Similarity and Induction
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DOI: 10.1007/s13164-009-0017-0
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- Weber, M. & Osherson, D. Rev.Phil.Psych. (2010) 1: 245. doi:10.1007/s13164-009-0017-0
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Abstract
We advance a theory of inductive reasoning based on similarity, and test it on arguments involving mammal categories. To measure similarity, we quantified the overlap of neural activation in left Brodmann area 19 and the left ventral temporal cortex in response to pictures of different categories; the choice of of these regions is motivated by previous literature. The theory was tested against probability judgments for 40 arguments generated from 9 mammal categories and a common predicate. The results are interpreted in the context of Hume’s thesis relating similarity to inductive inference.
1 Introduction
David Hume (2006) famously asserted a role for similarity in non-deductive inference. Here is the well-known passage.
In reality, all arguments from experience are founded on the similarity which we discover among natural objects, and by which we are induced to expect effects similar to those which we have found to follow from such objects. ... From causes which appear similar we expect similar effects.
Hume’s view is consistent with his predecessor Locke (1689), for whom analogy was “the great rule of probability.” Just what Locke and Hume meant by the term “probability” is open to discussion, but their thesis is clear. Similarity often lies behind inductive inference. The goal of the present essay is to sharpen this insight.
By “induction” we’ll understand a certain relation between a list of statements and some further statement. The first statements are called premises, the last the conclusion, and the ensemble an argument. The inductive strength of an argument for a given person will be identified with the subjective conditional probability she attaches to the conclusion given the premises. This definition raises questions about subjective probability in the minds of people who misunderstand chance. (Most college students can be led to incoherent estimates of probability; see Bonini et al. 2004; Tentori et al. 2004.) So we will just assume that the probability idiom conveys a familiar kind of psychological coherence condition. An argument is strong to the extent that the reasoner would find it odd to believe the premises without believing the conclusion. Squeezing this mental sensation into the unit interval and calling it probability provides a rough measure.
As mentioned, we also help ourselves to the unconditional probabilities of each premise and the conclusion of an argument. All that’s missing is the conditional probability of the conclusion given the premises, in other words, the inductive strength of the argument. Our project is thus to forge conditional probability from unconditional probability plus similarity. The criterion of success will be conformity to the estimates of conditional probability that people typically offer. This puts a descriptive spin on Hume’s thesis, which is consistent with his doubts about the normative justification of induction.
2 An Algorithm for Constructing Conditional Probability
Qc & Qa & Qb | Qc & Qa & \(\neg Qb\) |
Qc & \(\neg Qa\) & Qb | \(\neg Qc\) & Qa & Qb |
\(\neg Qc\) & \(\neg Qa\) & Qb | \(\neg Qc\) & Qa & \(\neg Qb\) |
Qc & \(\neg Qa\) & \(\neg Qb\) | \(\neg Qc\) & \(\neg Qa\) & \(\neg Qb\) |
It remains to test whether the scheme just presented approximates human intuition about chance.
