Bayes and Marr and conceptual motivation

There is a Bayes buzz in the psycho/cog world. Several have been arguing that Bayes provides the proper “framework” for understanding psychological/cognitive phenomena. There are several ways of understanding this claim. The more modest one focuses on useful tools leading to specific analyses that enjoy some degree of local justification (viz. this praises the virtues of individual analyses based on Bayes assumptions in the usual way (i.e. good data coverage, nice insights)). There is also a less modest view. Here Bayes enjoys a kind of global privilege (call this ‘Global Bayes’ (G-Bayes)). On this view, Bayesian models are epistemically privileged in that they provide the best starting point for any psychological/cognitive model. This assumption is often tied together with Marrian conceptions of how one ought to break a psycho/cog problem up into several partially related (yet independent) levels. It’s this second vision of Bayes that is the topic of this post. A version is articulated by Griffiths et. al. here. Many contest this vision. The dissidents’ arguments are the focus of what follows. Again, let me apologize for the length of the post. A lot of the following is thinking out loud and, sadly, I tend to ramble when I do this. So if this sort of thing is not to your liking, feel free to dip in and out or just ignore. Let’s begin.

Cosma Shalizi, (here) blogs about a paper by Eberhardt and Danks (E&D) (here) that relates to G-Bayes.  Shalizi’s blog post discusses E&D’s paper, which focuses on the problem that probability matching poses for Bayesian psycho theories.  The problem E&D identifies is that most of the empirical literature (which E&D surveys) ends with the subject pool “probability matching the posterior.”[1] Here’s the abstract:

Bayesian models of human learning are becoming increasingly popular

in cognitive science. We argue that their purported confirmation largely relies on a methodology that depends on premises that are inconsistent with the claim that people are Bayesian about learning and inference. Bayesian models in cognitive

science derive their appeal from their normative claim that the modeled inference is in some sense rational. Standard accounts of the rationality of Bayesian inference imply predictions that an agent selects the option that maximizes the posterior expected utility. Experimental confirmation of the models, however, has been claimed because of groups of agents that ‘‘probability match’’ the posterior. Probability matching only constitutes support for the Bayesian claim if additional unobvious and untested (but testable) assumptions are invoked. The alternative strategy of weakening the underlying notion of rationality no longer distinguishes

the Bayesian model uniquely. A new account of rationality—either for inference or for decision-making—is required to successfully confirm Bayesian models in cognitive science.

There are replies to this (here) by Tenenbaum & friends (T&F) and a reply to the reply (here) by Icard, who I think is (or was) a student of Danks’ at CMU.  The whole discussion is very interesting, and, I believe, important.

Here’s how I’ve been thinking about this translated into linguistiky terms. Before getting started, let me say up front that I may have misunderstood the relevant issues. However, I hope that others will clarify the issues (viz. dispel the confusion) in the comments. So, once again, caveat lector!

Griffiths et.al. describes Bayesian models as follows. They are intended as “explanations of human behavior.” The explanations are “teleological” in that “the [Bayesian-NH] solutions are optimal” and this optimality “licenses an explanation of cognition in terms of function.” The logic is the following: “the match between the solution and human behavior may be why people act the way they do” In other words, a Bayesian model intends to provide an optimal solution to “problems posed by the environment” with the assumption that if humans fit the model (i.e. act optimally) then the reason they do so is because this is the optimal solution to the problem. Optimality, in other words, is its own explanation. And it is the link between optimality and explanation that lends Bayes models a global kind of epistemic edge, at least for Marr level-1 descriptions of the computational problems that psycho theories aim to explain.

Curiously, Bayesians do not seem to believe that people actually do act/cognize optimally.[2] As T&F explains, Bayesian accounts do “not imply a belief that people are actually computing these optimal solutions” (p. 415).[3] Why not? Because it is recognized that such computations are intractable. Therefore, assuming that the models are models of what people do is “not a viable hypothesis.” How then to get to the actual psychological data points? To get from the optimal Bayesian model to the witnessed behavior requires the addition of “approximate algorithms that can find decent solutions to these problems in reasonable time” (p. 415-6). This seems to suggest that subjects’ behavior is approximately optimal.

