Abstract
Philosophers interested in the theoretical consequences of predictive processing often assume that predictive processing is an inferentialist and representationalist theory of cognition. More specifically, they assume that predictive processing revolves around approximated Bayesian inferences drawn by inverting a generative model. Generative models, in turn, are said to be structural representations: representational vehicles that represent their targets by being structurally similar to them. Here, I challenge this assumption, claiming that, at present, it lacks an adequate justification. I examine the only argument offered to establish that generative models are structural representations, and argue that it does not substantiate the desired conclusion. Having so done, I consider a number of alternative arguments aimed at showing that the relevant structural similarity obtains, and argue that all these arguments are unconvincing for a variety of reasons. I then conclude the paper by briefly highlighting three themes that might be relevant for further investigation on the matter.
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Notes
A reader might contest this, noting that numerous accounts of generative models as structural representations have been offered (e.g. Kiefer and Hohwy 2018, 2019; Wiese 2018). I am aware of the existence of such accounts. However, to me they all seem to presuppose the success of Gładziejewski’s (2016) original argument, to then improve on it in various ways.
Notice that this is a theoretical assumption, that can be theoretically contested (e.g. Orlandi, 2016).
Importantly, model inversion is not essentially an approximated process. So, by saying that a generative model is inverted one has not yet shown how the intractability problem is solved. Since the technical details are fairly complex (see Bogacz, 2017) and will not matter for my argument, I will not sketch them here. An anonymous referee has my gratitude for having noticed this issue.
Many thanks to the anonymous reviewer who noticed that the original formulation of this point was too strong.
And in fact, according to PP, action too requires the inversion of a generative model (see Friston 2011).
Here, “relevant” means “the one adopted by Gładziejewski”. Other definitions of structural similarity are surely possible (e.g. Shea, 2018, p. 117). However, since my focus here is Gładziejewski's argument, I will stick to the definition Gładziejewski favors.
Alternatively, structural representations can be defined as: “A collection of representations in which a relation on representational vehicles represents a relation on the entities they represent” (Shea, 2018, p. 118). This definition stresses the important fact that each element of the structural representation is also a representational vehicle, whose content is determined by the relevant structural similarity in which it participates. For instance, each object on a map stands for (i.e. represents) an environmental landmark, and spatial relations among objects on a map represent spatial relations holding among the corresponding landmarks. Notice that such a nesting of representational vehicles is entirely unproblematic: after all, both a sentence and the words forming it are representational vehicles in an entirely intelligible sense. Notice further that according to both Shea’s and Gładziejewski’s definition, the relevant structural representation is the entire structure of related objects, rather than any single part of that structure. That is, the elements (V and ℜV) of a structural representation need not be, on their own, structural representations.
This might or might not require a representational consumer. Gładziejewski asserts that a consumer is necessary in his (2015); but his (2016) does not mention consumers. Shea's definition of exploitable structural similarity (to which Gładziejewski adheres) does not require consumers, so I will skip them here. Notice that I adapted the notation in Shea's definition for the sake of orthographic consistency.
In its original formulation, the definition of decouplability also mentions representational consumers (see Gładziejewski, 2015). Here, I omit them for the reasons given in the previous footnote.
Notice that I'm not claiming that graphical models are not structurally similar to their targets. They are. As clarified above, a structural similarity might hold among any pair of entities. Yet, the relevant class of structural similarities that can be used to vindicate (a) is the class of structural similarities holding between representational vehicles and their targets; and graphs are not representational vehicles.
Or, at least, so it seems. To be honest, I believe that Kiefer is no longer committed to the claim that generative models are structural representations. Rather, it seems to me that Kiefer is committed to some form of functional role semantics. To be precise, Kiefer (2017, p. 12) seems to endorse the claim that generative models are structural representations. However, he seems to have quickly changed his mind about this point, as, in numerous later publications (Kiefer and Hohwy, 2018, p. 2393; 2019, p. 401–403; Kiefer, 2020, footnote 19) he takes the content of generative models to be determined by internal functional roles rather than by the structural similarity holding between a generative model and its target. I will more directly confront this issue in the main text, when dealing with the fourth alternative argument for (a). Many thanks to an anonymous referee for having pressed me on this issue.
Notice that the scope of my claim is restricted to PP and the usage of graphical models in the PP literature. I make no claim on how graphical models are used in the rest of cognitive neuroscience (and related disciplines). Many thanks to the reviewer who advised me to be more cautious on this point.