3 Behavioral Data
3.1 Eliciting Estimates of Probability
Bears | Camels | Cougars |
Dolphins | Elephants | Giraffes |
Hippos | Horses | Lions |
Form | Number of instances |
---|---|
\({{\textit Prob}}(Qc\mid Qa)\) | 5 |
\({{\textit Prob}}(Qc\mid \neg Qa)\) | 5 |
\({{\textit Prob}}(\neg Qc\mid Qa)\) | 5 |
\({{\textit Prob}}(\neg Qc\mid \neg Qa)\) | 5 |
\({{\textit Prob}}(Qc\mid Qa, Qb)\) | 10 |
\({{\textit Prob}}(Qc\mid Qa, \neg Qb)\) | 10 |
Average estimates of the 40 conditional and 9 unconditional probabilities
Argument | Rated prob | Argument | Rated prob |
---|---|---|---|
dolphins | horses | 0.516 | bears | hippos | 0.567 |
hippos | elephants | 0.678 | camels | giraffes | 0.677 |
lions | cougars | 0.758 | lions | \(\neg\)camels | 0.396 |
cougars | \(\neg\)horses | 0.421 | dolphins | \(\neg\)horses | 0.414 |
giraffes | \(\neg\)camels | 0.381 | elephants | \(\neg\)hippos | 0.383 |
\(\neg\)bears | horses | 0.388 | \(\neg\)dolphins | elephants | 0.482 |
\(\neg\)lions | cougars | 0.394 | \(\neg\)elephants | giraffes | 0.401 |
\(\neg\)bears | dolphins | 0.462 | \(\neg\)dolphins | \(\neg\)hippos | 0.596 |
\(\neg\)horses | \(\neg\)bears | 0.559 | \(\neg\)elephants | \(\neg\)hippos | 0.691 |
\(\neg\)camels | \(\neg\)lions | 0.597 | \(\neg\)giraffes | \(\neg\)cougars | 0.605 |
lions | bears, dolphins | 0.690 | camels | horses, giraffes | 0.714 |
dolphins | elephants, hippos | 0.570 | cougars | lions, giraffes | 0.723 |
elephants | dolphins, camels | 0.633 | camels | elephants, horses | 0.674 |
giraffes | cougars, hippos | 0.611 | hippos | horses, bears | 0.656 |
bears | cougars, lions | 0.696 | giraffes | horses, elephants | 0.763 |
cougars | lions, \(\neg\)bears | 0.654 | elephants | hippos, \(\neg\)dolphins | 0.662 |
giraffes | camels, \(\neg\)hippos | 0.622 | camels | bears, \(\neg\)dolphins | 0.510 |
horses | giraffes, \(\neg\)cougars | 0.573 | elephants | hippos, \(\neg\)bears | 0.626 |
elephants | lions, \(\neg\)camels | 0.455 | lions | cougars, \(\neg\)horses | 0.680 |
hippos | camels, \(\neg\)dolphins | 0.534 | horses | bears, \(\neg\)giraffes | 0.499 |
horses | 0.583 | hippos | 0.563 |
dolphins | 0.559 | bears | 0.588 |
elephants | 0.601 | giraffes | 0.565 |
camels | 0.550 | cougars | 0.564 |
lions | 0.633 |
Then we attempted to predict the conditional probabilities on the basis of similarity plus the unconditional probabilities, using the scheme described above. The unconditional probabilities are available from the data, having been directly elicited. But what shall we use as our measure of similarity?
3.2 Similarity Untainted by Inductive Inference
We could ask the students to provide numerical estimates of the similarity of pairs of species, using a rating scale. But such a procedure would not fairly test Hume’s idea. His thesis was that perceived similarity gives rise to judged probability. We must not inadvertently test the converse idea, that perceived probability gives rise to judged similarity. After all, it could be that lions and cougars seem similar because inferences from one to the other strike us as plausible. Then similarity would indeed be related to induction but not in the way Hume intended. To focus on Hume’s idea, we need to operationalize similarity without allowing probability estimates to play an implicit role.
For this purpose, we adopt the idea that similarity of categories—like horses and camels—is determined by their respective neural representations. To quantify neural similarity, we rely on functional magnetic resonance imaging (fMRI) to identify the patterns of activation that support the categories; proximity of categories is then measured in physical terms.
4 Neurophysiological Data
4.1 Obtaining Activation Maps
None of the fMRI subjects participated in the probability assessments. Also, no mention was made of similarity or probability either before or during scanning. The fMRI subjects simply verified the category of mammal images (or verified in control trials that # was absent).
The fMRI procedure parcels the brain into roughly 50,000 cubes called voxels, 3 millimeters on a side. For each voxel, we obtained a measure of the metabolic activity provoked by recognizing bears, another value for giraffes, and so forth. The measure is the β coefficient for a given mammal’s regressor in the best linear model of the voxel’s behavior in the experiment; see the Appendix. These values were averaged across the 12 subjects (after projection of each brain onto a common template). Average activations were also obtained when viewing phase-scrambled pictures of each mammal. For each mammal, the activations arising from viewing its scrambled version were subtracted from the activations produced by the verification task. The resulting distribution of corrected values (obtained from the subtraction) induces a “map” of activations over the brain. There is one such map for each mammal. We compared the maps for each pair of mammals to estimate similarity. The method of comparison will be explained shortly.
