Predicting Is Not Explaining: Targeted Learning of the Dative Alternation

2.1 What linguistics is about

Like biologists studying the structure, function, growth, evolution, distribution, and taxonomy of living cells and organisms, linguists study language. In this respect, doing linguistics means investigating the cognitive system which we identify as the knowledge of a language. This knowledge takes the form of a mental grammar. Therefore, understanding what it means to know a language is to understand the nature of such a grammar.

Despite aiming at an objective description of language, linguists have their theoretical preferences, depending on what they believe the true essence of mental grammar is. In this regard, two competing theories have shaped contemporary linguistics.

As described by Chomsky [5–7], transformational-generative grammar (henceforth “Chomskyan grammar/linguistics”) is based on the assumption that, like formal languages, the grammars of natural languages consist of (a) a set of abstract algebraic rules and (b) a “lexicon” that contains meaningful linguistic elements. The algebra is the innate core of grammar. It constitutes what Chomskyans call the “universal grammar”, common to all natural languages. It is therefore what Chomskyans truly look for. The lexicon is relegated to the periphery of grammar, along with what makes a language idiosyncratic (e.g. inter-speaker variation, cultural connotations, stylistic mannerism, non-standard usage, etc.). Central to Chomskyan grammar is the opposition between deep structure and surface structure. This opposition hinges on syntax, i.e. the way in which words are combined to form larger constituents such as phrases, clauses, or sentences. The deep structure of a sentence is its abstract syntactic representation. The surface structure of a sentence is its final syntactic representation in speech or text. For instance, the sentence students hate annoying professors has one surface structure but two alternative interpretations at the level of the deep structure: (a) students hate to annoy professors and (b) students hate professors who are annoying. Derivation is the process whereby a sentence is generated from abstract operations in the deep structure to a string of words in the surface structure. To sum up, Chomskyan grammar is a “top-down” approach to language: linguists study how algebraic rules “at the top” generate an infinite number of sentences “at the bottom”. One major problem with this approach is that by focusing on the top, linguists tend to look down upon the bottom.

Conversely, usage-based linguistics is a “bottom-up” approach. Its tenet is that actual language usage shapes the structure of language [8–11]. From a usage-based viewpoint, grammar is the product of usage varying from speaker to speaker and there are no hard and fast rules. Grammar is therefore derivative, not generative. It does not have a core and a periphery. It is instead a structured inventory of symbolic units. Not only does the inventory of a native speaker of English contain highly schematic constructions (the past tense, the ditransitive construction, the active construction, etc.), concrete words or phrases such as ritualized or formulaic expressions (double whammy, hang in there!), idioms (that didn’t go down well with the editor, he kicked the bucket, etc.), or non-canonical phrasal collocations (you’re getting to me these days), but also mixed constructions having both abstract and concrete elements (the more you drink, the smarter you think you are). We adopt the viewpoint of usage-based linguistics because (a) we believe it offers a psychologically realistic view of grammar and (b) such a view, unlike Chomskyan grammar, can be operationalized.

2.2 Corpus linguistics

Linguists must rely on the native speakers of a language acting as informants and providing data such as sentences. On the basis of such data, linguists test their hypotheses about the cognitive systems of native speakers. Chomskyans have relied heavily on introspecting judgments for data collection. Their motivation may date back to de Saussure, the father of modern linguistics, who delimited the object of study (langue) as a structured system disconnected from the vagaries of place and time wherein it is deployed. However, the method has been called into question because linguists’ intuitions are not always consonant with what they observe in the data.

With the rise of massive digital collections of texts, linguists who have been dissatisfied with the practice of using themselves as informants have found corpora to be far more useful than introspective judgments to test their hypotheses. Unsurprisingly, these linguists, who consider that genuine language use in all its complexity should be at the center of linguistic research, share the main tenets of the usage-based model.

All present-day digital corpora such as The Brown University Standard Corpus of Present-Day American English, the Switchboard corpus, or the British National Corpus conform to the following definition:

A corpus is a machine-readable collection of (spoken or written) texts that were produced in a natural communicative setting, and the collection of texts is compiled with the intention (a) to be representative and balanced with respect to a particular linguistic variety or register or genre and (b) to be analyzed linguistically [12].