Importantly, this seems exactly how Kiefer interpreted these models, see (Kiefer 2017, pp. 12–16).
The same two points seems to apply whether these models are intended to capture computational processes more generally, given that computational processes are often defined in terms of representations (see Fodor, 1981; Shagrir, 2001; Ramsey, 2007, pp. 68–77; Sprevak, 2010; Rescorla, 2012). This latter point, however, is not entirely uncontested (e.g. Piccinini, 2008).
Or both. The formulation in terms of “either (i) or (ii)” is due to the fact that it seems to me that one might interpret weighted connections either as parts of a structural representation (i.e. as members of V) or as relations among parts (i.e. as relations in ℜV).
Notice that I’m not denying that weight matrices encode the invariant relations that hold among the elements of the domain upon which the network has been trained to operate. I am only denying that there is a mapping from weight matrices (that is, from individual weights or sets of weights) to relations such that the mapping satisfies (i) or (ii). In simpler terms, I’m not denying that weight matrices represent invariant relations, I’m only denying that weight matrices represent invariant relations by being structurally similar to the target domain (or by participating in some relevant structural similarity with the target domain). Notice, importantly, that not all invariant relations need to be encoded in a vehicle that is structurally similar to its target. We might, for instance, stipulate that the sign “§” represents the fact that my father is n years older than me. If we do so, then “§” encodes an invariant relation holding between me and my father, and yet there just seems to be no structural similarity holding between “§” and the target it represents. Many thanks to an anonymous reviewer for having pressed me on this point.
Many thanks to an anonymous referee for having raised these objections.
And even if my confidence were misplaced, I would concede the point for the sake of discussion.
At this point, it might be tempting to wonder whether the relevant definition of structural similarity could be relaxed, so as to allow connections to be elements in the structural similarity in spite of the lack of any intelligible one-to-one mapping holding between them and the elements of the target domains. As an anonymous reviewer aptly noticed, O'Brien and Opie’s (2004) definition of structural similarity is not the only one on the market, and at least some alternative formulations do not require a one-to-one mapping (e.g. Kiefer and Hohwy, 2019, p. 400; Shea, 2018, p. 117). As far as I can see, the mapping can be relaxed so as to allow many elements of the vehicle to map onto one element of the target. However, I believe the mapping cannot be relaxed so as to allow one element of the vehicle to map onto many elements of the target. To see why this is the case, consider a minimal structural representation constituted by two objects a*and b* in a relation R*. Suppose that R* corresponds to a relation R, that a* corresponds to an element a and that b* maps onto two elements b and c. Now, given this mapping, the representation is accurate when aRb is the case. It is also accurate when aRc is the case. Hence, misrepresentation occurs only when both aRb and aRc are not the case. But, if this is correct, then the representation represents (aRb or aRc), and its content is disjunctive and thus indeterminate. Yet, it is widely assumed that a successful theory of content must deliver us determinate content. So, it seems to me that, in order for a structural-resemblance based theory of content to be successful, it must exclude one-to-many mappings. Now, the issue with weights in connectionist systems is that they seem to map one-to-many: each weight encodes information about many targets (see Clark, 1993, pp. 13–17; Van Gelder, 1991, pp. 42–47; Ramsey, Stich and Garon 1991, pp. 215–217 for early renditions of this point). Hence, it seems that each weight is bound to map onto many targets, generating the problem with content determinacy. Notice, importantly, that the same line of reasoning holds even when the relations map onto many. To see why, consider a modified version of the minimal structural representation considered above, in which a* maps onto a, b* maps onto b and R* maps onto two relations R and F. Again, given this mapping, misrepresentation occurs only when both aRb and aFb are not the case, and so the representation represents (aRb or aFb). In both cases, the disjunction problem is brought about by the claim that one-to-many mappings might constitute structural similarities, so as to circumvent the problems raised by superspositionality. Hence, we should not allow one-to-many to constitute structural similarities. Thanks to an anonymous referee for having pressed me on this point.
More precisely, it is common in the PP literature most heavily influenced by Friston’s free energy principle. Many thanks to an anonymous referee for having noticed this imprecision.