In other words, optimal Bayes considerations sets the problems to be solved by more tractable algorithms, which in fact do a pretty good job of coming close-ish to the best solution given resource constraints. So, what faces the tribunal of actual psychological evidence is the combination of (i) a Bayesian optimal solution and (ii) a good enough algorithm that approximates this optimal solution given other computational constraints. So, more specifically, in cases where we find individuals probability matching rather than all converging on the same solution (as the Bayes model would predict) the algorithm does the heavy lifting. Why? Because a Bayes account all by itself implies that all participants if acting in a rational/optimal Bayes manner should all make the same choice; the one with the highest posterior. So Bayes accounts need help in order to make contact with the data in such cases, for Bayes by itself is inconsistent with probability matching results.[4] Here’s Shalizi making this point:

Here's the problem: in these experiments (at least the published ones...), there is a decent match between the distribution of choices made by the population, and the posterior distribution implied plugging the experimenters' choices of prior distribution, likelihood, and data into Bayes's rule. This is however not what Bayesian decision theory predicts. After all, the optimal action should be a function of the posterior distribution (what a subject believes about the world) and the utility function (the subjects' preferences over various sorts of error or correctness). Having carefully ensured that the posterior distributions will be the same across the population, and having also (as Eberhardt and Danks say) made the utility function homogeneous across the population, Bayesian decision theory quite straightforwardly predicts that everyone should make the same choice, because the action with the highest (posterior) expected utility will be the same for everyone. Picking actions frequencies proportional to the posterior probability is simply irrational by Bayesian lights ("incoherent"). It is all very well and good to say that each subject contains multitudes, but the experimenters have contrived it that each subject should contain the same multitude, and so should acclaim the same choice. Taking the distribution of choices across individuals to confirm the Bayesian model of a distribution within individuals then amounts to a fallacy of composition. It's as though the poet saw two of his three blackbirds fly east and one west, and concluded that each of them "was of three minds", two of said minds agreeing that it was best to go east.

Now for a linguistics analogue: Acceptability data is what we largely use to adjudicate our competence theories.[5] Now, we recognize that acceptability and grammaticality do not perfectly overlap, there being some cases of grammatical sentences that are unacceptable and some cases of acceptable sentences being ungrammatical. However, this overlap is marginal (at least in a pretty big domain).[6] What E&D argues (and Shalizi emphasizes) is that this is false in the typical case of Bayesian psychological accounts. Typically the Bayes solution does not provide the right answer for in many cases we find that the tested population probability matches the posterior distribution of the Bayes model. And this is strongly inconsistent with what Bayes would predict. Consequently, what allows for the fit with the experimental data is not the Bayes vision of the computational problem but the added algorithm. In effect, it would be as if our competence theories mostly failed to fit our acceptability profiles and we explained this mismatch by adding parsing theories that took in the slack. So, for example, it would be as if our Gs allow free violations of binding principle A but our parsers say that such violations, though perfectly grammatical, are hard to parse and this is why they sound bad. 

Let me be clear here: syntacticians do in fact make such arguments. Think of what we say about the unacceptability of self-embedded sentences (e.g. ‘That that that Bill left annoyed Harry impressed Mary’). However, imagine if this were the case in general. I think that syntacticians would begin to wonder what work the competence theory was doing. In fact, I would bet that such a competence theory would quickly find its way to the nearest waste basket. This is effectively the point that Icard makes in his reply to T&F. Here’s the money quote:

It is commonly assumed that a computational level analysis

constrains the algorithmic level analysis. This is not always

reasonable, however. Sometimes, once computational costs

are properly taken into account, the optimal algorithm looks

nothing like the ideal model or any straightforward approximation

thereto” (p. 3) [my emphasis, NH].

This raises a question, which Shalizi (and many others) presses home, about the whole rationale behind Bayesian modeling in psychology. Recall, the explanatory fulcrum is that Bayes models provide optimal solutions, as it is this optimality that licenses teleological explanations of observed behavior. Shalizi argues that the E&D results challenge this optimality claim.

By hypothesis, then, the mind is going to great lengths to maintain and update a posterior distribution, but then doesn't use it in any sensible way. This hardly seems sensible, let alone rational or adaptive. Something has to give. One possibility, of course, is that is sort of cognition is not "Bayesian" in any strong or interesting sense, and this is certainly the view I'm most sympathetic to…

In other words, Shalizi and Icard are asking in what sense the Bayes level-1 theory of the computation is worth doing given that it’s description of the problem seems not to constrain the algorithmic level-2 account (i.e. Marr’s envisioned fertile link between levels seems to be systematically broken here). If this is correct, then it raises Icard’s question in a sharp form: in what sense is providing a Bayesian analysis even a step in the direction of explaining the psychological data? Or to put this another way: what good is the Bayesian computational level theory if E&D and Shalizi and Icard are right? Things would not be bad if most of the time subjects approximated the Bayes solution. What’s bad is that this appears to be the exception rather than the rule in the experimental literature that is used to argue for Bayes. Or so E&D’s literature review suggests is the case.