Note that the above definition covers general corpora as opposed to, say, language acquisition corpora or convenience corpora such as the Guardian data, as pointed out by one anonymous reviewer. Arguably, a combination of the usage-based methodology and corpus-driven research has led to a paradigm shift in linguistics.

Corpora have their limitations. One of the most frequent criticisms leveled against corpus linguistics by Chomskyans is that corpora do not indicate whether a given expression or use of an expression is impossible. From our usage-based perspective, the response to this critique is simple: because grammatical rules are mere generalizations about actual usage, negative evidence is of limited importance. A more serious criticism is the following: no corpus – however large and balanced – can ever hope to be representative of a speaker, let alone of a language. Supporters of introspective linguistics might argue that a corpus should not be a basis for linguistic studies if it cannot represent language in its full richness and complexity. Most corpus linguists rightly counter that they do not aim to explain all of a language in every study [13]. In this paper, we take a slightly different stance. While we acknowledge that no corpus can provide access to the true, unknown law of a language, we firmly believe that a corpus is a sample drawn from this law. We consider that ambitious corpus linguistics consists in bridging the gap between what we can observe and measure from a corpus, and what we do not know about a language. To achieve this, ambitious statistics is needed.

2.3 Why alternations matter to linguists

Verbs express events. Events involve participants. Participants occupy “places” in the clauses that verbs control. These places are called arguments. The verb drink, for example, is a two-argument verb because in a drinking event there is at least a drinker and a liquid that is drunk. In the active voice, the drinker occupies the subject position and the liquid the object position as in (i):

Suppose Bob wants to brush the whiskey off his breath before setting off to the linguistics lab. He is now holding his toothbrush in one hand and a tube of toothpaste in the other. Four participants are involved: Bob, his teeth, his toothbrush, and the toothpaste. In sentence (ii), three participants are assigned a semantic role by the verb: Bob is the agent (the initiator of the event), toothpaste is the theme (an inanimate entity undergoing a change of location from the brush to the teeth), and his teeth is the goal (the end-point of a motion). The toothbrush is not syntactically realized. The ensuing scene can be described using either (ii a) or (ii b):

In (ii a), his teeth is the object of the verb and toothpaste is the object of the preposition with. In (ii b), toothpaste is the object and his teeth is the object of the preposition onto. This means that our attention is brought to his teeth in (ii a) and to toothpaste in (ii b). The phenomenon of a verb exhibiting variation in its syntactic realization is called an alternation. Each alternating form is called an alternant.

Linguists have long believed that alternations were conditioned by verb meaning, two verbs with identical or similar meaning displaying similar alternating behavior [14]. To illustrate this, suppose now that Bob proceeds to shaving. The verb slather involves three participants: Bob, his face, and the shaving cream. Again, the scene can be described using either alternant in (iii):

Given that brush and shave are close in meaning (both associated events imply that a ‘thick mass’ is transferred from a container to a body part), we should not be surprised to see that they display the same alternation. Linguists might be enticed to include brush and slather in the same typological verb class, e.g. the class of “grooming verbs”. However, they are somehow different. For example, even though neither verb allows the theme argument to stand alone, as shown in (iv), only brush allows the goal to stand alone, as evidenced in (v): ^[1]

The question remains open as to whether one should (a) include both verbs in the same class, acknowledging that the class is heterogeneous, or (b) assign them to related but distinct classes, at the risk of demultiplying classes and blunting Ockham’s razor.

Such puzzles are central in an area of linguistics called argument realization, i.e. the study of the syntactic patterns that the arguments of a verb may enter [15]. In (ii) and (iii), the challenge is to explain why two apparently similar verbs display diverging behaviors and why the divergences takes these forms. Addressing these issues matters because of their far-reaching implications for our understanding of language. From a cognitive perspective, one may wonder how speakers classify store verbs in their mental inventories of linguistic units.

