Are Generative Models Structural Representations?

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.

1 Introduction

Predictive processing is a neurocomputational framework surrounded by a number of philosophical disputes. Some of these disputes concern foundational matters: is realism about the theoretical posits of predictive processing warranted (e.g. Colombo et al. 2018)? And, if yes, are these posits representations (e.g. Kirchhoff and Roberston 2018)? Other controversies concern what predictive processing (henceforth PP) entails: does PP support an internalist or externalist view of the mind (Clark, 2017; Hohwy, 2016)? Does it provide a complete account of cognition, or is it unable to account for the systematic nature of human thought and the curiosity so many intelligent animals blatantly manifest (Sims, 2017; Williams, 2018a)? Is consciousness really just the brain’s best guess (Dolega & Dewhurst, 2020; Hohwy, 2013)?

In this latter kind of disputes it is often assumed that PP is an inferentialist and representationalist theory of cognition. Rendered technically, the assumption is that PP revolves around approximated Bayesian inferences drawn by inverting a generative model operating under a predictive coding message-passing scheme (Clark, 2013; Hohwy, 2013). More mundanely, it is assumed that PP revolves around statistical inferences performed leveraging probabilistic models of the world. Models, in turn, are understood as structural representations: vehicles that represent their targets by mirroring the targets’ relational structure (e.g. Williams, 2017, 2018b). The tie between predictive processing and models runs so deep that some have suggested that predictive processing would not be humanly intelligible without them (Clark, 2015).

Here, I challenge this assumption. I claim that the only¹ argument offered to identify generative models with structural representations (Gładziejewski, 2016) is flawed, and that it cannot be easily ameliorated. By doing so, I hope to bring a small contribution to the disputes surrounding the philosophical foundations of PP.

The essay is structured as follows. In Sect. 2, I introduce the theoretical apparatus of PP. In Sect. 3, I introduce structural representations, and summarize the argument Gładziejewski offers to identify generative models with them. In Sect. 4, I turn from exposition to criticism, showing a flaw in Gładziejewski's argument, and claiming that it cannot be easily adjusted. A brief concluding paragraph follows.

2 A Brief Introduction to Predictive Processing

Here, I provide a short introduction to PP. Since PP is now largely known, I will sketch only its most fundamental aspects.²

To successfully orchestrate behavior, an agent’s brain must first determine in which environmental situation the agent is embedded; that is, what are the environmental causes of the energies impacting the agent's transducers. PP assumes that this task is burdened with uncertainty, as sensory states are under-determined in respect to their causes (Friston, 2005).³ Depending on the context, different causes might generate similar sensory states, just as a single object can generate an unruly manifold of different inputs.

To cope with this uncertainty, PP suggests that the brain resorts to a form of Bayesian inference, as it yields an optimal way to determine the most likely cause of a sensory state, given the incoming input and some prior knowledge of how environmental cause generate sensory states (Yuille and Kernsten 2006; Hohwy, 2013, pp. 13–40). Importantly, since exact Bayesian inferences are often computationally intractable, PP suggests that the brain approximates their results by inverting a generative model operating under a predictive coding processing regime.⁴

Generative models are data structures capturing the relations holding between some observable data (here, sensory states) and their hidden causes (here, worldly objects). These models are said to be generative, as the knowledge they embody can be leveraged “from the top-down” to generate expected instances of data (Hinton, 2007a; Danks, 2014, p. 44). Since real sensory states are generated by the nested interaction of multiple causes operating at different spatiotemporal scales, generative models need to be hierarchically organized to capture these nested causal relations. Importantly, this only requires that each hierarchical level l_N treats the hierarchically lower level l_N-1 as a data source, capturing the regularities it displays (see Hinton, 2007b). Lastly, these models must be probabilistic, embodying their causal knowledge formatted in terms of the probability density functions that are required to approximate Bayesian inferences (e.g. Knill & Pouget, 2004).

Predictive coding is a message passing scheme which deploys generative models as follows (Huang & Rao, 2011; Rao & Ballard, 1999). With the exception of the bottommost level, each level l_N of the model generates a prediction signal: an “expected” pattern of activity of l_N-1, which is then conveyed to l_N-1 through a set of descending connections. Hence, collectively, the levels of the model will generate a “downstream” flow of progressively spatiotemporally refined predictions about the incoming sensory inputs, ideally flowing from “higher” associative areas to primary sensory cortices (see Mesulam, 2008). As this signal is received, each level contrasts it with its own actual activity (or the incoming input in the case of the bottommost level), and computes the mismatch between the two. The magnitude of the mismatch, known as prediction error, is then conveyed “upwards”, from l_N to l_N+1, courtesy of a second set of ascending connections. As prediction error is received, each level changes the prediction signal conveyed downwards so as to minimize the incoming prediction error. This process is then iterated until the entire hierarchy reaches a global minimum of prediction error. Since the states of the generative model that best minimize the error correspond to the most likely causes of the incoming sensory signal (given the body of knowledge the generative model encodes), minimizing prediction error inverts the generative model, mapping the signal onto its most likely causes, implicitly realizing a form of Bayesian inference (Clark, 2013, 2016; Hohwy, 2013, 2019; Kiefer & Hohwy, 2019).

This crude sketch calls for significant amendments. Firstly, it is silent upon lateral connections, which allow different, explicitly coded, hypotheses to compete in the interpretation of incoming data (Friston, 2005).⁵ It is also silent on the expected precision of the incoming signals, which constantly modulates the message passing, determining the “impact” of error signals (Hohwy, 2013, pp. 59–74; Clark, 2016, pp. 53–82). Yet, the most significant amendment this sketch needs is the following: this mechanism is not just a mechanism of perception. It is also the engine of action (Hohwy, 2013, pp. 75–96; Friston, 2013a; Clark, 2016, pp. 111–137). To see how, consider the following two points.

