Abstract
Many approaches to evidence amalgamation focus on relatively static information or evidence: the data to be amalgamated involve different variables, contexts, or experiments, but not measurements over extended periods of time. However, much of scientific inquiry focuses on dynamical systems; the system’s behavior over time is critical. Moreover, novel problems of evidence amalgamation arise in these contexts. First, data can be collected at different measurement timescales, where potentially none of them correspond to the underlying system’s causal timescale. Second, missing variables have a significantly different impact on time series measurements than they do in the traditional static setting; in particular, they make causal and structural inference much more difficult. In this paper, we argue that amalgamation should proceed by integrating causal knowledge, rather than at the level of “raw” evidence. We defend this claim by first outlining both of these problems, and then showing that they can be solved only if we operate on causal structures. We therefore must use causal discovery methods that are reliable given these problems. Such methods do exist, but their successful application requires careful consideration of the problems that we highlight.
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Notes
We will be more precise about this claim in Sect. 3.
The two exceptions are (1) a constant-valued (and so not really variable) latent; and (2) a latent that is a cause of only one other variable (including its own future state) at each moment in time (and so essentially functions as an additional source of noise).
For example, suppose device \(D_1\) measures once every two seconds, and device \(D_2\) once per three seconds. In this case, the measurement timescale is every one second, but with numerous missing values (e.g., \(D_1\) missing values in every other timestep).
And different causal relations within a system can have different timescales. In general, we assume that the causal timescale of a system is the greatest common divisor of the timescales of the causal relations within the system.
Thanks to an anonymous reviewer for suggesting this idea.
Event-related fMRI designs can potentially get much tighter temporal resolution, though only under significantly stronger assumptions.
Thanks to an anonymous reviewer for noting that similar phenomena occur with resting state fMRI data.
For clarity, we omit the additional edges that represent connections due to common causes that are unobserved due to undersampling. For example, if \(u=2\), then 1 and 2 at the second (measurement) timestep have an unobserved common cause—namely, 1 at the previous (unmeasured) timestep.
Specifically, if the greatest common divisor of the lengths of the simple loops in the SCC is 1, then it will converge; otherwise, it will oscillate.
Technically, a conflict occurs between candidate \(\mathcal{G}^{1}\) and \({{\mathcal {H}}}\) when \(\forall u \{ \mathcal{G}^{u} \nsubseteq \mathcal{H} \}\).
The RASL output always contains at least as many graphs as the MSL output, as the latter considers only a single u. In practice, though, their output sets are frequently equal.
More precisely, not statistically significant from zero.
This would happen if, e.g., the underlying causal structure is: \(X \rightarrow C \rightarrow B \leftarrow A \leftarrow Y\).
IOD actually does perform a type of limited evidence amalgamation.
Technically minded readers might observe that there are usually infinitely many “extensions” in the static case as well. For example, if \(X \leftarrow L \rightarrow Y\) for latent L is possible, then the same graph with two latent common causes \(L_1, L_2\) will also fit the data. Thus, the dynamical case does not seem harder in this regard. However, static latents can typically be “collapsed” together (in a technical sense) in a way that these dynamical latents cannot. That is, underdetermination in the static case involves graphs that are the same in many key respects; in the dynamical case, the graphs are quite different.
We conjecture that this is actually an “if and only if” when \({{{\mathcal {G}}}}_{\mathbf{V } \cup \mathbf{L }}\) is a minimal extension.
ION would need significant modifications, though, since there is no a priori way to “match” newly introduced latents across graphs. A computationally hopeless algorithm would be to run ION on all possible relabellings of the latents, and then choose the result that minimizes a suitable criterion, such as number of resulting edges. Clearly, a more efficient version would need to be developed.
More generally, merging at the structural level will be superior whenever there is shared structure but varying parameters (Tillman and Spirtes 2011).
We omit details for reasons of space.
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Acknowledgements
Thanks to two anonymous reviewers for their valuable comments and feedback. Thanks to Jianyu Yang for his contributions to the MSL algorithm; Cynthia Freeman for collaboration on the RASL work; and Antti Hyttinen, Frederick Eberhardt, and Matti Järvisalo for their work on the constraint-centric formulation of the undersampling problem. Special thanks to Ian Beaver for helping with the code. The latent variable work was conducted in close collaboration with Erich Kummerfeld and Isaac Davis. DD was supported by NSF IIS-1318815 and NIH U54HG008540 (from the National Human Genome Research Institute through funds provided by the trans-NIH Big Data to Knowledge (BD2K) initiative). SP was supported by NSF IIS-1318759 and NIH R01EB006841. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
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Danks, D., Plis, S. Amalgamating evidence of dynamics. Synthese 196, 3213–3230 (2019). https://doi.org/10.1007/s11229-017-1568-8
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DOI: https://doi.org/10.1007/s11229-017-1568-8