Abstract
Michael Weisberg’s account of scientific models concentrates on the ways in which models are similar to their targets. He intends not merely to explain what similarity consists in, but also to capture similarity judgments made by scientists. In order to scrutinize whether his account fulfills this goal, I outline one common way in which scientists judge whether a model is similar enough to its target, namely maximum likelihood estimation method (MLE). Then I consider whether Weisberg’s account could capture the judgments involved in this practice. I argue that his account fails for three reasons. First, his account is simply too abstract to capture what is going on in MLE. Second, it implies an atomistic conception of similarity, while MLE operates in a holistic manner. Third, Weisberg’s atomistic conception of similarity can be traced back to a problematic set-theoretic approach to the structure of models. Finally, I tentatively suggest how these problems might be solved by a holistic approach in which models and targets are compared in a non-set-theoretic fashion.
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Notes
Parker expresses a similar interpretation of Weisberg’s account, claiming that “Ultimately, however, Weisberg’s account seems best characterized as an account of what underwrites scientists’ judgments of the extent to which models and targets are similar”. (2015, 271; author’s emphasis).
Weisberg’s account can be more strongly interpreted as aiming to “capture how similarity judgments are made by scientists”, for his similarity equation and its associated interpretation (as I will describe below) includes the way in which features are selected and weighted differently by scientists. This clearly involves the “how” problem: how do scientists select and weight these features so as to make appropriate similarity judgments? But to be charitable, in this paper I simply interpret his account as attempting to “capture similarity judgments made by scientists”. The latter interpretation is weaker than the former because it merely interprets Weisberg’s account as describing similarity judgments rather than the process by which they are made.
Weisberg also describes three other kinds of modelling practice which require different weighting parameters: hyperaccurate, how-possibly and mechanistic modeling. For details please see Weisberg (2013, pp. 150–152).
Assume there is such a data set regarding the variables in the model. Note this extremely simple case should be regarded as a textbook example used merely for introductory purpose.
This model is borrowed from Shipley (2002, pp. 47–48), though slightly different from the original one.
I will argue extensively in Sect. 4 that the applicability of Weisberg’s account is in fact in dispute.
Because we have already assumed that there is a causal mechanism between the two variables and that is just what the modellers aim to capture.
Though, for the limit of space, I will not demonstrate the claim that the toy model can be readily treated by many scientific testing methods, the following elaboration of the MLE regarding a more complicated model should lend support to this claim.
Variables on the right side of the equation are causes and on the left side are effects. ‘\(\varepsilon \)’ is an error variable representing other unmodelled cause of the variable E, or pure randomness.
The variance is the expected square of the deviation around a variable’s expected value, defined as: \(E=[(X_{i}-\mu _{X})^{2}]\), where \(\mu _{X}\) is \(X_{i}'\hbox {s}\) expected value. Simply put, it is the measure of distance between the expected and the measured values of a variable. The covariance is simply a generalization of the variance. If we have two different random variables \((X,\, Y)\) measured on the same observational units, then the covariance between them is defined as: \(E[(X_{i} -\mu _{X})(Y_{i} -\mu _{Y})]\), in which \((X_{i} -\mu _{X})\) and \((Y_{i}-\mu _{Y})\) mean deviations of X and Y from its mean \(\mu _{X}\) and \(\mu _{Y}\) respectively. (Shipley 2002, p. 74)
A free parameter is one that can be adjusted to make the model fit the data.
This is typically done by using the maximum likelihood chi-squared statistic. For details see Shipley (2002, pp. 110–114).
The maximum likelihood estimates (typically using the maximum likelihood chi-squared statistic) guarantee that, by iteratively choosing values for free parameters, the numerical values of the predicted covariance matrix become as close as possible to the actual covariances measured in the data. In essence, to increase the fit as close as possible is also to decrease the difference as much as possible. Hence at a certain point we obtain a minimum difference. The next step is to calculate the probability of having observed such a minimum difference. The value of probability can be calculated by some commercial computer programs, given certain maximum likelihood estimates and degrees of freedom.