First: prediction error minimization can occur under two directions of fit (Shea, 2013). One can change the predictions to make them fit the input, as sketched above. But one can also “keep the predictions still”, and force the input to fit them. Secondly, given that the agent’s body is, just like the external world, a source of sensory signals (and given that these two sources interact: moving towards an object will change the stimuli the object generates), the generative model must also model the agent's body (Hohwy, 2015). Mashing these two observations together yields the gist of how PP accounts for action: agents act by predicting specific bodily signals, to then cancel out the error relative to these predictions through movement (Adams, 2013). Actions appear thus to be generated by self-fulfilling predictions.

PP thus casts action and perception as two complementary sides of the same computational process of prediction error minimization. Given that processes of error minimization are inferential processes, as demonstrated by the brief analysis of the account of perception PP offers, this means that action is an inferential process too.⁶ This is why, in the PP literature, action is referred to as active inference. More specifically, an agent engaged in active inference tests its model of the world, seeking sensory evidence to confirm the predictions licensed by that model (Hohwy, 2015, 2016, 2017, 2018).

Importantly, however,active inference has a broader scope than action as usually understood (i.e. bodily movements fulfilling one's intentions). Nothing obliges the self-confirming expectations involved in active inference to be proprioceptive expectations, to be confirmed by bodily movements. They might be visual (i.e. exteroceptive) predictions, and elicit saccades (Friston et al. 2010). Or they might be interoceptive predictions, servicing an agent's homeostatic control (Seth, 2015) and emotional regulation (Seth & Friston, 2016). From this perspective, prediction error minimization is not primarily a tool for accurate perception and goal-directed action. Rather, it appears as a mean to the more fundamental end of maintaining an agent within its physiological bounds of viability. This line of reasoning connects PP to an ambitious framework in theoretical biology, namely the free energy principle (see Friston, 2013b, 2019; Allen & Friston, 2018). But the free energy principle will not be considered here,⁷ and the sketch of PP just proposed seems a sufficient introduction, given the task at hand.

In the next section, I briefly introduce structural representations and then summarize Gładziejewski's (2016) argument to identify generative models with them.

3 Structural Representations and Predictive Processing

3.1 Structural Representations

As hinted at in the introduction, structural representations are representations whose vehicles represent their targets by mimicking the inner relational structure of the targets. Consider, for instance, a cartographic map. It might depict a gulf being north of an isle by placing a certain element, corresponding to the gulf, above a second element, corresponding to the isle. Importantly, the PP literature on structural representations (e.g. Dolega, 2017; Hohwy, 2020; Kiefer & Hohwy, 2018, 2019; Ramstead et al. 2019; Wiese, 2017, 2018; Williams, 2017) points to a single formalized account of structural representations; namely Gładziejewski's (2015, 2016) account. According to Gładziejewski:

A state R of a system S is a structural representation of a target T only if:

• (a) R is structurally similar to T; &

• (b) R guides S's action aimed at T; &

• (c) R can satisfy (b) when decoupled from T; &

• (d) S can detect the representational error of R

Each point calls for clarification.

Point (a) clarifies that structural representations are iconic: their representational properties are (at least partially) grounded in the similarity holding between them and their targets. Yet notice that the relevant kind of similarity mentioned in (a) is structural similarity. The relevant⁸ definition of structural similarity is provided in (O'Brien and Opie 2004, p. 11):

Suppose S_V=(V,ℜ_V) is a system comprising of a set V of objects, and a set ℜ_V of relations defined on the members of V. The objects in V may be conceptual or concrete; the relations in ℜ_V may be spatial, causal, structural, inferential, and so on. […] We will say that there is a second-order resemblance between two systems S_V=(V,ℜ_V) and S_O=(O,ℜ_O) if, for at least some objects in V and some relations in ℜ_V, there is a one-to-one mapping from V to O and a one-to-one mapping from ℜ_V to ℜ_O, such that when a relation ℜ_V holds of objects in V, the corresponding relation ℜ_O holds of the corresponding objects in O.

There are several important things to highlight about this definition. One is that it can be straightforwardly applied to point (a) assuming that R is S_V and T is S_O. Another is that structural resemblance does not require first order resemblance to obtain. In fact, nothing in the definition requires S_V and S_O to have any common property. All they need to share is a common pattern of relations among their elements. Thirdly, the definition of structural similarity is tripartite. For S_V to be structurally similar to S_O, it must be the case that: (i) at least some of the objects of which S_V and S_O are constituted map one-to-one onto each other; and (ii) at least some of the relations holding among these objects map one-to-one onto each other; and (iii) corresponding objects stand in corresponding relations in both S_V and S_O. Notice that (i) to (iii) need to obtain in conjunction. Notice, lastly, that (i) to (iii) obtaining in conjunction entails that S_V is semantically unambiguous in respect to S_O. By this I mean that once the mapping rule is known, it is always in principle possible to determine, for all elements of S_V mapping onto S_O, to which element of S_O each element of S_V corresponds.

According to this definition, a structural similarity might hold among any two systems. However, the relevant structural similarity exhibited to satisfy point (a) must hold between a representational vehicle and the represented target (Kiefer & Hohwy, 2018; O'Brien, 2015). This is entailed by the definition of a structural representation: a representational vehicle that represents a target by being structurally similar to it.⁹ It is thus immediately clear that the relevant structural similarity holds between the vehicle (the concrete particular doing the representing) and the represented target. Therefore, R must be a representational vehicle: a concrete particular encoding content.

Point (b) establishes that structural representations are causally responsible for S's behavior (Gładziejewski & Miłkowski, 2017). Gładziejewski unpacks point (b) in terms of exploitable structural similarity (Gładziejewski, 2015, 2016; Gładziejewski & Miłkowski, 2017). Hence, the relevant structural similarity in (a) must be exploitable¹⁰ (Shea, 2014, 2018, p. 120). Put simply:

R's structural similarity with T is exploitable by S only if:

• (iv) ℜ_V is a set of relations S's downstream computational processing is systematically sensitive to; &

• (v) ℜ_O and O are of significance to S

Condition (iv) requires S to be sensitive, in its downstream processing, to the relevant relations ℜ_V in virtue of which R structurally resembles T. Given that computational processing is a mechanical affair, this requires the relations in ℜ_V, or at least the objects in V upon which ℜ_V is defined, to causally impact the processing of S in some systematic way. Condition (v) requires R to structurally resemble a target that matters to S's purposes; that is, a target which matters to S's computational functioning.