3.1 Theoretical issues

The dative alternation has been a fruitful research topic in many different theories. Substantial accounts of past research can be found for instance in Levin [16], Krifka [14] and Levin and Rappaport Hovav [15, chapter 7].

Chomsky [5, 6] suggests that an alternating verb has a single lexical entry for both forms. These forms have the same deep syntactic structure. Differences visible at the sentence level are explained by the fact that the surface structure of the basic form is a direct projection of the deep structure, whereas the surface structure of the derived form is the product of a transformation.

Subsequent Chomskyan studies holding a distinction between deep and surface structures debate over which variant of the dative alternation is transformationally derived from the basic argument realization. Conclusions differ. On the one hand, Fillmore [17], Hall [18], and Emons [19] contend that PD is basic whereas DO is derived. On the other hand, Burt [20] and Aoun and Li [21] argue for the opposite pattern of transformation: DO is basic whereas PD is derived.

Semantic restrictions to the dative alternation have challenged Chomskyan accounts. One restriction is that certain verbs alternate while others readily enter only one variant: ^[2]

Proponents of the Localist Hypothesis [22], according to whom locative expressions are seen as the source from which all more abstract expressions derive, construe the recipient as a spatial goal. They further argue that DO is possible in (xi b) if London refers metonymically to a person or an institution, in which case it differs from (xi a) where London is clearly a place:

A second restriction is the frequent lack of semantic equivalence between alternating forms in cases where the verb readily enters both variants [23, 24], as in (xii):

DO conveys a sense of completion in such a way that the teaching is successful in (xii b). Example (xii a) is more neutral in this respect. However, more recent studies warn that these semantic differences are intuitive and may be subject to contextual modulation [15, 25, 26].

Despite continuous efforts to maintain that alternating verbs have a single meaning underlying both formal variants [27–29], there is now cross-theoretical consensus that the two variants of the dative alternation have distinct semantic representations. According to Pinker [30] and Rappaport Hovav and Levin [31], caused motion underlies PD, whereas caused possession underlies DO, as schematized in (xiii):

In a similar fashion, Speas [32, pp. 88–89] schematizes the semantic representations of both variants as follows:

In the Construction Grammar framework, Goldberg [10] posits that PD is a subtype of the more general caused-motion construction (cf. the Localist Hypothesis), whereas DO expresses a transfer of possession:

The above finds empirical support in Gries and Stefanowitsch [33].

Given that the distribution of verbs across the dative variants is semantically constrained, and given the frequent lack of semantic equivalence between PD and DO for a given verb, a set of semantic factors have been recognized to influence the choice of PD vs. DO. Among the known lexical semantic restrictions applying to verbs in the dative alternation are the following:

1. Movement (PD) vs. possession (DO): in PD, the theme undergoes movement (literal or figurative) from an origin to a goal, whereas in DO the agent possesses the theme via the verb event.

2. Affectedness: as seen in (xii), the recipient of a dative verb is more likely to receive an affected interpretation when expressed as the first object in DO than in PD;

3. Continuous imparting of force: in PD, the verb can express a continuous imparting of force (e.g. haul, pull, push). DO shows a dispreference for such verbs (^??Will pushed Anthony the biscuits). Under certain conditions, exceptions occur [34].

4. Communication verbs: as opposed to speech act verbs (tell, read, write, cite, etc.) and verbs derived from nouns expressing communication means (fax, email, phone), which can occur in PD or DO, verbs that denote a manner of speaking (shout, yell, scream, whisper, etc.) have a strong dispreference for DO. Exceptions are listed in Gropen et al. [35].

5. Verbs of impeded possession: such verbs (deny, spare, cost) have a preference for DO.

6. Latinate verbs: due to their morphophonology, such verbs (donate, explain, recite, illustrate, etc.) disprefer DO, except when they express a future possession (guarantee, assign, offer, promise), as pointed out by Pinker [30, p. 216].

