Surprisal

Shannon (Reference Shannon1948) laid down the foundation of information theory in describing a mathematical theory of communication in terms of the expectancy of words in context. To ensure a successful transfer through a channel with limited capacity, the error rate of a signal can be reduced by adding redundancy in such a way that the channel capacity is optimally utilized (Shannon Reference Shannon1948). This is possible as words (or constituents) are not independent of one another (2).

On the one hand, cake is highly expected after bake in (2a). See, on the other hand, does not make cake very likely in (2b). Being, thus, less expected in (2b), cake carries more information there than in (2a). The information of an element and its probability are negatively correlated: The more likely an element is, the less information it carries and vice versa. This likelihood can be calculated as the surprisal value (Hale Reference Hale2001):

$$S\left( {unit} \right) = - {\log _2}p(unit|context)$$

The surprisal value is connected to processing difficulty (Hale Reference Hale2001). High surprisal values indicate more processing effort (e.g. Hale Reference Hale2001, Demberg & Keller Reference Demberg and Keller2008). Accordingly, surprisal has been used as an approximation to and prediction for processing difficulties in both modern (Levy & Jaeger Reference Levy, Florian Jaeger, Schölkopf, Platt and Hofmann2007, Demberg & Keller Reference Demberg and Keller2008, Gibson et al. Reference Gibson, Futrell, Piantadosi, Dautriche, Mahowald, Bergen and Levy2019, Boyce & Levy Reference Boyce and Levy2023 among others) and historical data (e.g. Degaetano-Ortlieb et al. Reference Degaetano-Ortlieb, Kermes, Khamis, Teich, Suhr, Nevalainen and Taavitsainen2018, Degaetano-Ortlieb & Teich Reference Degaetano-Ortlieb and Teich2022).

The surprisal values are, of course, also dependent on the amount of context, described as n-grams. Unigram-models, as used by Cuskley and colleagues (Reference Cuskley, Bailes and Wallenberg2021) and Wallenberg and colleagues (Reference Wallenberg, Bailes, Cuskley and Karl Ingason2021), only consider the frequency of a word. Unigram-surprisal can still be linked to processing as frequent words are easier to process (e.g. Divjak Reference Divjak2019, Díaz-Campos & Balasch Reference Díaz-Campos and Balasch2023).

Furthermore, “[t]here is ample evidence … that sentence comprehension is incremental: we do not wait until we have heard an entire sentence to start disambiguating and comprehending” (Levy Reference Levy2008:1129). Levy (Reference Levy2008) argues that recipients implicitly predict the upcoming word based on the input they have received thus far. After having comprehended only partial input, they keep various options about the sentence continuation in mind. Recipients assign different probabilities to these possible continuations. Cognitive capacity is consumed when a probable continuation is ruled out and the other possible clause continuations obtain a new (higher) probability assignment (Levy Reference Levy2008).

This has implications for word order. Most constituent functions are only assigned once per clause. Accordingly, having seen a subject eliminates the expectation of seeing another subject and the likelihood that any other object is presented next increases. This also holds for the verb in (S)OV-clauses (Levy Reference Levy2008). The more constituents have appeared in the clause, the more likely the verb becomes (Levy Reference Levy2008, Ferrer-i-Cancho Reference Ferrer-i-Cancho2024). However, predicting the verb lexeme is much harder in SOV languages like Japanese (Engelhardt, Filipović, & Hawkins Reference Engelhardt, Filipović and Hawkins2024). Engelhardt and colleagues (Reference Engelhardt, Filipović and Hawkins2024) also show that predicting a certain noun after having seen a verb is more successful.

To summarize: Processing difficulties are linked to the expectancy of words (or constituents) in a clause. The context can change how likely a word is; thus, the likelihood depends on the context, but also on the pure frequency of a specific word. The more likely or frequent a word is, the less cognitive effort is required to understand that word. The likelihood of a word either in a context or without a context can be calculated using the surprisal equation.

The Uniform Information Density hypothesis

Surprisal theory implies that elements differ in informativeness. This idea has been taken up by Aylett & Turk (Reference Aylett and Turk2004) in the Smooth Signal Redundancy hypothesis and Levy & Jaeger (Reference Levy, Florian Jaeger, Schölkopf, Platt and Hofmann2007) in the Uniform Information Density (UID) hypothesis. The two hypotheses basically state that speakers distribute the information in a clause as evenly as possible (see also ideas in, for instance, Fenk-Oczlon Reference Fenk-Oczlon1983). The advantages of UID are described by Collins (Reference Collins2014:656): “By transmitting the signal at a constant rate …, speakers ensure that their message is being communicated as rapidly as possible. At the same time, since every component of the utterance carries roughly the same amount of information, the signal is robust enough to withstand degradation from noise.”