Importantly, exploitable structural similarity is not a reflexive and symmetric relation (see Shea, 2014; Williams & Collings, 2017). For this reason, structural representations are immune to the objections that were fatal to iconic representations defined in terms of first order resemblance (see Goodman, 1969, p. 3–4). Notice further that insofar as exploitable structural similarity determines representational content, R's content is not just causally efficacious (Gładziejewski & Miłkowski, 2017): it is also intrinsic to R's material constitution as a representational vehicle (O''Brien & Opie, 2001; O’Brien, 2015; Lee, 2018), as it is literally inscribed in the physical form of R.

Condition (c) captures the idea that genuine representations function as stand-ins for their targets, enabling S to perform processes aimed at these targets even when they are absent (Grush, 1997; Pezzulo, 2008; Webb, 2006). Gładziejewski unpacks this idea in terms of decouplability, defining it as follows (Gładziejewski, 2015): R is weakly decoupled from T only if R and T are in no causal contact; R is instead strongly decoupled from T only if S and T are in no causal contact.¹¹

Lastly, (d) captures the idea that representations can be semantically evaluated by the systems leveraging them (see Bickhard, 1999). Importantly, representations are often semantically evaluated only indirectly, by assessing how successful they are in guiding action (Gładziejewski, 2015 pp. 78–79; 2016, p. 569). This indirect route of evaluation seems a natural outcome of exploitability. If, as (iv) entails, R's content determines downstream processing in S (and eventually S's behavior), then successful behaviors directly depend on R's content being correct (accurate and/or truthful). Pragmatic successes and failure thus appear as reliable indicators of the semantic status of a system's representational resources.

Before moving forward, it is important to clarify the scope of Gładziejewski’s account of structural representations. Following Chemero (2009: pp. 67–68), it is possible to distinguish between an epistemic representationalist claim and a metaphysical representationalist claim. Bluntly put, the epistemic representationalist claim is the claim that our best explanations of cognition need to posit representations. The metaphysical representationalist claim is instead the claim that cognitive systems contain components that really are representations. The two claims can in principle come apart. A fictionalist about representations, for instance, endorses the epistemic claim while denying the metaphysical one (Sprevak, 2013; Ramsey, 2020; see also Downey, 2018 for a fictionalist interpretation of PP). Gładziejewski’s account of structural representations aims at vindicating both claims (Gładziejewski, 2015: 70).¹² Thus, his account of structural representations succeeds just in case the relevant representational posits of PP (i.e. generative models) satisfy features (a) to (d) and these are the relevant sort of structures identified as representation by our best explanatory practices.

3.2 Generative Models as Structural Representations.

Gładziejewski (2016) holds that his account of structural representations straightforwardly applies to generative models. The general outlook of his argument is as follows:

(P1) Items satisfying conditions (a) to (d) in conjunction are structural representations

(P2) Generative models satisfy conditions (a) to (d) in conjunction; therefore

(C) Generative models are structural representations.

The argument needs little clarification. (P1) follows directly from the definition of structural representations. Thus, (P2) carries alone the whole weight of the argument. Here, I sketch the reasoning Gładziejewski provides to substantiate (P2). I will focus in particular on point (a), as it will be central in the following discussion.

To claim that generative models satisfy condition (a), Gładziejewski (2016, pp. 571–573) reasons as follows. Generative models can formally be treated as Bayesian nets. Bayesian nets are directed acyclic graphs, and thus are graphs: sets of nodes connected by edges. And, in general, graphs are always structurally similar to their targets, because: (i) each node stands for one, and only one, environmental variable, (ii) the edges connecting the nodes map onto some relation holding among environmental variables; and (iii) two nodes are connected by an edge if, and only if, the corresponding variables are in the relation of interest (see Danks, 2014, pp. 39–41). Notice that graphs are not just structurally similar to their targets: they are homomorphic to them; and homomorphism is a special, stronger, case of structural similarity (see O''Brien and Opie, 2004, p. 12).

Gładziejewski (2016, pp. 571–573) also suggests a specific way in which generative models structurally resemble their targets: the nodes of each hierarchical layer l_N in the model map onto the likelihoods of the corresponding sensory states. Interlevel connections between nodes mimic the temporal evolution of sensory states. Lastly, each node maps onto the prior probability of the corresponding worldly cause. The validity of the mapping on offer might be contested (e.g. Wiese, 2017), but the general point Gładziejewski is trying to make is clear: generative models are graphs and, as such, they are structurally similar to their targets. Therefore, they easily satisfy condition (a).

Point (b) is satisfied because generative models can engage in active inference. Recall: during active inference, the model “purposefully” generates a false prediction, to then cancel the prediction error it generates through movement. Thus, generative models are the engines of action, and guide behavior exactly as (b) requires.

To show that generative models satisfy condition (c), Gładziejewski exhibits the following evidence. To begin with, prediction error minimization seems responsible for a variety of paradigmatically offline cognitive phenomena, such as memory and imagination (e.g. Clark, 2016, pp. 84–107). Furthermore, early sensory cortices are involved in acts of imagination (e.g. Albers, 2013). Given that these cortices are commonly taken to be part of the neural machinery implementing our generative model, these studies suggest that generative models can in fact function when decoupled from their targets. Moreover, generative models must be counterfactually deep (Seth, 2014): to function effectively, they need to be able to predict how the sensory signals would change, were the agent to move in such-and-such a way. Given that counterfactual scenarios have no causal powers, our generative models will always encode some information about targets from which we are strongly decoupled.

Lastly, generative models clearly satisfy condition (d). When a hypothesis about the incoming sensory flow is selected and tested in active inference, the prediction error that the hypothesis generated is a measure of its inaccuracy. For instance, when I drink coffee, I (subpersonally) expect a flow of gustatory sensations which I try to bring about through the ingestion of coffee. But if the ingestion of the liquid does not bring about these sensations, the ensuing error will inform me that I'm in fact not drinking a coffee, prompting my generative model to revise its expectations. In this way, the error ensuing from failed active inferences indicates the representational inaccuracy of the tested hypothesis.