In scientific testing the acceptability criterion of 0.05 (or other values depending on different fields of study) is also called the significance level or P value, used by scientists to determine whether the null hypothesis should be rejected in favor of alternative hypothesis. In short, the significance level indicates whether there is a relationship between various variables represented in the model under testing, or whether the result can be explained by the null hypothesis. For example, if the probability is less than or equal to the significance level, then the null hypothesis is rejected and then the outcome is said to be statistically significant. An outcome is said to be statistically significant only if it can enable the rejection of the null hypothesis. On the other hand, if the probability is larger than the significance level, then the null hypothesis cannot be rejected.
There are reasons why we choose this model but not others. First, it is definitely a mathematical model and falls into the category of minimal modeling according to Weisberg’s terminology. Second, it is neither too complicated nor too simple to grasp. Third, and most important, it is a good exemplar in the discourse of causal modeling and related testing methods.
The figure is also drawn from Shipley (2002, p. 131), with only small modifications.
In most structural equations, the causal relations are assumed to be additively linear (Shipley 2002, p. 105), so we here follow this assumption. It also should be noted that since causal relations are asymmetric while these equations are symmetric, this translation is only a partial translation.
See footnote 13.
See Shipley (2002, pp. 110–114) for more details about these techniques.
This difference can be calculated using some formula. For more details, see Shipley (2002, pp. 113–114).
See footnote 14.
See Shipley (2002) for more details about the calculations of this value.
There might be other interpretations which set different threshold values for the acceptable probability, but our case discussed in this paper is neutral on exactly which value is actually chosen.
Parker recently made a similar point, asking “can feature weights really be assigned independently?” She says that “In his account of model-world similarity, Weisberg deviates from Tversky in restricting the weighting function such that it assigns weights to features independently of which other features are shared”, and that “Weisberg’s restriction seems to go too far, however. For surely the perceived significance of a feature ‘shared’ by a model and a target sometimes does depend on which other features are ‘shared”’. (Parker 2015, pp. 273–274).
For Weisberg, fidelity criteria “describe how similar the model must be to the world in order to be considered an adequate representation. There are two types of fidelity criteria: dynamical fidelity criteria and representational fidelity criteria.” (2013, p. 41).
MAXOUT is a representational ideal which guarantees that models are useful for predicting but gives no guarantee that the models are useful for explaining. (2013, p. 109).
The discussion surrounding representational strategies started from Levins (1966), who claimed that there are three types of representational strategies (i.e. realism, precision and generality) among which tradeoffs exist, and continued well into this century by Orzack and Sober (1993), Taylor (2000), Odenbaugh (2003, 2006), Orzack (2005), Weisberg (2006), Matthewson and Weisberg (2009), etc.
Weisberg himself also distinguishes various different representational ideals: completeness, simplicity, 1-causal, maxout, and P-general. For details please see his (2013, chapter 6, section 2).
The semantic view about models (or more generally about theories) has two versions, one is the set-theoretic predicate approach developed by Suppes (1957, 1960, 1962, 1967), Sneed (1971) and Stegmuller (1976), and the other is the state space approach developed by van Fraassen (1970, 1972, 1974) and Suppe (1974, 1977).
The semantic view of models, as Weisberg himself criticizes, takes mathematical models to be more or less equivalent to logicians’ models, and thus accounts for the model-world relation by tools (typically set theory) appropriate for logicians’ models. In particular, it takes model-world relation to be an isomorphism relationship (or some weakened versions), i.e. a mapping between two sets that preserves structure and relations. (Weisberg 2013, pp. 137–138).
I thank Arnaud Pocheville for letting me notice this.
I will say more about my conception of structures in the next section.
A proponent of similarity view can reply that the whole structure composed of various interconnected elements can be thought of as a single feature, and the similarity judgment is just made about the similarity of this single feature between the model and the target. But this way of reply makes Weisberg’s view descend into a version of holistic approach that he rejects.