Lexical semantic restrictions are sometimes overridden by information-structure factors (interalia [36–38]). Such factors have to do with how information is formally packaged in a sentence. The first factor is discourse givenness, that is to say the fact that the reference of an expression is present in the minds of speakers. In general, given material precedes new material. PD is expected when the theme is more given than the recipient, as in (xvi a), whereas DO is more likely when the recipient is more given than the theme, as in (xvii b):

The second factor is a corollary of the first: because recipients are typically human and themes typically inanimate, they are more likely to be given and thus to occur before themes. In this respect, DO is more frequent than PD. Bresnan and Nikitina [39] find empirical support for this, but they also find exceptions such as (xviii a):

Although peripheral, the third factor, heaviness, is correlated with information-structure considerations. Heaviness is characterized by the complexity and/or length of sentence constituents. Heavy material comes last, as exemplified below:

Because given material is generally shorter than non-given material (e.g. given recipients will generally occur in the form of pronouns), DO is the preferred realization of the dative alternation due to the last two factors.

Which factor(s) take(s) precedence over the other(s) is still theoretically unclear. Snyder [40] claims that information-structure factors are more important than heaviness, whereas Arnold et al. [36] treat all factors on equal footing. What is clearer is that what determines the dative alternation is a multifactorial problem whose full understanding is best resolved empirically. This is why we now turn to recent corpus-based, statistics-driven investigations of the dative alternation.

3.2 Corpus-based answers

Since Williams [41], the dative alternation has become a model construction for benchmarking predictive methods [2, 36, 42–44]. Focusing on DO, Williams [41] uses the logistic procedure to test on a two-part but limited data set (original data set, sample size is 168; aggregate data set, sample size is 59). The model construction includes 8 variables: syntactic class of verb, register, modality, givenness of goal, prosodic length of goal vs. theme, definiteness of goal, animacy of goal, and specificity of goal. Williams [41] finds that not all independent variables are predictors of the position of the goal. Only three reach a relatively high level of significance in the model: the prosodic length of goal vs. theme (the length of the goal is shorter than the length of the theme), syntactic class of verb (ditransitive), and register (informal).

Arnold et al. [36] investigate the effects of newness and heaviness on word order in the dative alternation. Their data consists of debate transcriptions from the Canadian parliament (the Aligned-Hansard corpus). Utterances are manually annotated for: constituent order (non-shifted vs. shifted; prepositional vs. double object), heaviness (three categories of relative length measured as follows: number of words in the theme minus number of words in the recipient), and newness (given, inferable, or new). Arnold et al. [36] conclude that heaviness and newness are significantly correlated with constituent order. DO is preferred when the theme is (a) newer and (b) heavier than the goal.

Gries [43] uses linear discriminant analysis to investigate the effect of multiple variables on the choice of PD vs. DO. in the British National Corpus. Gries observes that all properties of NPgoal along with morphosyntactic variables have the highest discriminatory power. However, (a) discriminant analysis makes distributional assumptions that are seldom satisfied by the data, and (b) Gries [43] concedes that the data set is limited: being part of a larger project, it consists of only 117 instances of the dative alternation.

To circumvent assumptions about the data distribution and to control for the influence of multiple variables on a binary response, Bresnan et al. [42] use (mixed-effects) logistic regression, like Williams [41] and Arnold et al. [36]. Unlike those previous works, Bresnan et al. [42] predicting’s data set is relatively large, consisting of 2,360 dative observations from the 3M-word Switchboard collection of recorded telephone conversations. More importantly, the authors also address the question of circular correlations, which are largely ignored in former statistical models, e.g.:

1. personal pronouns are short, definite and have animate, discourse-given referents;

2. animate, discourse-given nominals are often realized as personal pronouns, which are short and definite.

Bresnan et al. [42] ’s dative data set is annotated for 14 explanatory variables whose influence on the choice of the dative variants is considered likely: modality, verb, semantic class of verb use, and length, animacy, ^[3] definiteness, ^[4] pronominality, and accessibility of recipient/theme; see also Section 4.1. One of their logistic regression models predicts which variant of the dative alternation is used with high accuracy.