This concludes the exposition of Gładziejewski's argument. In the next section, I will attack it, claiming that it does not succeed in equating generative models to structural representations.

4 Generative Models and Structural Representations: An Unjustified Identification

I divide this section in two sub-sections. The first section examines Gładziejewski's argument for (a), and argues that the argument Gładziejewski offers fails in vindicating (a). The second section examines some alternative arguments for (a).

4.1 A Problem in Gładziejewski's Argument for Point (a)

Recall Gładziejewski’s (2016) argument for (a). The argument is as follows: generative models can be thought of as Bayesian graphs, Bayesian graphs (and graphs in general) are structurally similar to their targets; therefore, generative models are structurally similar to their targets. Therefore, (a) obtains.

Yet, I believe that this line of reasoning cannot vindicate (a). This is because the relevant structural similarity exhibited to vindicate (a) must hold between a representational vehicle and a represented target. But graphical models are not representational vehicles. So the argument Gładziejewski provides does not substantiate (a).¹³ Let me unpack.

If one is a realist about representations (as Gładziejewski surely is, see Gładziejewski, 2015, 2016; Gładziejewski & Miłkowski, 2017), then one is committed to the claim that representations are concrete particulars encoding content (e.g. Shea, 2018, pp. 25–43). Accordingly, structural representations are defined as concrete particulars (i.e. representational vehicles) which carry content in virtue of a relevant exploitable structural similarity that holds between them and their representational targets. They are representational vehicles that do the representing by structurally resembling. Hence, the relevant structural similarity must hold between a representational vehicle (a concrete particular) and a represented target. This is why the content of structural representations is supposed to be intrinsic to their material constitution (e.g. O'Brien & Opie, 2001; O’Brien, 2015; Williams & Colling, 2017; Lee, 2018). Their content is intrinsic because it is inscribed in the physical form of the representational vehicle (the concrete particular that does the representing).

But Bayesian nets, and graphs in general, are mathematical objects (e.g. Danks, 2014, p. 40; Leitgeb, 2020). They are defined as a finite set of nodes connected by a finite set of edges (Koski & Noble, 2009, p. 41) and sets, nodes and edges are mathematical objects. Mathematical objects might or might not be particulars (it is irrelevant for the purpose of the argument), but definitely are not concrete. So they cannot be representational vehicles, given that representational vehicles are concrete particulars. Hence, the structural similarity Gładziejewski exhibits cannot be used to vindicate point (a). It just isn’t what point (a) requires.

The same issue can also be appreciated from another point of view. Thus, consider condition (b). Recall that, in order to satisfy (b), the relevant structural similarity that satisfies (a) must be exploitable. Exploitability is partially defined by condition (iv): ℜ_V is a relation S's downstream computational processing is systematically sensitive to. But, as clarified above, for a computation to be sensitive to ℜ_V, either ℜ_V or the objects V upon which ℜ_V is defined, must possess the relevant causal powers needed to systematically orient S's processing. Computational processing is, at the end of day, a physical affair, which is ultimately responsive only to the physical properties of the computational states (e.g. Williams & Collings, 2017, p. 1945). However, the only structural similarity Gładziejewski provides to vindicate (a) is defined over mathematical objects. But mathematical objects do not seem to have the causal powers needed to systematically influence S's downstream processing. Hence, the structural similarity that Gładziejewski exhibits to satisfy (a) cannot be exploitable.

This places Gładziejewski's line of argument in a dire situation: either the structural similarity he shows does not vindicate (a), or it does. But if it does, then (b) fails to obtain. Either way, the argument seems to fail, leaving the claim that generative models are structural representations unjustified.

As I understand it, the overall flaw in Gładziejewski's argument is the following. Structural representations are defined in terms of their vehicle properties. Hence they should be identified at the level of the physical machinery doing the computation (what Marr would call the implementation level). Structural representations are bits of an information processing system literally resembling their target according to an appropriate mapping rule. But graphs, wherever they sit in the explanatory hierarchy of cognitive science, surely do not sit at the level of the physical machinery doing the computing (Danks, 2014, pp. 13–37; 218–221). Hence, Gładziejewski's argument seems to be pitched at the wrong level of the relevant explanatory hierarchy. For this reason, I conclude that Gładziejewski's argument does not justify the claim that generative models are structural representations.

But there might be other means to justify that claim.

4.2 Alternative Arguments for (a)

4.2.1 Alternative Argument 1

The claim that graph theoretic notions cannot sit at the level of the physical machinery might not be warranted. For instance, graph theoretic notions are used to map the connections between different neural regions (e.g. Sporns, 2010). Thus, graph theoretic notions can sit at the level of the physical machinery. This suggests an alternative way to vindicate (a): if graphs are structurally similar to their targets, and these graphs can be transparently mapped onto cortical structures and/or neural activity patterns, then, since structural similarity is a transitive relation, the relevant cortical structures/activity patterns are structurally similar to the target of the graphs. Some (e.g. Kiefer, 2017) defend structural representations along precisely these lines.¹⁴

This alternative argument for (a) is fairly attractive, as it is maximally conservative over the structure of Gładziejewski's original argument. It also nicely integrates with the existing scientific literature on PP, at least insofar some generative models, rendered as Bayesian nets, have been mapped onto cortical structures (e.g. Bastos, 2012; Friston, 2017a). Isn't this sufficient to show that at least these generative models are structurally similar to their targets?

I believe that a negative answer is warranted. This is because the graphs presented in (Bastos et al. 2012; Friston et al. 2017a) and a number of similar publications in the PP literature do not model any worldly target. There is thus no specific worldly target that they represent. So, even if the cortical machinery is in some relevant sense structurally similar to these graphs, there is no third element to which the cortical machinery can be structurally similar to by being structurally similar to these graphs. For this reason, it seems to me correct to conclude that the alternative argument for (a) provided above fails.