One might contend this verdict is premature. For the elements (i.e. objects of V and relations of ℜV) of structural representations are representational vehicles in their own right (e.g. Shea, 2018, p.118; Ramsey, 2007, p. 79, footnote 3). Thus, claiming that the brain as a whole is a structural representation might in principle justify the claim that the relevant elements of the structural similarity (i.e. patterns of activation) are representations too, leading to a vindication of epistemic representationalism. I believe that the problem with this line of reasoning is the following: the brain-world structural similarity Friston envisages is not defined over patterns of activation in the brain. Rather, it is defined over the anatomical structure of the brain. The relevant elements in the structural similarity are not patterns of activation. Hence, this way of vindicating (a) fails to properly vindicate the epistemic representationalist claim.
Here, one might be tempted to simply reject condition (c) and accept that entire brains are structural representations of the environment. As far as I can see, this is a legitimate move. However, it seems quite an ad hoc move. There are good independent reasons to hold that representations are necessarily decouplable from their targets (see Grush, 1997; Webb, 2006; Pezzulo, 2008: Orlandi, 2014, pp. 120–134). Moreover, abandoning (c) would likely make Gładziejewski’s account of structural representations far too liberal, as Gładziejewski himself acknowledges (Gładziejewski, 2016, p. 571).
To be clear, Kiefer and Hohwy do not explicitly set out to defend “whole brain” representations. However, it seems to me that their account entails that the whole brain is a structural representation, at least insofar they take the entire causal network instantiated by the brain to be the relevant structural representation. A reviewer noticed that this characterization of Kiefer and Hohwy’s position might be too ungenerous, since, strictly speaking, Kiefer and Howhy speak only of connections among cortical regions. Hence, their position is best described as a form of “whole cortex”, rather than “whole brain” representationalism. However (and the reviewer seems to agree) noticing this does not substantially alter the dialectical situation. So, I will continue to speak of Kiefer and Hohwy as endorsing a form of “whole brain” representationalism, mainly for the sake of simplicity.
Notice, importantly, that Kiefer and Hohwy seem to consider decouplability a necessary feature of representations, see (Kiefer and Hohwy 2019, p. 400).
Of course, individual patterns of activation can be decoupled from the individual target they represent in virtue of the overall brain-world structural similarity. However, to be satisfied, point (c) requires that the entire vehicle of structural representation (in this case, the whole brain) is decoupled from its target (in this case, the world). Thus, noticing that in some cases (e.g. during dreaming) certain patterns of activation are tokened in a way that is functionally independent from the incoming sensory stimulation is not sufficient to vindicate point (c). This is because individual patterns of activations are not the entire vehicle of the structural representations, but rather elements of that vehicle. Thanks to an anonymous referee for having pressed me to clarify this point.
One might object that Kiefer and Hohwy (2018, 2019) should be counted as defending structural representations because they stress that the relevant structural similarity is relevant for the system’s success. As I understand it, the problem with this line of argument is that the same holds true also for causal theories of content (see Nirshberg and Shapiro, 2020, pp. 6–7; Facchin 2021, pp. 9–12).
To be precise, Friston suggests that the dorsal horn of the spinal cord embodies an inverse model. But an inverse model still seems to me to count as a model.
Presumably, single, well identified, regions of the cortical hierarchy.
Arguably, Kiefer and Hohwy’s (2018, 2019) account is one such account, given Kiefer and Hohwy’s commitment to functional role semantics. However, given that they seem to take (wrongly, in my opinion) functional role semantics as a kind of structural resemblance, it is very hard to evaluate their proposal as an alternative to structural representations-based accounts of PP.
A reviewer noticed that structural representations are less popular in the philosophy of mind, where teleosemantic theories of content still appear to dominate. It might be worth noticing, at this point, that teleosemanticists are increasingly willing to incorporate some forms of structural similarity in their accounts (e.g. Millikan, 2020; Neander, 2017). Moreover, the standard notion of exploitable structural similarity has been elaborated within a roughly teleosemantic framework (Shea, 2018). Yet, nothing, in my argument, hinges over this.
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Acknowledgments
The author wishes to thank the participants to the IUSS WIP seminars for useful feedback on the essay. Thanks also to Niccolò Negro and Giacomo Zanotti for their useful comments on some previous version of this essay. A special thanks goes to Eleonora, for her moral support.
Funding
This work has been funded by the PRIN Project “The Mark of Mental” (MOM), 2017P9E9N, active from 9.12.2019 to 28.12.2022, financed by the Italian Ministry of University and Research.
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Facchin, M. Are Generative Models Structural Representations?. Minds & Machines 31, 277–303 (2021). https://doi.org/10.1007/s11023-021-09559-6
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DOI: https://doi.org/10.1007/s11023-021-09559-6