Using Bresnan et al.’s data set, Baayen [2] tests naive discriminative learning (henceforth NDL) on the dative alternation. Baayen compares NDL to other well-established statistical classifiers such as logistic regression [42, 45], memory-based learning [44, 46], analogical modeling of language [47], support vector machines [48], and random forests [49]. He addresses two questions:

1. how can statistical models faithfully reflect a speaker’s knowledge without underestimating or overestimating what a native speaker has internalized?;

2. how do occurrence and co-occurrence frequencies in human classification compare to such frequencies in machine classification?

Like memory-based learning, NDL stands out because it reflects human performance. Unlike parametric regression models, it is unaffected by collinearity issues. When two or more predicting variables are highly correlated, multiple regression models may indicate how well a group of variables predicts an outcome variable, but may not detect (a) which individual predictor(s) improve the model, and (b) which predictors are redundant. Unlike memory-based learning however, NDL does not need to store exemplars in memory to capture the constraint networks that shape linguistic behavior. Such exemplars are merged into the weights [54, p. 320].

Baayen [54] corpus fits a NDL model with the following predictors: verb, semantic class of verb use, and length, animacy, definiteness, accessibility, and pronominality of recipient and theme. NDL provides a very good fit to the dative data set, which compares well to predictions obtained with other classifiers such as memory-based learning, mixed-effects logistic regression and support vector machine.

The prediction of the dative alternation is now a well-travelled path in quantitative linguistics, as evidenced by the high accuracy of the most recent methods. Yet, the community is in midstream. There is far more to the dative alternation than its prediction, since predicting is not explaining. We believe that this distinction is worth maintaining both at the conceptual and the operational levels. This idea is the backbone of our article.

4.1 Data

We used the dative data set available in the languageR package [42, 55]. It contains 3,263 observations consisting of 15 variables. The variables divide into:

1. speaker, a categorical variable with 424 levels, including NAs;

2. modality, a categorical variable with 2 levels: spoken vs. written;

3. verb, a categorical variable with 75 levels: e.g. accord, afford, give, etc.;

4. semantic class, a categorical variable with 5 levels: abstract (e.g. give in give it some thought), transfer of possession (e.g. send), future transfer of possession (e.g. owe), prevention of possession (e.g. deny), and communication (e.g. tell);

5. length in words of recipient, an integer valued variable;

6. animacy of recipient, a categorical variable with 2 levels: animate vs. inanimate;

7. definiteness of recipient, a categorical variable with 2 levels: definite vs. indefinite;

8. pronominality of recipient, a categorical variable with 2 levels: pronominal vs. nonpronominal;

9. length in words of theme, an integer valued variable;

10. animacy of theme, a categorical variable with 2 levels: animate vs. inanimate;

11. definiteness of theme, a categorical variable with 2 levels: definite vs. indefinite;

12. pronominality of theme, a categorical variable with 2 levels: pronominal vs. nonpronominal;

13. realization of recipient, a categorical variable with 2 levels: PD vs. DO;

14. accessibility of recipient, a categorical variable with 3 levels: accessible, given, new;

15. accessibility of theme, a categorical variable with 3 levels: accessible, given, new.

We considered speakers coded NA as mutually independent speakers, also independent from the set of identified speakers. About 80% of the identified speakers contribute more than one construction. This is a source of dependency between observations.

The approach we develop below takes this dependency into account. For the sake of clarity, we describe our approach in the context of independent observations. However, our results were obtained considering dependency.

4.2 Predicting and explaining the dative alternation

Our goal is to both predict and explain the dative alternation in English. In the next two subsections, we rephrase these two challenges in statistical terms. In a unifying probabilistic framework reflecting subject-matter knowledge, we specifically elaborate two statistical parameters targeted toward the above two goals. By “subject-matter knowledge” we mean what has been operationalized from what linguists know about the dative alternation and, more specifically, our data set. The parameters differ substantially because the two goals are radically different.