But what, then, is the purpose of the graphs in (Bastos et al. 2012; Friston et al. 2017a)? The answer, if I understand the literature correctly, is the following: these graphs are, in a sense, didactic tools, aimed at showing, with a fair degree of approximation, that the cortical machinery is arranged in a way such that it can easily perform the inferential processes PP revolves around (see Bastos et al. 2012, p. 703; Friston et al 2017a, p. 393). In fact, it seems to me that, within the PP literature, graphical models are often deployed to capture the message passing within the brain; that is, how inference is performed (see, for instance, de Vries & Friston, 2017; Friston, 2017b; Friston, 2017c; Donnarumma, 2017; Matsumoto & Tani, 2020; see also Hinton & Sejnowski, 1983 for an earlier model).¹⁵ I believe that this is an important point to notice for two distinct reasons.

First, if these graphical models are intended to be models of the relevant message passing, it seems more natural to suppose they will map onto the cortical machinery performing the inferences, rather than on the representational vehicles manipulated in inferential processes.¹⁶ Secondly, and relatedly, if those graphical models are accurately characterized as portraying the inferential message passing in the brain, it seems to me that they presuppose the presence of some relevant representational vehicle, as inferences are defined over representations.¹⁷ These representations might (but, as far as I can see, need not) be structural representations. However, as these graphical models seem to presuppose the presence of representations, it seems to me that they cannot be invoked to justify one’s representationalist claim, on pain of circularity.

Importantly, I do not believe that the considerations offered above rule out in any way the possibility of using graphical models to justify (a). As far as I can see, one might still resort to a graphical model to argue that at least some representational vehicles in the brain are structurally similar to their targets using the argument by transitivity sketched above. However, to do so, one would need a graphical model depicting some specific worldly target. And, to the best of my knowledge of the PP literature, no such graphical model has yet been proposed.

4.2.2 Alternative Argument 2

Artificial neural networks might provide a different way to leverage graph theoretic notions to vindicate (a). As formal objects, artificial neural networks are graphs. But they are also somewhat plausible sketches of the physical machinery implementing or realizing some given computational process of interest (see Haykin, 2009, pp. 1–18; Rogers & McClelland, 2014). Moreover, at least some artificial neural networks encoding generative models (such as Helmholtz machines) are Bayesian graphs (e.g. Dayan & Hinton, 1996). Therefore, even if these artificial neural networks cannot prove that generative models in the brain are structural representations, they can show that generative models can be structural representations, thereby providing circumstantial evidence in favor of (a) obtaining. If our plausible sketches of the physical machinery encoding generative models are graphs (or at least graph-like), then we have a solid reason to believe the real physical machinery encoding generative models is graph-like. And, given that graphs are structurally similar to their targets, we have a solid reason to believe that (a) obtains. However, I think such a belief would be misplaced. Indeed, it seems to me that a closer consideration of artificial neural networks provides a reason to believe that (a) does not obtain.

To see why, consider first that artificial neural networks are often said to encode generative models in their weighted connections (e.g. Dayan & Hinton, 1996; Hinton, 2014; Spratling, 2016, p. 3).¹⁸ But weighted connections (or, more precisely, weight matrices) are typically considered to be superposed representations. And the vehicles of superposed representations are not structurally similar to their targets. As a consequence, if considering artificial neural networks provides evidence regarding the status of (a), the evidence they provide is not in favor of (a) obtaining.

To elaborate a little, consider first the notion of a superposed representation. A representation R is said to be a superposed representation of two targets T and T' when R encodes information about T and T' using the same set of physical resources. When applied to weight matrices, the idea is that weight matrices superpositionally represent their targets when each individual weight is assigned a value such that the network can exhibit the functionality needed to operate on all its targets (Clark, 1993: pp. 17–19, see Van Gelder, 1991, 1992 for further discussion). For instance, a single net can be first trained to recognize (or generate) instances of T. If the network is then trained so as to recognize (or generate) both instances of T and T', then the weights of the net will encode information about both representational targets, and the weight matrix will represent both in a superposed fashion.

However, in weight matrices: “Each memory trace is distributed over many different connections, and each connection participates in many different memory traces” (McClelland & Rumelhart, 1986, p. 176). So, it seems each individual weight maps onto many different representational targets (or aspects thereof). But if this is the case, then either condition (i) or (ii)¹⁹ of structural similarity are blatantly violated, since they require a one-to-one mapping. As a further proof of their violation, recall that the obtaining of (i) to (iii) in conjunction entails semantic unambiguity (see Sect. 3.1). That is, if (i) to (iii) jointly obtain, it is always in principle possible to tell, for each “bit” of the representational vehicle, which “bit” of the represented target it corresponds to. However, in superposed representations: “It is impossible to point to a particular place where the memory of a particular item is stored” (Rumelhart & McClelland, 1986, p. 70). Superposed representations are thus not semantically unambiguous. Therefore, at least one condition among (i) and (iii) is not met. As a consequence, superposed representations do not support the claim that (a) obtains.²⁰

The argument outlined above can be challenged in two ways.²¹ First, generative models are not encoded in connections alone; they are jointly encoded by connections and activity vectors (e.g. Buckley, 2017, p. 57). Secondly, the definition of structural similarity relevant to the obtaining of (a) quantifies only over some. Thus, noticing that connections do not participate in any one-to-one mapping does not provide an argument to the effect that (a) does not obtain: connections might simply be excluded from the objects V or relations ℜ_V of R participating in the structural similarity. I address these challenges in turn.

