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Edited by: David R. Mandel, Defence Research and Development Canada, Toronto Research Centre, Canada

Reviewed by: David E. Over, Durham University, UK; Tania Lombrozo, University of California, Berkeley, USA

*Correspondence: Igor Douven, Sciences, Normes, Décision, Paris-Sorbonne University, Maison de la Recherche, 28 rue Serpente, 75006 Paris, France

This article was submitted to Cognition, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

There has been a probabilistic turn in contemporary cognitive science. Far and away, most of the work in this vein is Bayesian, at least in name. Coinciding with this development, philosophers have increasingly promoted Bayesianism as the best normative account of how humans ought to reason. In this paper, we make a push for exploring the probabilistic terrain outside of Bayesianism. Non-Bayesian, but still probabilistic, theories provide plausible competitors both to descriptive and normative Bayesian accounts. We argue for this general idea via recent work on explanationist models of updating, which are fundamentally probabilistic but assign a substantial, non-Bayesian role to explanatory considerations.

There has been a probabilistic turn in the cognitive sciences, a development most prominently marked by the emergence of the “Bayesian paradigm” in the psychology of human learning and reasoning (e.g., Evans and Over,

There are, nonetheless, various ways in which a theory might be probabilistic without being Bayesian. Most obviously, theories can draw upon probabilities interpreted in non-Bayesian ways (e.g., Gigerenzer and Hoffrage,

We focus our sights on the question of how humans update their confidences when confronted with new information^{1}

_{1} and _{2}, an agent's credences are to be updated so as to satisfy the equality Pr_{t2}(_{t1}(_{t1}(

Here, _{t1}—representing the agent's credences at time _{1}—and Pr_{t2}—representing the agent's credences at later time _{2}—are defined, and Pr_{t1}(_{1}) conditional probability of

The Bayesian account thus requires updates to be determined

There are two crucially distinct ways one can interpret any theory of updating: as providing norms that updates rationally ought to satisfy, or as a descriptive model of how people in fact update. At the same time that cognitive scientists focusing on the descriptive interpretation have increasingly turned to probabilistic models, more and more philosophers have come to regard Bayesianism as providing the norms of both rational action and rational belief (e.g., Maher,

A central contention of this paper is that some probabilistic models of updating that conflict with Bayes's Rule constitute strong, plausible competitors to Bayes's Rule, whether the models in question are interpreted descriptively or normatively. We make a case for this claim by focusing on a particular family of non-Bayesian, probabilistic models of updating, namely explanationist models. We argue that explanationist models may be predictively more accurate than Bayesianism (Section 3) without being normatively defective in any way (Section 4). Probabilistic alternatives to Bayesianism accordingly deserve more explicit attention in cognitive science and philosophy than they have thus far received. Before making our case, however, in the next section we offer a general description of explanationism.

Deductive inference plays a key role in human reasoning. It is unsurprising, therefore, that this form of inference has been amply studied by psychologists (see, e.g., Evans,

What

To illustrate, consider the following famous anecdote about the invasion of the Thames by the Dutch fleet in 1667—also known as “the Raid on the Medway”—and Sir Isaac Newton, who was a Fellow at Trinity College, Cambridge, at the time:

Their guns were heard as far as Cambridg, and the cause was well-known; but the event was only cognizable to Sir Isaac's sagacity, who boldly pronounc'd that they had beaten us. The news soon confirm'd it, and the curious would not be easy whilst Sir Isaac satisfy'd them of the mode of his intelligence, which was this; by carefully attending to the sound, he found it grew louder and louder, consequently came nearer; from whence he rightly infer'd that the Dutch were victors. [William Stukeley,

The “mode of intelligence” referred to here, which according to Westfall (

Abduction has been identified as playing a central role in scientific reasoning by various historians and philosophers of science (e.g., McMullin,

A more modern name for abduction is “Inference to the Best Explanation” (IBE), and most statements of abduction to be found in the literature are rather straightforward unpackings of that name. In Musgrave's (

In recent years, researchers have become interested in a version of abduction that is probabilistic in nature and even has Bayes's Rule as a limiting case (Douven, _{i}_{i≤n} is a set of self-consistent, mutually exclusive, and jointly exhaustive hypotheses, this version of abduction models human learning as an act of updating one's degrees of belief on new evidence in accordance with

_{1} and _{2}, an agent's credences are to be updated so as to satisfy the equality
_{t1}(

It is easy to verify that probabilistic abduction concurs with Bayes's Rule if ℰ is set to be the constant function 0, meaning that no bonus points for explanatory bestness are ever attributed. It is not much more difficult to verify that probabilistic abduction concurs with Bayes's Rule

Naturally, as stated here, probabilistic abduction is really only a schema as long as ℰ has not been specified. For present purposes, this matter can be left to the side. In fact, for this paper, the rule only serves to show that there are versions of abduction that are direct contenders to Bayes's Rule. But one can think of many more probabilistic update rules that explicate the broad idea that explanatory considerations have confirmation-theoretic import—the central idea underlying abduction. Rather than advocating any particular such rule, we now proceed to argue that the Bayesian model of updating—whether construed descriptively or normatively—may plausibly be improved in various ways by taking into account explanatory considerations, leaving the details of how exactly to account for such considerations for another occasion.