4.2 Choosing Neural Regions
First we address the question: which structure of the brain should be mapped, that is, where are mammal categories located? It has been observed that lesions to the left temporal lobe are sometimes associated with specific deficits in knowledge of biological categories including mammals, vegetables, and fruit, sparing knowledge of human artifacts like furniture and tools (Warrington and Shallice 1984; Saffran and Schwartz 1994; Capitani et al. 2003). Also, single cell recording from inferior temporal cortex in monkeys reveals neurons that are responsive to natural categories (although their specificity may be influenced by size and position in the visual field, among other features; see Zoccolan et al. 2007). Partially converging information is available from human neuroimaging. A review of studies by Martin (2001) points to activity (often bilateral) in the lateral fusiform gyrus, medial occipital cortex, and superior temporal sulcus when subjects are asked to identify and name pictures of animals. Inferior regions of the left occipital cortex seem also to be recruited when viewing pictures of animals in contrast to tools (Martin et al. 1996). Broadly consistent findings emerge from studies of processing category-names (e.g., Perani et al. 1999, who report left fusiform gyrus activations for animals). Kounios et al. (2003) reach similar conclusions in their summary of the literature. There are, however, many inconsistent findings in both the clinical and neuroimaging literature (Caramazza 2000; Joseph 2001; Gerlach 2007). It is also unclear whether any given brain locus holds an integrated animal representation rather than perceptual or abstract features associated with it (e.g., visual properties in the left fusiform gyrus; see Thompson-Schill et al. 2006).
Moreover, structures beyond the temporal and occipital lobes have been implicated in the manipulation of conceptual knowledge. For example, Freedman et al. (2001) report categorical responding to pictures of cats and dogs by neurons in the lateral prefrontal cortex of monkeys.^{3} Human neuropsychology and neuroimaging also implicate the premotor cortex of the frontal lobe in object categorization, especially of manipulable objects such as fruit, tools, and clothing, although such evidence is not univocal (e.g. Martin et al. 1996; Chao et al. 1999; Gerlach et al. 2002; for reviews, see Gainotti 2000) and Martin 2007). There has been no similar report for mammal categories, however.
4.3 Comparing Activation Maps
The nine mammals give rise to 36 \(= \dbinom{9}{2}\) such computations of dissimilarity. To convert them to similarity, each is first inverted (divided into 1). Then the 36 resulting numbers are linearly scaled to run from \(\frac13\) to \(\frac23\). Occupying just the middle of the unit interval leaves room for pairs less similar than ours (e.g., moles compared to dolphins), as well as pairs more similar (e.g., camels versus dromedaries).
Similarities computed from left ventral-temporal cortex and from left BA 19
Mammals | Ventral temporal | BA 19 | |
---|---|---|---|
lion | cougar | 0.650 | 0.591 |
hippo | elephant | 0.612 | 0.569 |
giraffe | camel | 0.605 | 0.641 |
giraffe | horse | 0.581 | 0.617 |
camel | horse | 0.628 | 0.632 |
dolphin | horse | 0.365 | 0.422 |
lion | bear | 0.656 | 0.628 |
horse | elephant | 0.592 | 0.611 |
elephant | camel | 0.593 | 0.583 |
hippo | giraffe | 0.602 | 0.622 |
bear | cougar | 0.658 | 0.510 |
dolphin | hippo | 0.418 | 0.463 |
bear | horse | 0.620 | 0.581 |
giraffe | dolphin | 0.395 | 0.401 |
dolphin | elephant | 0.333 | 0.333 |
elephant | giraffe | 0.593 | 0.598 |
camel | hippo | 0.649 | 0.651 |
hippo | bear | 0.653 | 0.590 |
hippo | horse | 0.608 | 0.605 |
horse | cougar | 0.608 | 0.576 |
dolphin | bear | 0.409 | 0.362 |
bear | elephant | 0.656 | 0.564 |
horse | lion | 0.603 | 0.666 |
camel | cougar | 0.659 | 0.590 |
cougar | giraffe | 0.660 | 0.579 |
elephant | lion | 0.646 | 0.636 |
camel | lion | 0.630 | 0.667 |
bear | giraffe | 0.636 | 0.535 |
camel | bear | 0.639 | 0.596 |
elephant | cougar | 0.630 | 0.500 |
hippo | cougar | 0.644 | 0.557 |
dolphin | cougar | 0.433 | 0.451 |
dolphin | camel | 0.382 | 0.453 |
lion | hippo | 0.667 | 0.652 |
lion | giraffe | 0.620 | 0.661 |
dolphin | lion | 0.404 | 0.467 |
5 Predicting Conditional Probabilities
Our neural measure is uncontaminated by use of strength-of-inference as an index of similarity; for, the neural measure was obtained from mammal-stimuli individually, with no mention of similarity or probability. Relative to the model of inductive strength advanced above, a pure test of Hume’s thesis is therefore possible. It suffices to enter neural similarity into the model, along with the unconditional probabilities culled directly from subjects. The predictions generated thereby can then be compared to the results of the probability experiment.