Are generative models really jointly encoded by activity vectors and weighted connections as the first challenge suggests? As far as I can see, the answer is positive; and focusing only on weighted connections (as I did above) is a mistake. But, to my excuse, it is a mistake the that the PP literature encourages:

We allowed the network to learn a hierarchical internal model of its natural image inputs by maximizing the posterior probability of generating the observed data. The internal model is encoded in a distributed manner within the synapses of the model at each level. (Rao & Ballard, 1999, p. 80, emphasis added)

The representation at any given level attempts to predict the representation at the level below; at the lowest level this amounts to a prediction of the raw sensory input. It is the backward connections, therefore, that instantiate the generative model. (Shipp, 2016, p. 3, emphasis added)

The prediction error minimization (PEM) framework in cognitive science is an approach to cognition and perception centered on a simple idea: organisms represent the world by constantly predicting their own internal states. [...] Cascades of predictions are matched against the incoming sensory signals, which act as negative feedback to correct a generative model encoded in the top-down and lateral connections. (Kiefer & Hohwy, 2019, p. 384, emphasis added)

The generative model, which in theories such as hierarchical predictive coding is hypothesized to be implemented in top-down cortical connections, specifies the Umwelt of the organism, the kinds of things and situations it believes in independently of the current sensory data [...] (Kiefer, 2020, p. 2, emphasis added)

Sadly, this excusation does not address the first challenge. Importantly, however, it seems to me that the first challenge is really no challenge at all. Allowing (so to speak) activity patterns to participate in the relevant structural similarity alongside weighted connections does not change the fact that, at least prima facie, weighted connections do not appear to map one-to-one onto any target. Thus, simply counting activity patterns as elements of V does not vindicate (a). This is because weighted connections are still considered elements of either V or ℜ_V, and, as a result, at least some elements of the vehicle do not map one-to-one onto elements of the target as required by (a). Counting activity vectors in, on its own, is not enough: one must also be able to exclude that weighted connections participate in the relevant structural similarity.

This brings me to the second challenge. Is it possible to define some relevant vehicle target structural similarity without involving weighted connections? I think that the correct answer is negative.

To start, it is surely possible to define some relevant network-target structural similarity without appealing to weighted connections. There is nothing particularly new in this claim: Paul Churchland’s state-space semantics is the most obvious example of a network-target structural similarity that does not involve connections (see Churchland, 1986, 2012; see also O'Brien and Opie, 2004). In his view, the entire activation space of the hidden layers of a network structurally resembles the target domain upon which the network has been trained to operate. And I’m fairly confident that a similar structural similarity can be found by considering artificial neural networks encoding generative models.²²

Isn’t this just conceding that (a) obtains? I do not think so. For activation spaces (the first relevant relatum of the structural similarity) are not vehicles, because they are not concrete particulars. They are abstract mathematical spaces that are used to account for the systematic behavior of artificial neural networks. So, they fail to vindicate (a) for the same reasons Gładziejewski's argument fails to vindicate (a).

Notice that I’m not claiming that the activation space-target domain structural similarity cannot determine the content of each individual activity vector. The relevant structural similarity holding between the activation space and the target might be enough to determine the content of each individual vector (i.e. of each individual element of V, using O’Brien and Opie’s notation). However, the fact that each individual vector acquires content in virtue of the relevant structural similarity holding between the entire activation space and the target domain does not entail that each individual vector is a structural representation. This is because the elements of the vehicle (i.e. the objects of V and relations of ℜ_V) involved in a structural similarity need not be structurally similar to elements of the target (i.e. the members of O and ℜ_O) they correspond to. Given that, to my knowledge, only individual vectors are tokened in connectionist systems, the relevant structural similarity holding between the state space of a network and the network’s target domain is insufficient to substantiate the claim that structural representations are tokened within the system.

Moreover, I honestly doubt that it is possible to rightfully exclude weighted connections from the relevant structural similarity. To see why, consider the following: if a vehicle represents in virtue of the structural similarity it bears to a target, then the more the vehicle and the target are structurally similar, the more the representation will be accurate. The accuracy of a structural representation non-accidentally increases when (and, at least prima facie, only when) the elements of the vehicle are rearranged in a way that increases the extent to which the vehicle is structurally similar to the target.

If this is correct, then there seems to be a solid reason to deny that we can exclude connections from the relevant network-target structural similarity, for modifications of weighted connections made in accordance to the learning algorithm do improve the representational accuracy of connectionist systems. Thus, if these systems represent by means of structural similarity, it seems that weighted connections must be counted among the elements participating in the similarity. Surely, the relevant definition of structural similarity provided when unpacking condition (a) quantifies only over some, but that “some” seems to include weighted connections. However, if my argument based on superpositionality is correct, then weighted connections do not map one-to-one on their targets as (a) requires. In short: if artificial neural networks deploys structural representations, connections must be involved. Yet, their involvement seems to prevent the obtaining of (a).²³

4.2.3 Alternative argument 3

Alternative argument 3: maybe one does not need to look at artificial neural networks to vindicate the claim that generative models are structurally similar to their targets. In fact, PP theorists often point to a relevant structural similarity one might leverage to vindicate (a). Friston (2013a, p. 133) provides one clear example, worth quoting at length:

[...] every aspect of our brain can be predicted from our environment. [...] A nice example is the anatomical division into what and where pathways in the visual cortex. Could this have been predicted from the free-energy principle? Yes – if the anatomical structure of the brain recapitulates the causal structure in the environment, then one would expect independent causes to be encoded in functionally segregated neuronal structures.

Since points (i) and (ii) in the definition of structural similarity quantify only over some, this quote by Friston provides us a structural similarity sufficient to vindicate (a): if Friston is right, there is a structure-preserving mapping form some cerebral regions onto some environmental targets. Furthermore, examples like the one highlighted in the quote seem fairly easy to multiply. It might be pointed out, for instance, that the anatomical segregation of visual and auditory cortices reflects the fact that visual and sensory input can have different worldly causes. So there is, I submit, a relevant brain-world structural similarity. Therefore, if the whole brain is the generative model (a claim that is not uncommon in the PP literature,²⁴ e.g. Bastos et al. 2012, p. 702), then condition (a) is met.

