intervening events is significantly higher than it would have been had the immediately preceding event in the set not happened. This effectively requires the chance function to be monotonically increasing, and turns out to be an implausibly strong condition; Menzies (1996) himself disavows even an amended version of this theory. As I draw out below, the point probability framework and this continuity bind often force one to choose between stability in the chance function and such overly strong constraints on it.
Hitchcock (2004, 414) reports Ned Hall’s suggestion that one evaluate the probability of an effect shortly before the time at which the effect occurs; Hitchcock also outlines there a related proposal of his own.
The original version is still available online at http://web.mit.edu/gradphilconf/2008/A%20Probabilistic%20Analysis%20of%20Causation.pdf.
The term “imprecise probability” traces back to Walley’s (1991) foundational work in the area.
Other prominent orderings are center-point and radius less than, center-point less than and radius greater than, lower point and center point less than, upper point and center point less than. See Guerra and Stefanini (2011).
Even if the quantum jump from r to 1 at time \(t_y\) in Fig. 3 is defined to be the interval value [r, 1], the function is still not path-continuous. Were there compelling motivations, there are ways to accommodate such jumps within a path continuous framework, e.g., by relaxing it to require only left or right path continuity or by defining P(t) to be [r, 1] at \(t_y\) and all subsequent times, but as things stand gaplessness works equally well without the complications.
For example, one may weaken the definition of Lower SC by requiring only that one (rather than all) elements in the domain set at a point have converging sequences. So an interval function \(F: {\mathbb {R}} \rightarrow {\mathbb {IR}}[0,1]\) is Weak LS Continuous at \(x_0 \in {\mathbb {R}}\) if and only if there is a \(y_0 \in F(x_0)\) such that for any sequence \(x_i \in {\mathbb {R}}\) with \(\{x_i\} \rightarrow x_0\) there exists a sequence of elements \(y_i \in F(x_i)\) with \(\{y_i\} \rightarrow x_0\).
It should be noted that semicontinuity for real valued functions like \(\underline{f}\) and \(\overline{f}\) is distinct from, though not unrelated to, semicontinuity for generalized set-valued functions and interval-valued functions like F. In particular, semicontinuous real-valued functions may well be gappy in a way that is precluded in \({\mathbb {IR}}[0,1]\) or more general set-valued spaces.
An interesting question for further work is whether the G-continuity of \(F=[\underline{f},\overline{f}]\) is equivalent to [(\(\underline{f}\) being LSC) and (\(\overline{f}\) being USC)] or [(\(\underline{f}\) being USC) and (\(\overline{f}\) being LSC)], but not both of \(\underline{f}\) and \(\overline{f}\) being one of LSC or USC.
I stress that it can be ignored only if P is continuous; it is not true that the inequality holds only if P is continuous.
It would be an interesting project in itself to recast all of the particular idiosyncratic details of the various competing probabilistic accounts of causation in term of imprecise probabilities, including reassessing each of the examples and arguments they employ.
Even when the temporal index is explicitly expressed as in (1) (as opposed to placed out of sight within a “variables taking on values” approach), as far as I can tell the temporal index is simply “carried along,” that is, the continuity properties of chance as a function of time are not addressed. I note too that as mentioned above, Menzies’ (1989) account does indirectly rule out the possibility of any (and therefore any discontinuous) drops in chance, but at the cost of an implausibly strong monotonicity requirement, which in part leads him to disavow the account altogether (Menzies 1996).
Of course the tradeoff with this move is that it would rule out certain kinds of “gapless” functions, i.e., those that are LSC and not USC. Recall Sect. 5.2.
References
Anguelov, R., & Markov, S. (2007). Numerical computations with hausdorff continuous functions. In T. Boyanov, S. Dimova, K. Georgiev, & G. Nikolov (Eds.), Numerical methods and applications, lecture notes in computer science (Vol. 4310, pp. 279–286). Berlin: Springer.
Anguelov, R., Markov, S., & Sendov, B. (2006). The set of hausdorff continuous functions–The largest linear space of interval functions. Reliable Computing, 12(5), 337–363.
Aubin, J., & Frankowska, H. (1990). Set-valued analysis. Birkhäuser: Systems & Control.
Augustin, T., Coolen, F. P., de Cooman, G., & Troffaes, M. C. (2014). Introduction to imprecise probabilities. New York: Wiley.
Eells, E. (1991). Probabilistic causality. New York: Cambridge University Press.