Specifically, the experiment produced 40 numbers, corresponding to the arguments shown in Table 1. Each is an average estimate of conditional probability, to be paired with the corresponding probability calculated from our model based on neural similarity. Because similarity was calculated twice (on the basis of two neural regions), predictions are evaluated separately for left VTC and left BA19.
The predictive accuracy of the model cannot be attributed exclusively to the use of neural similarity; the model also rests on estimates of unconditional probability, and these were obtained from the same subjects whose conditional probabilities are at issue. To isolate the role of similarity, we substituted for neural similarity 36 random numbers drawn uniformly from \(\left[\frac13, \frac23\right]\). In 60 random trials, the average correlation between predicted and observed conditional probabilities was .405. No random trial reached r = .716.
6 Discussion
6.1 Other Avenues to Neural Similarity
We calculated similarity from left VTC and left BA 19 because these areas are suggested by the previous literature devoted to the neural representation of natural categories like animals. Certain other areas, however, yield equally good results. For example, when similarities are computed from the left primary visual cortex, the correlation between predicted and observed estimates of conditional probability is .707.
Squared deviation is perhaps the simplest approach to neural similarity but at least one other technique works as well. Given a neural region R and mammal m, the alternative assigns a point in three-dimensional space that reflects the overall position of the activations in R in response to m. The similarity between two mammals is then measured by the Euclidean distance between the points assigned to them (with inversion and linearization to \(\left[\frac13, \frac23\right]\), as before). Used in the rhinal sulcus of the temporal lobe, this index of similarity predicts conditional probability at r = .728. On the other hand, most other neural structures have no predictive success under either of the approaches to similarity discussed here. Understanding how and where similarity is coded in the brain is a topic of current investigation (Weber et al. 2009).
6.2 A Lower Bound for Conjunctions Based on Independence
In our theory of inductive strength, the value of \({{\textit Prob}}(Qb_{\!1} \ \&\ \cdots \ \&\ Qb_{\!n})\) is situated in the interval from \( \max\{0, 1 \!-\! n\! +\! \sum_{i=1}^n {{\textit Prob}}(Qb_i)\}\) to \(\min\{{{\textit Prob}}(Qb_{\!1}),\cdots, {{\textit Prob}}(Qb_{\!n})\}\). Similarity is used to choose a point in the interval, with low similarity pushing the point to the lower bound. Let us consider changing the lower bound to the product of the probabilities of the conjuncts: \(\prod_{i=1}^n {{\textit Prob}}(Qb_i)\). The latter bound embodies the idea that low similarity signals the stochastic independence of the conjuncts rather than their incompatibility. This might correspond better to what Hume had in mind since he seems to take the absence of similarity to reflect no reason for belief rather than reason for disbelief (Cohen 1980).