5.1 Machine learning prediction

We consider first the inference of Q(P0) as defined in eq. (2). The literature on the topic of classification, both from the theoretical and applied points of view, is too vast to select a handful of outstanding references. Instead of choosing one particular approach, we advocate for considering all our favorite approaches, seen as a library of algorithms, and combining them into a meta-algorithm drawing data-adaptively the best from each of them. Many methods have been proposed in this spirit, now gathered under the name of “ensemble learners” (see to cite only a few seminal works, with an emphasis on methods using the cross-validation principle [65–69]). Specifically, we choose to rely on the super-learning methodology [3, 70].

We now give a nutshell description of the super-learning methodology. Say that we have n independent observations O1=(W1,Y1),…,On=(Wn,Yn) drawn from P0 and an arbitrarily chosen partition of {1,…,n}, i.e. a collection of sets {T(ν)⊂{1,…,n}:1≤ν≤V} such that ∪ν=1VT(ν)={1,…,n} (their union covers {1,…,n}) and for each 1≤ν1≠ν2≤V, T(ν1) ∩ T(ν2)=∅ (the sets are pairwise disjoint). For convenience, we introduce the notation Pn,S to represent the subset {Oi:i∈S} of the complete data set, represented by Pn, corresponding to these observations index by i∈S⊂{1,…,n}. We use the data to infer the best combination of K algorithms Qˆ1,…,QˆK which map any subset of the data set to a function from W to [0,1]. For instance, Qˆ1(Pn,T(2))(W) is the predicted conditional probability that Y=1 given W according to the first algorithm trained on {Oi:i∈T(2)}. Among a variety of possible ways to combine Qˆ1,…,QˆK we decide to resort to convex combinations: thus, for each α∈A={a∈R+K:∑k=1Kak=1}, we define Qˆα=∑k=1KαkQˆk, the meta-algorithm mapping any subset Pn,S of the data set to the function Qˆα(Pn,S)=∑k=1KαkQˆk(Pn,S) from W to [0,1]. Note that if every Qˆk produces functions mapping W to [0,1] then so does Qˆα for any α∈A.

Recall that the risk RP0L(Qˆα(Pn,S))=EP0{L(Qˆα(Pn,S),O)} quantifies how close Qˆα(Pn,S) is to Q(P0), the parameter of P0 that we wish to target. Of course, we cannot compute RP0L(Qˆα(Pn,S)) in general because we do not know P0. Its estimator

is known to be over-optimistic, since the same data are involved in the construction of Qˆα(Pn,S) and in the evaluation of how well it performs. Cross-validation offers a powerful way to circumvent this: the cross-validated estimator

(we slightly abuse notation) accurately evaluates how good are the estimators of Q(P0) produced by the α-indexed meta-algorithm Qˆα. They key is that in each term of the RHS of eq. (13), the subset of data used to “train” Qˆα, represented by Pn,T(ν), and the subset used to evaluate its performances, represented by Pn,T(ν)c, are disjoint. This motivates the introduction of

the minimizer of the cross-validated risk, which finally yields the super-learner

by training Qˆαn on the complete data set. It can be shown that, if every Qˆk produces functions mapping W to [0,1] then the super-learner performs almost as well as the so-called “oracle” (since it cannot be inferred without knowing the true law P0) best algorithm in the library. We refer the reader to Section A.2.1 for a more accurate mathematical statement of this remarkable fact.

5.2 Targeted minimum loss explanation

We now turn to the estimation of Ψ1(P0), Ψ2(P0) and Ψ3(P0) as defined in eqs (6), (8) and (12). We take the route of TMLE, a paradigm of inference based on semiparametrics and estimating functions (see Chapter 25, for recent and comprehensive introductions [71, 72]). Introduced by van der Laan and Rubin [4], TMLE has been studied and applied in a variety of contexts since then (we refer to for an overview [3]). An accessible introduction to TMLE is given in (Sections 12, 13 and 14 [58]).

It is apparent in eqs (6), (8) and (12) that the parameters Ψ1(P0), Ψ2(P0) and Ψ3(P0) all depend on Q(P0). Let us assume that we have already built an estimator of Q(P0), which we denote by Qninit– that could be, for instance, the super-learner Qˆαn(Pn) whose construction we described in Section 5.1. Here, the superscript “init” indicates that we think of Qninit as an initial estimator of Q(P0) built for the sake of predicting, not explaining.