I must confess that, to me, this way of vindicating (a) seems to lead to a Pyrrhic victory at best. To begin with, claiming that the whole brain is a structural representation seems to prevent Gładziejewski’s account from vindicating the epistemic representationalist claim. This is because, in our best explanatory practices, “representation” typically denotes states of information processing systems (e.g. Kandel et al. 2012 p. 372), rather than entire information processing systems.²⁵ The structural similarity presented above seems to enable us to vindicate only metaphysical representationalism about the whole brain (i.e. the claim that the whole brain really is a “big” representation). Given that Gładziejewski’s account of structural representations aims at vindicating both metaphysical and epistemic representationalism, this way of vindicating (a) seems to lead to a partial failure of his account.²⁶

Secondly, a complaint about content. What would such a “whole brain” structural representation represent? If I understand Friston correctly, the brain is supposed to recapitulate the causal structure of the world. Thus, the relevant structural similarity holds between the anatomical structure of the brain and the causal structure of the world. But a structural representation represents the target whose structure is mirrored in the structure of the vehicle, and here such a target is the world (see Wiese, 2018, p. 219; Williams, 2018a, 2018b, p. 154–155). This is not the kind of content naturalistic theories of content are supposed to deliver, for the world is not the kind of content invoked in the scientific explanations of our cognitive capacity, nor the kind of content relevant to our personal-level mental states. This isn’t a knockdown objection against alternative argument 3. But it surely shows that the argument has some very undesirable consequences.

Lastly, and, I believe, most importantly, this way of vindicating (a) seems to prevent (c) from obtaining. If the entire brain is a single gigantic representation representing the world, it is very hard to see how decouplability might obtain. There is always some sort of causal contact between brains and worlds. Since point (c) spells out decouplability in terms of causal contact, this way of vindicating (a) seems to prevent the obtaining of (c).²⁷

4.2.4 Alternative argument 4

Alternative argument 4: perhaps there is a way to make “whole brain” representations work. Thus, consider Kiefer and Hohwy’s (2018, 2019) defense of generative models as structural representations.²⁸

On the view Kiefer and Hohwy favor, we should conceive the brain as a complex causal network. If I understand them correctly, we should interpret each node in such a network as a definite pattern of neuronal activity, and the arrows connecting the nodes as causal relations between patterns (i.e. if node a is connected to node b, then pattern a causes pattern b). This network of causal relations, on the account Kiefer and Hohwy propose, structurally resembles the causal structures of the world as captured by “material inferences”; that is, inferences such as that from “It’s raining” one infers “The street is wet” (see Kiefer & Hohwy, 2018, pp. 2392–2393). In this way, the entire brain (which instantiates the causal network), comes to reflect, and hence to represent, the causal structure of the world.

Kiefer and Hohwy’s account of “whole brain” structural representations seems to me a significant improvement from the previously scrutinized one. For one thing, given that in this view the relevant elements of the structural representation are patterns of activation, and given that the elements of a structural representations can be counted as representations in their own right, Kiefer and Hohwy’s proposal would allow one to substantiate the epistemic representationalist claim. Moreover, it can also assign each individual pattern of activation a determinate content, depending on its causal embedding within the network. However, it seems to me that relying on Kiefer and Hohwy’s proposal to vindicate (a) has serious drawbacks.

To start, the problem with (c) is not solved by Kiefer and Hohwy’s account.²⁹ If the brain is a complex causal network mirroring the causal structure of the world, it is correct to say that the relevant structural representation (i.e. the brain) represents the world. And I simply do not see how one could sever the constant brain-world causal contact so as to vindicate (c).³⁰

Secondly, Kiefer and Hohwy’s account raises a puzzle about the inferential status of brain processes. If causal relations holding among patterns of activation are members of ℜ_V, it seems to me that it logically follows that they are part of the vehicle. After all, according to O’Brien and Opie’s (2004) definition of structural similarity, both the objects of V and the relations in ℜ_V are parts of S_V. But in order for the relevant structural similarity to satisfy (a), S_V must be a vehicle. Hence, Kiefer and Hohwy’s proposal seems to imply that causal relations among patterns of activations are part of the vehicle. But if this is the case, then it seems to me that these causal relations cannot be inferential processes, for inferential processes seem to be distinct from the representational vehicles upon which they operate. So, it seems that if Kiefer and Hohwy’s (2018, 2019) account of structural similarity is accepted, causal interactions among neural activity patterns cannot be rightfully called inferences. And this seems a problem, given that the inferentialist reading of PP tends to go hand in hand with the claim that generative models are structural representations (e.g. Gładziejewski, 2017; Hohwy, 2018; Kiefer, 2017).

Lastly, a wholesale acceptance of Kiefer an Hohwy’s (2018, 2019) account might, paradoxically, force one to abandon the claim that generative models are structural representations. The point is subtle but important. According to Kiefer and Hohwy:

The contents of parts of a structural representation are (at least in the case of causal generative models of an environment) in effect determined by their internal functional roles. (Kiefer & Hohwy, 2018, p. 2393; see also Kiefer & Hohwy, 2019, p. 402; Kiefer, 2020, endnote 17)

But this is not how the parts (i.e. objects and relations) of a structural representation acquire their contents. The content of a structural representation is determined by the relevant structural similarity it bears to a target; and the content of the parts (i.e. the elements of V and ℜ_V) of a structural representation is determined by the way in which they participate in the relevant structural similarity; that is, by the way in which they map onto a corresponding element of the target. The relevant relation determining the contents of the elements of a structural representation is the structural similarity holding between the vehicle and the target; not the relations ℜ_V holding among the various members of V. Surely, since structural similarity is structural, it must, in some relevant sense, be sensitive to the relevant members of ℜ_V. But this does not entail that the content of the elements of a structural representation is determined by their relations ℜ_V.

Another, perhaps more perspicuous, way to put the same point is this: were the content of the elements of V determined by the relations ℜ_V holding between them, then the elements of V would have content whether the entire vehicle is structurally similar to something or not. Moreover, even in cases in which the whole vehicle S_V is structurally similar to some target S_O, there is, as far as I can see, no prior guarantee that the content assigned to each element in V in virtue of the relations in ℜ_V would match the content each element of V would bear, were their contents determined by the relevant mapping from S_V to S_O constituting the structural similarity. To put it bluntly, what I’m trying to point out is this: Kiefer and Hohwy espouse a form of functional role semantics. But functional role semantics and structural similarity have no essential connections, pace Kiefer and Hohwy. It thus seems to me that a wholesale adoption of Kiefer and Hohwy’s proposal ends up undermining the broader structural-representationalist claim. Kiefer and Hohwy might provide a way to vindicate (a); but a wholesale acceptance of their proposal seems to make such a vindication redundant. If one adheres to functional role semantics,one does not need a structural similarity.