In cases where no causal relation (direct or indirect) holds between two variables, the situation could be slightly different, for weighting one would not affect weighting the other. For example, if there is no causal relation between \(X_{3}\) and \(X_{4}\) in the leaf gas-exchange model, then weighting the causal relation between \(X_{3}\) and \(X_{5}\) would not affect weighting the one between \(X_{4}\) and \(X_{5}\). But the general claim still applies here, for many (if not most) variables in the model still bear causal relations on one another.
A whole body of literature falls into this category of non-set-theoretic approach, such as the “mediating models movement” led by Morgan and Morrison (1999), Hughes’s “denotation-demonstration-interpretation” account (1997), Suárez’s “inferential account” (2004, 2015a, b), etc. Suárez, in some sense, even anticipates the “anti-set-theoretic” criticism developed in this article, as he claims, when criticizing the Tversky-Weisberg similarity account, that “the idea of context-relative description presupposes that some antecedent notion of representation is already in place, since it assumes that sources and targets are represented as having particular sets of features in context” (Suárez 2015a, 8; my emphasis).
Least squares is a method of fitting a model to data by minimizing the squared differences between observed and predicted values (Johnson and Omland 2004, p. 102). Whether or not the least squares method is a holistic approach is less obvious but it is at least not set-theoretic, for any change in a parameter of the predicted curve will possibly affect all the deviations to be squared (i.e. the distance of each point to the curve).
Suárez’s use-based inferential account is especially pertinent, expressing that “A represents B if (i) the representational force of A points towards B, and (ii) A allows competent and informed agents to draw specific inferences regarding B” (2004, p. 773). Note that the reference to the presence of agents and purposes of inquiry is crucial for this account: “First, the establishing and maintaining of representational force in (i) requires some agent’s intended uses to be in place; and these will be driven by pragmatic considerations. Second, the type and level of competence and information required in (ii) for an agent to draw inferences regarding B on the basis of reasoning about A is a pragmatic skill that depends on the aim and context of the particular inquiry” (2004, p. 773). This account is deflationist because it construes representation as means, rather than constituents, of inference-making, implying that representation can be fulfilled by any means of inference (such as induction, deduction, abduction, etc.), without committing to any constitutive relationship (such as similarity, isomorphism, partial isomorphism, homomorphism, etc.) between the target and the model.
Though my account is developed using a mathematical model as an example, I do not think it cannot be applied to concrete models. It will be worth developing another article to support this claim, but let me just mention several points relevant to our current purpose. First, I agree (with Weisberg) that concrete models are concrete structures (plus certain construal). Second, these concrete structures, like their mathematical counterparts, are organized entities in which elements bear relationships to each other. Third, I think a set-theoretic approach of comparison (or similarity) cannot capture these organized entities.
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Acknowledgments
I am grateful to a number of friends and colleagues for feedback on early drafts of this work, including Pierrick Bourrat, Mark Colyvan, Paul Griffiths, Qiaoying Lu, John Matthewson and Arnaud Pocheville. Special thanks is due to Paul Griffiths and Arnaud Pocheville, who gave me extremely useful help and encouragement over the course of developing this work. Thanks to the National Social Science Fund of China (Grant Number: 14ZDB018).
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This study was funded by the National Social Science Fund of China (Grant Number: 14ZDB018).
Appendices
I thank Arnaud Pocheville for giving me this demonstration.