Accordingly, speakers aim to avoid peaks as well as troughs in the information profile. Levy & Jaeger (Reference Levy, Florian Jaeger, Schölkopf, Platt and Hofmann2007) focus on that-omission at the onset of a relative clause. They show that the relative pronoun is omitted when the first word of the relative clause is highly expected (Levy & Jaeger Reference Levy, Florian Jaeger, Schölkopf, Platt and Hofmann2007). In this case, the first word of the relative clause must carry its own meaning and indicate the relative clause. This double task is too much to process when the first word of the relative clause is itself highly informative (Levy & Jaeger Reference Levy, Florian Jaeger, Schölkopf, Platt and Hofmann2007). They claim that an overt or omitted relative pronoun changes the information profile and helps to smooth out the information profile of the whole (relative) clause. The UID hypothesis has been successfully applied in various studies, but there have also been mixed and negative results (see Juzek (Reference Juzek2024) for an overview). Recently, UID has also been employed in studies about word order variation.

Collins (Reference Collins2014:652) “reviews three psycholinguistic models of the relationship between syntactic alternations and processing: Maximum Per Word Surprisal (MPWS) … Uniform Information Density (UID) (Levy & Jaeger Reference Levy, Florian Jaeger, Schölkopf, Platt and Hofmann2007; inter alia), and Dependency Length Minimization (DLM)” using a context-free grammar of English. He hypothesizes that a more uniform word order is preferred over a less frequent one and asks whether this smoother information profile is achieved through “great global variability, but minimal local variability” or a profile “centered globally around a mean” (Collins Reference Collins2014:657). Whether UID in both forms can replicate DLM predictions or explain cases where DLM makes none is tested in a choice experiment. Participants favor the alternative with a more uniform information profile, regardless of its form (Collins Reference Collins2014:672). Meister and colleagues (Reference Meister, Pimentel, Haller, Jäger, Cotterell and Levy2021) come to similar conclusions.

In contrast to Collins (Reference Collins2014), Cuskley and colleagues (Reference Cuskley, Bailes and Wallenberg2021) and Wallenberg and colleagues (Reference Wallenberg, Bailes, Cuskley and Karl Ingason2021) investigate the influence of word order variation on the information profile using word and lemma probability. Cuskley and colleagues (Reference Cuskley, Bailes and Wallenberg2021) state that a uniform information profile is one with as little clustering of low or high surprisal values as possible. They develop DORM to calculate this uniformity. Basically, they “take the arithmetic mean of every adjacent, overlapping pair of information content values (i.e. a rolling mean) … then compute the sample variance of the resulting list of means (an unbiased estimator, using denominator n − 1) and arrive at a single number, the DORM” (Cuskley et al. Reference Cuskley, Bailes and Wallenberg2021:4). The calculation of the DORM is described in section 4.2.

To sum up: The Uniform Information Density hypothesis states that information should be distributed as uniformly as possible over a sentence, that is, one should avoid peaks and troughs. Originally it was used to describe and explain the variation of single words. Lately, word order variations and their influence on the uniformity of a whole sentence have come into focus.

Information structure in terms of givenness

One of the most influential factors to cause the Acc>Dat order is information structure (Féry & Krifka Reference Féry, Krifka and van Sterkenburg2008), especially givenness (Lenerz Reference Lenerz1977, Rauth Reference Rauth2020). The given>new constraint states that information that is presumed to be familiar to the recipient is placed before new information. Still, the constraint should not be interpreted as always requiring given information to precede new, but rather as a guideline that can sometimes override the default constituent order of Dat>Acc, as in (3).

In (3), the carrots have already been mentioned in the context sentence and, thus, are given, while the horse has not been mentioned in the context. It is new information. Placing carrots before horse violates Dat>Acc, but it keeps the preference for given>new.

Following Féry & Krifka (Reference Féry, Krifka and van Sterkenburg2008), Prince (Reference Prince and Cole1981), and Gundel and colleagues (Reference Gundel, Hedberg and Zacharski1993), referents that have not been mentioned in the previous discourse are considered new. Besides given and new referents, there are also inferable referents (e.g. Prince Reference Prince and Cole1981). These referents need not have been mentioned in the discourse, but they relate to other events or referents: For example, every school class has a teacher. Accordingly, the accommodation of a teacher in a school context is expected. Teacher would be annotated as inferable.