Contrary to what the growing popularity of Bayesianism among psychologists might lead one to expect, studies regularly find that people update in ways inconsistent with the Bayesian model; see, for instance, Phillips and Edwards (^{2}

The typical reaction to such findings is to look on departures from Bayesian reasoning as a complication or problem, and subsequently to hunt for explanations for why people are ostensibly straying from the proper rational norms. A far less explored option is to question whether Bayes's Rule (and with it Bayesianism) describes the appropriate normative standard for updating. We ask the normative question in the next section. In this section, we explore whether probabilistic models that take into account explanatory considerations might do better at describing people's updating behavior than Bayes's Rule.

The non-Bayesian, probabilistic models that we examine are related to research reported in Douven and Schupbach (

Good's (

and Schupbach and Sprenger's (

It is to be noticed that, while all three measures have 0 as the “neutral point,” they are not all on the same scale. In particular, Popper's and Schupbach and Sprenger's measures have range [−1, 1] while Good's measure has range (−∞, ∞). However, Schupbach (

which do all have range [−1, 1]. Below, we use “_{a}

Schupbach (_{A}); (ii) to do the same for the hypothesis that urn B had been selected (_{B}); and (iii) to assess how likely it was in the participant's judgment that urn A had been selected, given the outcomes at that point. The participant had to answer the questions about explanatory goodness by making a mark on a continuous scale with five labels at equal distances, the leftmost label reading that the hypothesis at issue was an extremely poor explanation of the evidence so far, the rightmost reading that the hypothesis was an extremely good explanation, and the labels in between reading that the hypothesis was a poor/neither poor nor good/good explanation, in the obvious order.

The data obtained in this experiment allowed Schupbach to calculate, for each participant and for each of the measures that he considered, the explanatory power of _{A} and _{B} after each draw the participant had witnessed, where either objective probabilities or credences could be used for the calculations. The results of these calculations were compared with the actual judgments of explanatory goodness that the participant had given after each draw. The results somewhat favored Schupbach and Sprenger's (

In Douven and Schupbach (

To be more precise, Douven and Schupbach (_{A} into a third variable (“A”), and the judgments of explanatory goodness of _{B} into a fourth (“B”). They then fitted a number of linear regression models, with S as response variable and with all or some of O, A, and B as predictor variables. The most interesting comparison was between the Bayesian model (called “MO” in the paper), which had only O as a predictor variable, and the full, explanationist model (“MOAB”), which had O, A, and B as predictor variables. In this comparison, as in the general comparison between all models that had been fitted, the explanationist model clearly came out on top. The difference in AIC value between MO and MOAB was over 120 in favor of the latter. Also, MOAB had an ^{2} value of 0.90, while MO had an ^{2} value of 0.83. A likelihood ratio test also favored MOAB over MO: χ^{2}_{(2)} = 124.87,

In short, the explanationist model MOAB was much more accurate in predicting people's updates than the Bayesian model MO, strongly suggesting that, at least in certain contexts, agents's explanatory judgments play a significant role in influencing how they update. Note that, by accepting this conclusion, one is not leaving the probabilistic paradigm: conditional probabilities figure as a highly significant predictor in MOAB as well. The conclusion

The previous research showed that, in a context in which one is trying to predict people's updated credences, if next to objective probabilities one has access to people's explanatory judgments, one is well-advised also to take the latter into account. In reality, however, we rarely know people's explanatory judgments. Does explanationism suggest anything helpful in contexts in which only objective probabilities are available? It may well do so. Provided we have all the probabilistic information at hand that is required as input for the measures of explanatory power stated above, we can use the output of those measures in combination with objective probabilities and try to predict someone's updates on that combined basis. Given that Schupbach (

In Douven and Schupbach (_{A}, and degrees of explanatory goodness of _{B} as predictors. Values of the last two predictors were determined in five distinct ways: using Popper's measure, using three separate rescalings of Good's measure (_{0.5}, _{1}, _{2}), and using Schupbach and Sprenger's measure. In the following, variable “Y_{X}” represents degrees of explanatory goodness for hypothesis _{Y} (Y ∈ {A, B}) calculated using measure X ∈ {P, G1, G2, G3, SS}, where “P” stands for Popper's measure, “G1” for _{0.5}, which is the first rescaled version of Good's measure, and so on. Similarly, “MXYZ” names the model with predictors X, Y, and Z.