Appendix 1: The leaf gas-exchange model
Box 1. The structural equations for the leaf gas-exchange model | |
|---|---|
SLM \((\hbox {X}_{1}) = \hbox {N} (0, {\upsigma }_{1})^{\mathrm{a}}\) | \({\upvarepsilon }_{2}= \hbox {N} (0, {\upsigma }_{2})\) |
Leaf nitrogen concentration \((\hbox {X}_{2}) = \hbox {a}_{1}\hbox {X}_{1}+ \hbox {b}_{2}{\upvarepsilon }_{2}\) | \({\upvarepsilon }_{3}= \hbox {N} (0, {\upsigma }_{3})\) |
Stomatal conductance \((\hbox {X}_{3}) = \hbox {a}_{2}\hbox {X}_{2}+ \hbox {b}_{3}{\upvarepsilon }_{3}\) | \({\upvarepsilon }_{4} = \hbox {N} (0, {\upsigma }_{4})\) |
Net photosynthetic rate \((\hbox {X}_{4}) = \hbox {a}_{3}\hbox {X}_{3} + \hbox {b}_{4}{\upvarepsilon }_{4}\) | \({\upvarepsilon }_{5}= \hbox {N} (0, {\upsigma }_{5})\) |
Internal \(\hbox {CO}_{2}\) concentration \((\hbox {X}_{5}) = \hbox {a}_{4}\hbox {X}_{3} + \hbox {a}_{5}\hbox {X}_{4} + \hbox {b}_{5}{\upvarepsilon }_{5}\) | |
\(\hbox {Cov}(\hbox {X}_{1},\, {\upvarepsilon }_{2})=\hbox {Cov}(\hbox {X}_{1},\, {\upvarepsilon }_{3})=\hbox {Cov}(\hbox {X}_{1},\, {\upvarepsilon }_{4})=\hbox {Cov}(\hbox {X}_{1},\, {\upvarepsilon }_{5})=\hbox {Cov}(\hbox {X}_{2},\, {\upvarepsilon }_{3})=\hbox {Cov} (\hbox {X}_{2},\, {\upvarepsilon }_{4})= \hbox {Cov}(\hbox {X}_{2},\, {\upvarepsilon }_{5})=\hbox {Cov}(\hbox {X}_{3},\, {\upvarepsilon }_{4})=\hbox {Cov}(\hbox {X}_{3},\, {\upvarepsilon }_{5})=\hbox {Cov}(\hbox {X}_{4},\, {\upvarepsilon }_{5})=\hbox {Cov}({\upvarepsilon }_{2},\, {\upvarepsilon }_{3})=\hbox {Cov}({\upvarepsilon }_{2},\, {\upvarepsilon }_{4})= \hbox {Cov}({\upvarepsilon }_{2},\, {\upvarepsilon }_{5})=\hbox {Cov}({\upvarepsilon }_{3},\, {\upvarepsilon }_{4})=\hbox {Cov}({\upvarepsilon }_{3},\, {\upvarepsilon }_{5})=\hbox {Cov}({\upvarepsilon }_{4},\, {\upvarepsilon }_{5})=0\) | |
Box 2. Predicted population variances and covariances for variables described in Box 1 | |||||
|---|---|---|---|---|---|
X1 | X2 | X3 | X4 | X5 | |
\(\hbox {X}_{1}\) | \(\hbox {Var}(\hbox {X}_{1})\) | \(\hbox {a}_{1}\hbox {Var}(\hbox {X}_{1})\) | \(\hbox {a}_{1}\hbox {a}_{2}\hbox {Var}(\hbox {X}_{1})\) | \(\hbox {a}_{1}\hbox {a}_{2}\hbox {a}_{3}\hbox {Var}(\hbox {X}_{1})\) | \((\hbox {a}_{1}\hbox {a}_{2}\hbox {a}_{4}+\hbox {a}_{1}\hbox {a}_{2}\hbox {a}_{3}\hbox {a}_{5})\hbox {Var} (\hbox {X}_{1})\) |
\(\hbox {X}_{2 }\) | \(\hbox {Var}(\hbox {X}_{2})\) | \(\hbox {a}_{1}^{2}\hbox {a}_{2}\hbox {Var}(\hbox {X}_{1})\) | \(\hbox {a}_{1}^{2}\hbox {a}_{2}\hbox {a}_{3}\hbox {Var}(\hbox {X}_{1})\) | \((\hbox {a}_{1}^{2}\hbox {a}_{2}\hbox {a}_{4}+ \hbox {a}_{1}^{2}\hbox {a}_{2}\hbox {a}_{3}\hbox {a}_{5})\hbox {Var}(\hbox {X}_{1})\) | |