While Jachmann and colleagues (Reference Jachmann, Drenhaus, Delogu and Crocker2024) found that inferable referents are regarded more like new referents, Rauth (Reference Rauth2020) describes a similar behavior for inferable and given referents in the way that they are more likely to precede new referents. In general, givenness is described by Rauth (Reference Rauth2020) as one of the most relevant factors for the object order in dialects. When the accusative object is given and the dative object is new, the Acc>Dat order is favored, as in (3). This factor is stable throughout MHG and ENHG. Givenness seems, however, to become less crucial over time (Rauth Reference Rauth2020:314).

Length

Behaghel (Reference Behaghel1932:6) has shown the importance of length in his Gesetz der wachsenden Glieder (‘law of increasing constituents’). If there is a length difference, the longer argument is placed behind the shorter one. In (4), the accusative object has only two words, but the dative object has six words. Therefore, the Acc>Dat order can be used.

Heine & König (Reference Heine and König2010) discussed the importance of the constituent length for the deviation from the unmarked constituent order also for other languages with a more restricted word order than German: “Whenever there is a heavy argument then it follows the light one” (Heine & König Reference Heine and König2010:102), but still, they also say that information-structural notions have a stronger influence.

In general, variable object order can be found in many languages. Especially English word order variation is connected to heaviness and offers the possibility to put DOs before IOs (Dative Alternation; see e.g. Bresnan Reference Bresnan, Featherston and Sternefeld2007), which must then be represented as prepositional phrases (PPs). The same type of alternation is also, for example, possible in Dutch (Colleman Reference Colleman2009) and northern languages like Swedish (Valdeson Reference Valdeson2021).

Information Theory

Ortmann and colleagues (Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024) have proposed an information-theoretic account for the variable object order in the German middle field. Their hypothesis is that a specific object order is chosen to create a smooth flow of information while the alternative word order would have led to a more uneven information profile. They conducted a corpus study using the SdeWaC corpus (Faaß & Eckart Reference Faaß, Eckart, Gurevych, Biemann and Zesch2013) from which they automatically extracted more than 13,000 object pairs. These pairs are both located in the same middle field, have the same length, and one definite and one indefinite article. The latter serves as an approximation to givenness (see e.g. Heim (Reference Heim1982) or Hauenschild (Reference Hauenschild, Jacobs, von Stechow, Sternefeld and Vennemann1993) for the connection between givenness and definiteness). Lemma-based bigram models were used to estimate the probability of words. Then, the DORM values for each sentence were calculated on the mean surprisal value of the constituents.

To test their hypothesis, the authors created a variant for each sentence with the reversed object order (Acc>Dat if the original sentence held Dat>Acc and vice versa) and recalculated both the bigram-surprisal values and the DORM values. The DORM values of the variant were subtracted from the DORM values of the original clauses to create “DORM_diff” (Ortmann et al. Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024:71). The authors expected that value to be significantly below zero if the not-produced variant had a less uniform information profile. However, they did not find the expected significant result for the Acc>Dat order, though the DORM_diff was negative on average but not significantly different from zero (Ortmann et al. Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024:73). For the Dat>Acc order, they were able to show the expected results (Ortmann et al. Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024:73).

Next, the authors conducted general logistic regression to predict the object order in the original sentences (Ortmann et al. Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024:75–79). They found: “A higher DORM_case reduces the likelihood of observing DAT > ACC. … Hence, a more uniform, smoother distribution for the order ACC > DAT increases the likelihood of observing this marked order in the sentence” (Ortmann et al. Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024:77). Furthermore, DORM_giv can also predict the order of definite and indefinite objects (Ortmann et al. Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024:79). The known influence of definiteness is also a relevant predictor for the object order (Ortmann et al. Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024:77–79): The authors show that the definite object is mostly placed before the indefinite one.

The current study uses methods from Ortmann and colleagues (Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024), namely the application of DORM and the methods to automatically find dative and accusative objects in the same middle field.