Flores-Franulic, A., Chalco-Cano, Y., & Roman-Flores, H. (2013). An ostrowski type inequality for interval-valued functions. In IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS), 2013 Joint, pp. 1459–1462.
Glynn, L. (2010). Deterministic chance. British Journal for the Philosophy of Science, 61(1), 51–80.
Glynn, L. (2011). A probabilistic analysis of causation. British Journal for the Philosophy of Science, 62(2), 343–392.
Glynn, L. (2014). Unsharp best system chances. http://philsci-archive.pitt.edu/10239/.
Guerra, M., & Stefanini, L. (2011). A comparison index for interval ordering. In IEEE symposium on foundations of computational intelligence (FOCI), 2011, pp. 53–58.
Hájek, A. (2003). What conditional probability could not be. Synthese, 137(3), 273–323.
Hall, N. (2004). Two concepts of causation. In J. D. Collins, E. J. Hall, & L. A. Paul (Eds.), Causation and counterfactuals (pp. 181–276). Cambridge, MA: MIT Press.
Hitchcock, C. (2004). Do all and only causes raise the probabilities of effects? In J. D. Collins, E. J. Hall, & L. A. Paul (Eds.), Causation and counterfactuals (pp. 403–418). Cambridge, MA: MIT Press.
Ismael, J. (2011). A modest proposal about chance. Journal of Philosophy, 108(8), 416–442.
Kvart, I. (2004). Causation: Probabilistic and counterfactual analyses. In J. D. Collins, E. J. Hall, & L. A. Paul (Eds.), Causation and counterfactuals (pp. 359–386). Cambridge, MA: MIT Press.
Kyburg, H.E. (1999). Interval-valued probabilities. In The society for imprecise probability: Theories and applications. http://www.sipta.org/documentation/interval_prob/kyburg.
Lewis, D. (1986). Postscripts to ‘causation’. In D. Lewis (Ed.), Philosophical papers (Vol. II, pp. 172–213). Oxford: Oxford University Press.
Li, F.C., & Li, J. (2010). Ordering method of interval numbers based on synthesizing effect. In International conference on machine learning and cybernetics (ICMLC), 2010, (vol. 1, pp. 108–112).
Menzies, P. (1989). Probabilistic causation and causal processes: A critique of Lewis. Philosophy of Science, 56(4), 642–663.
Menzies, P. (1996). Probabilistic causation and the pre-emption problem. Mind, 105(417), 85–117.
Moore, R. (1966). Interval analysis. Prentice-Hall series in automatic computation. Englewood Cliffs: Prentice-Hall .
Moore, R. (1979). Methods and Applications of Interval Analysis. Studies in Applied and Numerical Mathematics. Society for Industrial and Applied Mathematics.
Moore, R., Kearfott, R., & Cloud, M. (2009). Introduction to Interval Analysis. SIAM e-books. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104).
Noordhof, P. (1999). Probabilistic causation, preemption and counterfactuals. Mind, 108(429), 95–125.
Northcott, R. (2010). Natural-born determinists: A new defense of causation as probability-raising. Philosophical Studies, 150(1), 1–20.
Peressini, A. (forthcoming) Causation, probability, and the continuity bind.
Rosen, D. A. (1978). In defense of a probabilistic theory of causality. Philosophy of Science, 45(4), 604–613.
Sober, E. (2010). Evolutionary theory and the reality of macro probabilities. In E. Eells & J. Fetzer (Eds.), The place of probability in science (pp. 133–162). Netherlands: Springer.
Walley, P. (1991). Statistical reasoning with imprecise probabilities. New York: Chapman and Hall.
Weichselberger, K. (2000). The theory of interval-probability as a unifying concept for uncertainty. International Journal of Approximate Reasoning, 24(2–3), 149–170.
Acknowledgments
Key parts of this work were done on sabbatical in Berlin 2012-13; I thank Dörte and Frieder Middelhauve and Jodi Melamed for helping make the sabbatical possible (and fun) along with Michael Pauen and the Berlin School of Mind and Brain for providing a place to work, and a stimulating and gemütlich community. A version of this paper was presented at the Imprecise Probabilities in Statistics and Philosophy Conference at Ludwig-Maximilians-Universität München, June 28, 2014. I am grateful for the comments and suggestions I received from participants. I thank also this journal's anonymous reviewers for their helpful comments.
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Peressini, A.F. Imprecise Probability and Chance. Erkenn 81, 561–586 (2016). https://doi.org/10.1007/s10670-015-9755-9
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DOI: https://doi.org/10.1007/s10670-015-9755-9