This is not to deny that Kiefer and Hohwy (2018, p. 2393; 2019, p. 402) stress that the relevant (i.e. content constituting) functional relations in ℜ_V mirror, in the relevant sense, the causal structure of the world (i.e. ℜ_O): in their view, functional role semantics entails a relevant structural similarity. But this surely does not allow us to count Kiefer and Hohwy as defenders of structural representations. For, as many have stressed, causal/informational theories of content entail a relevant vehicle-target structural similarity too (Morgan, 2014; Nirshberg & Shapiro, 2020; Facchin 2021). But surely causal/informational theories of content are not structural resemblance-based theories of content, for at least three reasons. First, the claim that content is determined by causal/informational factors is logically distinct from the claim that content is determined by a relevant vehicle-target structural similarity. Secondly, it is not the case that the obtaining of all vehicle-target structural similarities hinges upon some appropriate causal/informational relation holding the vehicle and the target (see Shea, 2018, p. 139–140). Lastly, in the case of genuine structural representations, the relevant structural similarity is what determines the relevant content. Hence, it should not be a “side effect” of some other content-determining factor (see Gładziejewski & Miłkowski, 2017 for further discussion).³¹ It seems to me that one needs only to substitute “causal/informational relations” with “functional relations” to make the same remarks about Kiefer and Hohwy’s proposal.

4.2.5 Alternative argument 5

Alternative argument 5: one might further try to vindicate (a) by claiming that, since generative models can be rendered as Bayesian nets, and that Bayesian nets are computationally useful because they are structurally similar to their target (Danks, 2014 p. 39), whatever piece of machinery is instantiating the relevant generative models must also be structurally similar to the target to be computationally useful. This way of vindicating (a), however, seems flawed. Generative models can be run by everyday personal computers: Von Neumann architectures computing over arbitrary symbols. And symbols surely aren't structural representations: in fact, the two are typically contrasted (O'Brien & Opie, 2001; Williams & Collings, 2017).

There might be other ways to vindicate (a). But I'm not claiming that the proposition “generative models are structural representations” is false. I'm only claiming that it is presently unjustified. And the discussion above seems sufficient to substantiate that claim. If the arguments provided above are correct, Gładziejewski’s original argument fails to substantiate the claim that generative models are structural representations; and, as far as I can see, there is no “rough and ready” way to vindicate that claim on offer in the current philosophical market.

5 Concluding Remarks

In this essay, I have tried to argue that the identification of generative models with structural representations is, at present, unjustified. Here, I wish to single out some recurrent themes that emerged in the discussion above, as they are likely to be important to understand the theoretical commitments of PP.

One issue that repeatedly emerged from the discussion above concerns the vehicles of generative models. The PP literature, I believe, is fairly confusing on this point. The word “model” is in fact applied to a variety of quite distinct things, including: (i) the whole brain (e.g. Bastos et al., 2012, p. 702), (ii) axonal connections (e.g. Shipps, 2016, p. 3), (iii) functionally specialized networks of neural areas (e.g. the mirror system as a model of bodily kinematics, see Kilner et al., 2007), (iv) neuronal responses and connections (Buckley et al. 2017, p. 57), (v) the spinal cord³² (Friston, 2011, p. 491), (vi) single hierarchical levels³³ (e.g. Kiefer & Hohwy, 2019, p. 387) and I would not be surprised if this list is not complete. This liberal, almost casual, usage of “model” makes it very difficult to understand what piece of neural machinery should be taken as the vehicle of the relevant representation. It also makes unclear what sort of structural similarity would be appropriate to vindicate a structural representationalist claim. I believe that clarifying this point should be a priority for philosophers (and empirical scientists) interested in defending the claim that generative models are structural representations. This is because structural representations are defined in terms of a (relational) vehicle property; namely, structural similarity. Hence, determining what counts as the relevant vehicle is essential in order to vindicate the claim that the relevant vehicle is the vehicle of a structural representation.

A related problem is that it is unclear whether the candidate vehicle (that is, the candidate piece of neural circuitry) is supposed to be (or embody) a model or merely to encode a model.³⁴ As far as I can see, “being/embodying a model” and “encoding a model” are used roughly as synonyms in the PP literature. Yet, there seems to be an obvious difference between the two claims: the hard drive of my PC surely encodes numerous early drafts of this very essay, but my hard drive is not an early draft of this essay. If the core claim PP makes is that the relevant candidate vehicle is a model, and models really are structural representations, then some structural similarity must be found; otherwise, PP would be in trouble. But if the core claim PP makes is just that the relevant candidate vehicle simply encodes a model, then the absence of any relevant vehicle-target structural similarity might be entirely unproblematic (supposing that not each and every form of encoding entails a structural similarity).

This latter point also suggests that, where the core claim of PP that the brain only encodes a generative model, the reading of generative models as structural representations would not be mandatory. For this reason, I believe that philosophers willing to defend a representationalist account of PP need not necessarily commit themselves to a “structuralist” account of representations. Representations in cognitive science need not necessarily be structural representations; and it would be interesting to explore whether a representationalist account of PP not based on structural representations is viable.³⁵

Lastly, a point about the rhetoric of the philosophical discussion surrounding PP. Many philosophers defending a representational reading of PP based on structural representations seem to hold that the “representation wars” are over, since PP has conclusively resolved the issue in favor of (structural) representationalism (e.g. Clark, 2015; Gładziejewski, 2016; Williams, 2017). Now, I find it sincerely hard to deny that structural representations are increasingly popular in cognitive neuroscience (e.g. Poldrack, 2020; Williams & Colling, 2017).³⁶ However, if the argument I have provided here is correct, the newfound popularity of structural representations might have very little to do with predictive processing. Identifying the factors that actually contribute to the popularity of structural representations might prove useful to fully understand the strength and merits of both representationalism and antirepresentationalism in cognitive science.