It is therefore worth reporting that the revised model with multiplicative lower bound underperforms the original model. In every neural region examined (and with both measures of similarity), correlations between observed and predicted probabilities are about 0.1 lower for the revised model. Also, the independence bound implies \({{\textit Prob}}(Qb\ \&\ \neg Qb) > 0\) whenever \(0 < {{\textit Prob}}(Qb) < 1\), a coherence violation.
6.3 Limitations and Extensions
Hume’s thesis about induction has here been examined through the lens of a particular model of probability judgment, which starts from unconditional probability and pairwise similarity. The model cannot generate arbitrary distributions. A joint distribution over the statements Qb_{1}, Qb_{2} ⋯ Qb_{n} (where Q is a predicate and b_{1} ⋯ b_{n} are objects) requires 2^{n} − 1 numbers in general. The scheme described here specifies distributions based on only \(n + \dbinom{n}{2}\) numbers (n unconditional probabilities and all pairwise similarities). It follows that our method must omit many potential distributions. This kind of compression, however, may be compatible with describing aspects of human judgment, which likely chooses distributions from a limited set in most circumstances.^{5}
Even if our method corresponds to the distributions that describe human judgment, we have not provided evidence that reasoning proceeds by constructing probabilities for conjunctions. Without such evidence, our model should be interpreted as describing just an input-output relation (unconditional probabilities and similarities in, conditional probability out).
Other challenges arise when arguments display distinct predicates in premise and conclusion, or involve relations like preys on. Inferences involving non-natural kinds—like artifacts and political parties—bring fresh distinctions to light. Accommodation is also needed for the tendency of even well-educated respondents to issue probabilistically incoherent estimates of chance, or to judge similarity asymmetrically (Tversky 1977; but see also Aguilar and Medin 1999). Confronting these complexities is inevitable for the development of any theory of human inductive judgment. The data presented here suggest merely that progress will involve similarity in something like the sense Hume had in mind.
6.4 Hume
In the foregoing discussion we’ve interpreted Hume’s thesis as a psychological claim, namely, that inductive inference (as people actually perform it) is driven by similarity. Our formal rendition of this claim enriches the determinants of inductive strength by appealing to the prior probabilities of premises and conclusion. Such additions notwithstanding, the predictive success of the model (limited though it be) supports Hume’s thesis.
To test the thesis as Hume intended it, we relied on a measure of similarity that is free from contamination by inferential reasoning. The measure rests on comparison of the neural representations of mammal-categories, in the absence of judgments of similarity or probability. This is not to deny that over many years, inferences about the properties of mammals might affect how they are ultimately coded in the brain. Thus, lions and cougars may be represented via a common pattern because they are perceived to share many properties. Neural similarity could therefore depend on inference via this route. Nonetheless, our measure of similarity is directly mediated by the mental representation of concepts rather than accessing the machinery of inductive cognition. This seems a fair way of making Hume’s claim precise. So perhaps our results sustain his claim that similarity provides a partial foundation for understanding inductive inference.
The model presented here is an alternative to “QPf,” described in Blok et al. (2007). It relies on some of the same concepts.
Participants were interviewed singly. Questions were posed in individualized random order via computer interface. Responses were made with a slider that controlled a field displaying numbers in the unit interval. The concept of conditional probability was reviewed prior to testing.
The same categoricity, however, was observed when the monkeys were trained on concepts involving cat/dog mixtures. Note that the LPFC is directly interconnected with inferior temporal cortex (Webster et al. 1994).
The results reported below are virtually identical if the activations for each mammal in a given region are “mean-centered.” To mean-center mammal M in region R, the average activation (β) in the map for M in R is subtracted from all the activations in the map prior to computing the sum of squared deviations between M and any other mammal.
A natural generalization of our model replaces (binary) similarity with the homogeneity of sets of n ≥ 2 objects. To illustrate, such a measure might assign greater homogeneity to { camels, horses, giraffes } compared to { camels, bears, lions }. All distributions over Qb_{1}, Qb_{2} ⋯ Qb_{n} can be generated by a model like ours that relies on n-ary homogeneity.
Acknowledgements
We thank Sergey Blok, James Haxby, Douglas Medin, and Lawrence Parsons for discussion and assistance in various stages of this work.