Table ^{2} column in Table ^{2} values obtained in the tests were all significant, this is a first indication that any of the explanationist models provides a better fit with the data than the Bayesian model. Naturally, the better fit might be due precisely to the fact that the explanationist models include more predictors than MO. For that reason, it is worth looking also at the AIC metric, which weighs model fit and model complexity against each other and penalizes for additional parameters. Burnham and Anderson (

^{2} |
^{2} |
|||||
---|---|---|---|---|---|---|

MO | 3 | 202.39 | −398.77 | 48.06 | 0.83 | |

MOA_{P}B_{P} |
5 | 222.64 | −435.27 | 11.55 | 40.50^{***} |
0.85 |

MOA_{G1}B_{G1} |
5 | 216.72 | −423.43 | 23.40 | 28.66^{***} |
0.85 |

MOA_{G2}B_{G2} |
5 | 211.27 | −412.53 | 34.29 | 17.76^{**} |
0.84 |

MOA_{G3}B_{G3} |
5 | 228.41 | −446.83 | 0.00 | 52.06^{***} |
0.86 |

MOA_{SS}B_{SS} |
5 | 208.27 | −406.53 | 40.30 | 11.76^{*} |
0.84 |

_{2} is the squared correlation between the fitted and observed values

Furthermore, we see that it makes a large difference which measure is used to calculate degrees of explanatory goodness. In particular, the model which includes next to O also A_{G3} and B_{G3} as predictors—so degrees of explanatory goodness obtained via _{2}—does best: it has the lowest AIC value of all models, the difference each time being greater than 10, and it has the highest _{2} value (although in this respect all models are close to each other). This is confirmed by applying closeness tests for non-nested models to pairs of models consisting of MOA_{G3}B_{G3} and one of the other explanationist models. Using Vuong's (_{G3}B_{G3} is significantly preferred over any of the other explanationist models (in each case, _{G1}B_{G1}; in a comparison of MOA_{G3}B_{G3} with MOA_{G1}B_{G1}, Vuong's test has no preference for either model. On the other hand, using Clarke's (_{G3}B_{G3} is preferred over _{G3}B_{G3}. That O, A_{G3}, and B_{G3} are all highly significant buttresses Douven and Schupbach's (

_{G3}B_{G3}

Intercept | 0.33 | 0.02 | 14.90 | <0.0001 | |

O | 0.40 | 0.04 | 0.56 | 9.72 | <0.0001 |

A_{G3} |
0.24 | 0.03 | 0.30 | 7.48 | <0.0001 |

B_{G3} |
−0.13 | 0.03 | −0.15 | −3.67 | 0.0002 |

Finally, it is worthwhile comparing MOA_{G3}B_{G3} (the best model with degrees of explanatory goodness determined via _{2}) with MOAB [the best model from Douven and Schupbach (_{2} value of MOAB equals 0.90. Its AIC value equals −519.64. So, on both counts, MOAB does better. MOAB is also preferred over MOA_{G3}B_{G3} according to Vuong's test (

In fact, if judgments of explanatory goodness are available, one can even consider constructing a model that includes _{2}. Doing this for the present case, we find that in a model with all of O, A, B, A_{G3}, and B_{G3}, as predictors, B_{G3} is no longer significant. However, the model with the remaining variables as predictors does significantly better than MOAB in a likelihood ratio test: χ^{2}_{(1)} = 9.12, _{2} value is the same (0.90) for both models.

Summing up, we have found evidence that, at least in some contexts, explanationism is descriptively superior to Bayesianism: by taking explanatory considerations into account, next to conditional probabilities, we arrive at more accurate predictions of people's updates than we would on the basis of the objective conditional probabilities alone. Naturally, the kind of context we considered is rather special, and more work is needed to see how far the results generalize. Nonetheless, our results weigh against the generality of the increasingly popular hypothesis that people tend to update by means of Bayes's Rule.

Here is a natural response to the findings of the previous section: “Surely people's updates do indeed break with Bayes's Rule. But this is unsurprising. Bayes's Rule is best interpreted as a norm of proper or rational updating in the light of new evidence. It is an idealization that actual agents can at best hope to approximate, to the extent that they are reasoning as they should. Even if experimental evidence calls descriptive Bayesianism into question then, it does nothing to invalidate Bayesianism as an ideal, normative theory.” In this section, we challenge this idea, summarizing recent work that compares Bayes's Rule with explanationist models of updating in order to clarify their respective roles in a full normative theory of rational updating.