To fix ideas, consider the following abstract state of affairs: Subjects consistently miss target A and hit target B. One theory is that they are really aiming for A, but are missing it and consistently hitting B because trying to hit A is too hard for them. Moreover, it is too hard in precisely a way that leads to consistently hitting B. This account might be correct. However, it does not take a lot of imagination to come up with an alternative: the subjects are not in fact aiming for A at all.  This is the logic displayed by the literature that E&D discusses. It is not hard to see, IMO, why some might consider this less than powerful evidence in favor of Bayes accounts.

Let me further embroider this point as it is the crux of the criticism. The Bayes discussions are often cloaked in Marrish pieties. The claim is that Bayes analyses are intended to specify the “problem that people are solving” rather than provide a “characterization of the mechanisms by which they might be solving it” (p. 2 Griffiths et. al.). That is, Bayes proposals are level-1, not level-2 theories. However, what makes Marrish pieties compelling is precisely the intimation that specifications of the problems to be solved will provide a hint about the actual computations and representations that the brain/mind uses to solve these problems.  Thus, it is fruitful to indulge in level-1 theorizing because it sheds light on level-2 processes. In fact, one might say that Marr’s dubbing level-1 theories as ‘computational’ invites just this supposition, as does his practice in his book. Problems don’t compute, minds/brains do. However, a computational specification of a problem is useful to the degree that it suggests the magnitudes the brain is computing and the representations and operations that it uses to compute them. The critique above amounts to saying that if Bayes does not commit itself to the position that by and large cognition is (at least approximately) optimal, then it breaks this Marrian link between level-1 and level-2 theory and thereby looses its main conceptual Marrian motivation. Why do a Bayes level-1 analysis if it in no way suggests the relevant brain mechanisms or the variables mental/brain computations juggle or the representations used to encode them? Why not simply study the representations and algorithms directly without first detouring via a specification of a problem that will not shed much light on either?

Here’s one more try at the same point: Griffiths et. al. try to defend the Bayes “framework” by arguing that it is a fecund source of interesting hypotheses. That’s what makes Bayes stories good places to start. But this just seems to beg the question critics are asking; namely why should we believe that the framework does indeed provide good places to start? This is the point that Bowers and Davis (B&D) seem to be making in their reply to Griffiths et. al. here and the one that Shalizi, E&D and Icard are making as well.

Let me provide an analogy with contemporary minimalism. In proposing a program it behooves boosters to provide reasons for why the program is promising (i.e. why adopting the program’s perspective is a good idea). One, of course, hopes that the program will be fecund and generate interesting questions and analyses (i.e. models), but although the proof of a program’s pudding is ultimately in the eating, a program’s proponents are required to provide (non-dispositive) reasons/motivations for adopting the program’s take on things. In the context of MP this is the role of Darwin’s Problem. It provides a rationale for why MP generated questions are worth pursuing that are independent of the fruits the program generates. Darwin’s Problem motivates the claim that the questions MP asks are interesting and worth pursuing because if answered they promise to shed light on the basic architecture of FL. What is the Bayes analogue of Darwin’s problem? It seems to be the belief that considering the properties of optimal solutions to “problems posed by the environment” (Griffiths et. al. p. 1) will explain why the cognitive mechanisms we have are the way they are (i.e. a Bayes perspective will provide answers to a host of why questions concerning the representations and algorithms used by the mind/brain when it cognizes). But why believe this if we don’t also assume that a specification of the computational level-1 problems will serve to specify level-2 theories of the mechanisms? All the critiques in various ways try to expose the following tension: if Bayes optimality is not intended to imply anything about the mind/brain mechanisms then why bother with it and if it is so intended then the evidence suggests that it is not a fruitful way to proceed for more often than not it empirically points in the wrong direction. That’s the critique.