Taking a closer look at eqs (6), (8) and (12), we see that it is easy to estimate Ψ1(P0), Ψ2(P0) and Ψ3(P0) by relying on Qninit. Consider eq. (6): if we substitute Qninit for Q(P) in the formula, then only the marginal law of W−1 is left unspecified. The simplest way to estimate the latter, which can be shown to be the most efficient too, is to use its empirical counterpart. That means estimating the marginal law of W−1 by the empirical law under which W−1=Wi−1, the ith observed value of W−1 in the data set, with probability 1/n. Substituting the empirical marginal law of W−1 for its counterpart under P in eq. (6) yields an initial estimator of Ψ1(P0), say ψn1,init, writing as

[Correction added after online publication 11 December 2015: “Substituting the empirical marginal law of..” should read “Substituting the empirical marginal law of W−1”]

Likewise, the parameter Ψ2(P0) can be simply estimated by ψn2,init=(ψk,n2,init:1≤k≤K) with

while the parameter Ψ3(P0) can be estimated by

Interestingly, the optimization problem eq. (15) can be solved easily, see Section A.3.3.

Arguably, ψn1,init, ψn2,init and ψn3,init are not targeted toward Ψ1(P0), Ψ2(P0) and Ψ3(P0) in the sense that, although they are obtained by substitution, the key estimator Qninit which plays a crucial role in their definitions was built for the sake of prediction and not specifically tailored for estimating either Ψ1(P0), Ψ2(P0) or Ψ3(P0). In this respect, the targeting step of TMLE can be presented as a general statistical methodology to derive new substitution estimators from such initial estimators so that the updated ones really target what they aim at.

Targeting is made possible because Ψ1, Ψ2 and Ψ3, seen as functions mapping M to [−1,1], [−1,1]K and Θ, respectively, are differentiable, see Section A.3.1. In these three cases, the resulting gradients (derivatives), denoted by ∇Ψ1, ∇Ψ2 and ∇Ψ3, drive our choices of estimating functions. Targeting the parameter of interest consists in (a) designing a collection {Qn,εinit:ε∈E} of functions mapping W3 to [0,1] conceived as fluctuations of |Qninit=Qn,εinitε=0 in the direction of the parameter of interest, and (b) identifying that specific element of the collection which better targets the parameter of interest, see Section A.3.2. Let us denote by Qn1,targ=Qn,εn1init, Qn2,targ=Qn,εn2init and Qn3,targ=Qn,εn3init the three a priori different fluctuations of Qninit that respectively target Ψ1(P0), Ψ2(P0), and Ψ3(P0). They finally yield, by substitution, the three estimators

The optimization problem (19) can be solved easily just like eq. (15), see Section A.3.3.

The above estimators satisfy ψn1,targ=Ψ1(Pn1,targ), ψk,n2,targ=Ψk2(Pn2,targ), ψn3,targ=Ψ3(Pn3,targ) for three empirical laws Pn1,targ,Pn2,targ,Pn3,targ∈M. They are targeted in the sense that they satisfy EPn{∇Ψ1(Pn1,targ)(O)}=0, EPn{∇Ψ2(Pn2,targ)(O)}=0, EPn{∇Ψ3(Pn3,targ)(O)}=0, three equalities which are the core of the theoretical study of their asymptotic properties. The two main properties concern the consistency of the estimators and the construction of asymptotic confidence intervals. An estimator is consistent if it converges to the truth when the sample size goes to infinity. The targeted estimators defined in eqs (17), (18) and (19) are double-robust: the stronger requirement for them to be consistent is that either the corresponding targeted estimator of Q(P0), say Qntarg, converge to Q(P0)or the conditional law of the variable whose importance is sought given the other components of W, say g(P0), be consistently estimated by, say, gn. Furthermore, the stronger requirement to make it possible to build asymptotically conservative confidence intervals is that the product of the rates of convergence of Qntarg to Q(P0) and of gn to g(P0) be faster than 1/n. Finally, we wish to acknowledge that it is possible to target all parameters with a single, specifically designed, richer collection of fluctuations. Targeting all parameters at once enables the construction of simultaneous confidence regions that better take the mutual dependency of the estimators into account. In a problem with higher stakes, we would have gone that bumpier route.