\(+\hbox {a}_{2}\hbox {b}_{2}^{2}\hbox {Var}(\varepsilon _{2})\) | \(+\hbox {a}_{2}\hbox {a}_{3}\hbox {b}_{2}^{2}\hbox {Var}(\varepsilon _{2})\) | \(+(\hbox {a}_{2}\hbox {a}_{4}\hbox {b}_{2}^{2}+ \hbox {a}_{2}\hbox {a}_{3}\hbox {a}_{5}\hbox {b}_{2}^{2})\hbox {Var}(\varepsilon _{2})\) | |||
\(\hbox {X}_{3}\) | \(\hbox {Var}(\hbox {X}_{3})\) | \(\hbox {a}_{1}^{2}\hbox {a}_{2}^{2}\hbox {a}_{3}\hbox {Var}(\hbox {X}_{1})\) | \((\hbox {a}_{1}^{2}\hbox {a}_{2}^{2}\hbox {a}_{4}+ \hbox {a}_{1}^{2}\hbox {a}_{2}^{2}\hbox {a}_{3}\hbox {a}_{5})\hbox {Var}(\hbox {X}_{1})\) | ||
+\(\hbox {a}_{3}\hbox {b}_{3}^{2}\hbox {Var}(\varepsilon _{3})\) | \(+(\hbox {a}_{4}\hbox {b}_{3}^{2}+\hbox {a}_{3}\hbox {a}_{5}\hbox {b}_{3}^{2})\hbox {Var}(\varepsilon _{3})\) | ||||
\(\hbox {X}_{4}\) | \(\hbox {Var}(\hbox {X}_{4})\) | \(\hbox {a}_{1}^{2}\hbox {a}_{2}^{2}\hbox {a}_{3}^{2}\hbox {a}_{5}\hbox {Var}(\hbox {X}_{1})+\hbox {a}_{3}\hbox {a}_{4}\hbox {Var}(\hbox {X}_{3})\) | |||
\(+\hbox {a}_{3}^{2}\hbox {a}_{5}\hbox {b}_{3}^{2}\hbox {Var}(\varepsilon _{3})+\hbox {a}_{5}\hbox {b}_{4}^{2}\hbox {Var}(\varepsilon _{4})\) | |||||
\(\hbox {X}_{5}\) | \(\hbox {Var} (\hbox {X}_{5})\) | ||||
Appendix 2: Derivation of Weisberg’s similarity account from the JSC
I thank Arnaud Pocheville for giving me this demonstration.
One can show that Weisberg’s similarity index is the weighted average of the similarity coefficients upon mechanisms and attributes. The demonstration goes as follows:
The last equation is just Weisberg’s formula. If the sets of mechanisms and attributes are disjoint (which in fact is an assumption made by Weisberg), that is, if \(M_m \cap M_a =T_m \cap T_a =M_m \cap T_a =T_m \cap M_a =\emptyset \), then Weisberg’s similarity index is the Jaccard similarity index on the union of mechanisms and attributes:
Proof of this equation follows directly from:
Let us prove the numerator:
The denominator:
This means that if attributes and mechanisms are disjoint, similarity on mechanisms and attributes can be thought of somehow as independent. If mechanisms and attributes were not disjoint, however, elements in the intersection would be counted twice in the average and the Weisberg’s similarity index would be simply wrong.
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Fang, W. Holistic modeling: an objection to Weisberg’s weighted feature-matching account. Synthese 194, 1743–1764 (2017). https://doi.org/10.1007/s11229-016-1018-z
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DOI: https://doi.org/10.1007/s11229-016-1018-z