Please take the above with a large grain of salt. I really am no expert in these matters so this is an attempted reconstruction of the arguments as a non-expert understands them. But, I believe that I got the main points right and if things are as Shalizi, E&D, Icard and B&D describe them to be then it stands as a critique of the assumption that Bayes based accounts are prima facie reasonable places to start if one wants to account for some cognitive phenomenon (i.e. it seems like a strong critique of G-Bayes). Or to put this in more Marrian terms: it is not clear that we should default to the position that Bayes theories are useful or fecund Level-1 computational analyses as they do not appear to substantially constrain the Level-2 theories that do all (much) of the empirical heavy lifting. This is the critique. It suggests, as Shalizi puts it, acting like a Bayesian is irrational. And if this is correct, it seems important for it challenges the main normative point that Bayes types argue is the primary virtue of their way of proceeding.

Let me end with two pleas. First, the above, even if entirely correct, does not argue against the adequacy of specific Bayesian proposals (i.e. against local Bayes). It merely argues that there is nothing privileged about Bayesian analyses (i.e. G-Bayes is wrong) and there is nothing particularly compelling about the Bayes framework as such. It is often suggested (and sometime explicitly stated) that Rational Analyses of cognitive function (and Bayes is a species of these) enjoy some kind of epistemologically privileged status due to their normative underpinnings (i.e. the teleological inferences provided by optimality). This is what these critical papers undercut if successful. None of this argues against any specific Bayes story. On its own terms any such particular account may be the best story of some specific phenomenon/capacity etc. However, if the above is correct, a model gains no epistemic privilege in virtue of being cast in Bayesian terms. That’s the point that Shalizi, E&D, Icard and B&D make.  Specific cases need to be argued on their merits, and their merits alone.

Second, I welcome any clarifications of these points by those of you out there with a better understanding of the current literature and technology. Given the current fashion of Bayes story telling (it is clearly the flavor of the month) in various parts of cognition (including linguistics) it is worth getting these matters sorted out. It would be nice to know if Bayes is just technology (and if so justified on application at a time) or is it a framework which comes with some independent conceptual motivation. I for one would love to know.

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[1] Griffiths et. al. agree that there is a lot of this in the literature and agree it is a serious problem. They offer a solution based on Tenebaum & friends that is discussed below.

[2] Optimality of cognitive faculties is a pretty fancy assumption. I know because Minimalists sometimes invoke similar sentiments and it has never been clear to me how FL or cognition more generally could attain this perfection. As Bob Berwick never tires of telling me, this is a very fancy assumption in an evolutionary context. Thus, if we are really perfect, this may be more of a problem requiring further explanation than an explanation itself.

Griffiths et. al. seem to agree with this. They tend to agree that human’s are not optimal cognizers nor even approximately optimal cognizers. Nonetheless, they defend the assumption that we should look for optimal Bayes solutions to problems as a good way to generate hypotheses. Why exactly is unclear to me. The idea seems to be the tie in between optimal solutions and the teleological explanations they can offer to why questions. Nice as this may sound, however, I am very skeptical for the simple reason that teleological explanations are to explanations what Christian (and political?) Science is to science. Indeed, these were precisely the kinds of explanations that 16^th century thinkers tried to hard to excise. It is hard to see how this functional fit, even were it to exist, is supposed to explain why a certain system has the properties it has absent non-teleological mechanisms that get one to these optimal endpoints.

[3] Note that whether people compute optimal solutions is consistent with the claim that people act optimally. I mention this for the teleological account explains just in case the subjects are acting according to the model. Only then can “the match between the solution and human behavior” teleologically explain “why people act the way they do.” If however, subjects don’t act this way, then it is hard to see how the teleological explanation, such as it is, can gain a footing. I confess that I don’t follow the Griffiths et. al. reasoning here. The only way out I can see is if they assume that subjects do in fact act (approximately) optimally even if they don’t compute using Bayes like algorithms. But even this seems incorrect, as we shall see below.

[4] It is important to note that it is the posterior probability (a number that is the product of prior Bayesian calculation) that is being matched, as Griffith’s et. al. observe.

[5] Actually, acceptability under an interpretation.  Acceptability simpliciter is just the limiting case where some bit of linguistic data is not acceptable under any interpretation.

[6] Why the caveat? If Gs generate an unbounded number of Ss then most will be beyond the capacity for judging acceptability. What we assume is that where an acceptability judgment can be made grammaticality and acceptability will largely overlap.