6.1 Categorical contextual information variables

Let us now comment on the results of Table 1. We disregard the estimates whose p-values are large, because they correspond to insignificant results. We arbitrarily set our p-value threshold to 1%. An estimate ψn of the effect of setting W=w1 as opposed to setting W=w0 can be interpreted as follows: all other things being equal, the probability of obtaining a PD construction increases/decreases additively by ψn when W is set to w1 as opposed to. Ranked by decreasing magnitude of the estimates, we obtain:

1. a 38.24% decrease when accessibility of recipient switches from accessible to new;

2. a 16.57% increase when semantic class switches from abstract to communication meaning;

3. a 14.71% decrease when semantic class switches from abstract to future transfer of possession meaning;

4. a 13.98% decrease when pronominality of recipient switches from nonpronominal to pronominal;

5. a 11.68% decrease when pronominality of theme switches from nonpronominal to pronominal, see examples (xxii) and (xxiii);

6. a 11.52% increase when semantic class switches from abstract to transfer meaning;

7. a 9.38% increase when animacy of recipient switches from animate to inanimate, see example (xx);

8. a 9.28% decrease when semantic class switches from abstract to prevention of possession meaning;

9. a 8.43% increase when animacy of theme switches from animate to inanimate;

10. a 7.82% decrease when accessibility of theme switches from accessible to new;

11. a 5.68% decrease when definiteness of theme switches from definite to indefinite, see example (xxi);

12. a 3.95% increase when definiteness of recipient switches from definite to indefinite.

Consider for instance example (xx): under the constraint “set the animacy of recipient to inanimate”, the speaker selects either (xx a) or (xx b); under the constraint “set the animacy of recipient to animate”, she selects either (xx c) or (xx d). What matters is the extent to which the probability to select the PD construction is altered when one switches from one constraint to the other. Even if linguists might find (xx d) slightly more natural than (xx c), (xx a) is undoubtedly more natural than (xx b). This is consonant with our result, which states that the probability of the PD construction increases when the animacy of recipient is set from animate to inanimate.

Illustrating the inferred statement about the effect of definiteness of theme is challenging. We see this as a welcome opportunity to emphasize the singularity of our statistical approach. To produce a convincing example, we have to choose a longer theme than before. Indeed, linguists know for a fact that when the theme is long, PD is dispreferred. In example (xxi), one can conceive that the preference of (xxi d) over (xxi c) is slightly stronger than that of (xxi b) over (xxi a). This is consonant with our result, which states that the probability of the PD construction decreases slightly when the definiteness of theme is set from definite to indefinite.

Example (xxi) is clearly counterintuitive to linguists used to interpreting results from logistic regression models. This is a common pitfall. It is due to the belief that the interpretation of a fitted logistic regression still holds even when the true law does not belong to the logistic model. This is never the case. From a mathematical point of view, the parameter matching definiteness of theme in a logistic regression model is a very awkward function of the true law. No matter how awkward the function is, no sensible interpretation can be built without it. In contrast, the parameter we define and estimate to assess the effect of definiteness of theme is a rather simple function of the true law. Moreover, its simple statistical interpretation is buttressed by a causal interpretation, at the cost of untestable assumptions. The above lines epitomize the approach defended in this article.

How do statisticians intuit then? Denote W1 the definiteness of theme (W1=1 for indefinite and W1=0 for definite), W2 the length of theme, and consider this baby model, tweaked for demonstration purposes. Say, contrary to facts, that the true difference P0(Y=1|W1=1,W−1)−P0(Y=1|W1=0,W−1) depends on W−1 only through a thresholded version of W2. More precisely, say that