Consider the so-called dynamic Dutch Book argument, which has convinced many philosophers that Bayes's Rule is the only rational update rule^{3}

There are at least three reasons for being dissatisfied with this argument. First, Douven (

Second, even if non-Bayesian updating did make one vulnerable to dynamic Dutch books, it would not follow that such updating is necessarily irrational. For the possibility has

Third, even many Bayesians have become dissatisfied with the dynamic Dutch book argument. Above, it was said that the argument heavily depends also on what decision-theoretic principles are assumed. However, such principles would seem out of place in debates about

Motivated by this concern, Bayesians have sought to give an altogether different type of defense of their update rule. The alternative approach starts from the idea that update rules, like epistemic principles in general, are to be judged in light of their conduciveness to our epistemic goal(s), and that it is epistemically rational to adopt the update rule that is most likely to help us achieve our epistemic goal(s). The defense adopts inaccuracy minimization as the preeminent epistemic goal; update rules are accordingly epistemically defensible to the extent that they allow us to minimize the inaccuracy of our credences—where inaccuracy is spelled out in terms of some standard scoring rule(s). And according to Bayesians, it is their favored update rule that does best in this regard^{4}

It has recently been noted, however, that the goal of inaccuracy minimization, as it is used in the previous defense, is multiply ambiguous (Douven,

What has effectively been shown is that Bayes's Rule minimizes inaccuracy in the first sense. However, no argument has been provided for holding that minimizing inaccuracy in that sense trumps minimizing inaccuracy in one of the other senses. So, in light of results showing that, given these other interpretations of our epistemic goal, certain versions of abduction outperform Bayes's Rule in achieving that goal (Douven,

The upshot is that there is currently no good reason to hold that Bayesianism describes the unequivocally superior normative theory of updating. Both arguments that implore us to believe otherwise—the dynamic Dutch book argument and the inaccuracy minimization argument—fail in this regard. Bayes's Rule may be the uniquely best at enabling us to achieve one particular epistemic goal (minimizing expected inaccuracy in the long run). But there are other epistemic goals that we might have, which also involve the minimization of inaccuracy and which seem equally legitimate. Relative to some of these, abduction proves to be more conducive than Bayes's Rule. Results reported in Douven (

Nothing that we have said here calls into question the value of the probabilistic turn in recent cognitive science. We do, however, take issue with the narrowness of the focus of work in this vein. While we think that there is much fruit to be gleaned from modeling (actual and ideal) credences using probabilities, doing so does not necessitate using a Bayesian account. We have strived here to exemplify a promising way to expand fruitful research being pursued in cognitive science and philosophy today: namely, by exploring the probabilistic terrain outside of Bayesianism.

Doing so, we found strong support for explanationism, both as a descriptive and normative theory. At least in certain contexts, people do seem to base their updates partly on explanatory considerations; and at least with respect to certain plausible epistemic ends, that is what they ought to do. The present Research Topic (in which this article has been placed) centers around the question of how to improve Bayesian reasoning. This question could be taken to presuppose that Bayesianism is the one apt model of uncertain reasoning, and that all departures from Bayesianism are in need of improvement, repair, or explaining-away. In the above, we have challenged these presuppositions. Our findings suggest that when people update their credences partly on the basis of explanatory considerations and thereby flout Bayesian standards of reasoning, that can be because doing so puts them in a better position to achieve their epistemic goals. So, at least in some contexts, we can improve

We suspect that the answer to this question will depend sensitively on context and on the specific epistemic goals that are most salient for an epistemic agent. More research is thus needed to explore when exactly people are non-Bayesians and when exactly they should be. Specifically, do people tend to rely on some version of abduction mostly in those contexts in which it is best for them to do so, and similarly for Bayes's Rule? Bradley (

While the above is not a call to abandon Bayes's Rule across the board—in some contexts, it may be exactly the right rule to follow—our present findings do go straight against Bayesianism as philosophers commonly understand that position, namely, as the position that any deviance from Bayesian updating betokens irrationality. It is to be emphasized, however, that there is no apparent incompatibility between our findings and much of the work in psychology that commonly goes under the banner of Bayesianism. There is nothing in the writings of Chater, Evans, Oaksford, Over, or most of the other researchers commonly associated with the Bayesian paradigm in psychology that obviously commits them either to Bayes's Rule as a universal normative principle or to the hypothesis that, as a matter of fact, people generally do obey the rule^{5}

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are greatly indebted to Tania Lombrozo and David Over for valuable comments on a previous version of this paper.

^{1}In this paper, we use “update” in the general sense of belief change. It is worth noting that some authors in the Bayesian camp (e.g., Walliser and Zwirn,

^{2}This is not to deny that there is also evidence in support of the descriptive adequacy of Bayesianism. See in particular Griffiths and Tenenbaum (

^{3}The dynamic Dutch book argument was first published by Teller (

^{4}See Rosenkrantz (

^{5}While Bayes's Rule has a very central place in the work of Griffiths, Tenenbaum, and their collaborators (see, e.g., Griffiths and Tenenbaum,