The topological field model

Ortmann and colleagues (Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024), Rauth (Reference Rauth2020), and the current study focus on the relative order of the dative and accusative objects in the so-called middle field, the position between the Linke Satzklammer (‘left sentence bracket’, LSB) and the Rechte Satzklammer (‘right sentence bracket’, RSB) (Wöllstein Reference Wöllstein2014). The description of German clauses using the Topologisches Feldermodell ‘topological field model’ captures the difference between the position of the finite and lexical verb best (Haftka Reference Haftka, Jacobs, von Stechow, Sternefeld and Vennemann1993, Wöllstein Reference Wöllstein2014, Telljohann et al. Reference Telljohann, Hinrichs, Kübler, Zinsmeister and Beck2017, Höhle Reference Höhle, Müller, Reis and Richter2019).Footnote ⁶

There, German clauses are described in terms of the position of the finite verb. This does not, however, suffice with reference to the position of the lexical verb. Only in the synthetic verb forms present and preterit active are the lexical and finite verbs the same and they stand in the LSB in independent main (V2) clauses. All other tenses and passive voice are analytic. Thus, the finite verb can be placed in the LSB, but the lexical verb is in the RSB. Independent of synthetic or analytic tenses and voices, both finite and infinite verbs are in the RSB in subordinate clauses that are opened by a relative, a wh-pronoun, or a subjunction, or that do not have an overt opening, excluding conditional subordinate V1-clauses. Thus, the lexical verb and its valency information are not presented until near the end of these VL-clauses, which can cause processing difficulties in German (e.g. Konieczny Reference Konieczny1996:80–100).

The position of the lexical verb is in all those cases determined by grammar. Variation is possible only in a few cases such as substituting the synthetic preterit with the analytic perfect tense in V2-clauses (Hennig Reference Hennig2000:29–31). Since there is no real variation possible both in V2- and VL-clauses and in both clause types the same principles of prediction should apply, it is possible to count main and subordinate clauses together as is done here.

To describe the position of the lexical verb in the clause that contains the double object construction, as in (10), I apply the notation FV-V2 (‘full verb in V2’), where the full verb occupies the second position, independent of the overtness of the prefield. Clauses with the full verb in the RSB, as in (11), are called FV-VL (‘full verb in VL’). The latter notation is independent of the classical description of the position of the finite verb.

Processing dependent on the verb position

Levy (Reference Levy2008) generally claims that recipients predict upcoming material which implies that a grammatical category like the predicate becomes more expected the more constituents precede it. Konieczny (Reference Konieczny1996) shows these differences in processing and that they depend on the position of the lexical verb. In the case of attachment ambiguities of NPs in German, for example, processing difficulties were smaller for FV-V2 clauses than for FV-VL clauses.

Building on this, I argue that the object order is related to prediction and to the position of the full verb. The full verb crucially carries the valency information. Only in processing it does the recipient know which arguments are needed in the sentences. In FV-VL clauses such as (11), however, this information is delayed. The recipient is left with uncertainty about the arguments in the clause. This uncertainty can be reduced by choosing an object order which reduces the amount of possible full verbs. For German, this object order is the Dat>Acc order.

When the dative object is placed before the accusative object, recipients should discard sentence continuations with a transitive verb and rank those with a ditransitive or intransitive verb higher, following Levy (Reference Levy2008). Since ditransitive or intransitive verbs are less frequent than transitive verbs (e.g. De Knop & Mollica Reference De Knop, Mollica, Ibáñez, Oyón and Pérez Sobrino2017:208), fewer sentence continuations are possible then. This limitation should facilitate the processing of the whole clause. Moreover, it might also be beneficial for the sentence production. Thus, in the VF-VL case, the Dat>Acc order should be (even more) favored.

On the other hand, there are the FV-V2 clauses where the verb valency is known to the recipient early. According to Fenk-Oczlon (Reference Fenk-Oczlon1983:30), who described the SVOFootnote ⁷ order of simple sentences as the most natural in terms of information distribution, the presentation of the verb before the object “already facilitates the reduction of uncertainty for the recipient to such an extent that further means of generating redundancy can be dispensed with” (translation, S.V.). Thus, if the full verb is presented in the LSB (FV-V2), Acc>Dat should be more possible because the recipients and speakers know that a dative and an accusative object must be placed in the clause at some point. A certain order of the two objects is less crucial to reduce uncertainties.

However, free datives (Mollica Reference Mollica, Schierholz and Giacomini2018), which are not distinguishable from dative objects by their form (De Knop & Mollica Reference De Knop, Mollica, Ibáñez, Oyón and Pérez Sobrino2017), pose a problem for that hypothesis. They are not required for the grammaticality or semantic correctness of a clause and can, hence, be combined with transitive verbs (e.g. Hole Reference Hole, Alexiadou, Kiss and Müller2012).Footnote ⁸ If my aforementioned assumption is correct, placing a free dative before an accusative object would trigger the prediction of a di-/intransitive verb and its valency structure for the whole clause. Accordingly, processing difficulties would then occur at the latest on the lexical verb in the RSB (12a,b), which should not be the case in the correctly predicted structure. Nonetheless, the processing difficulties on the RSB should also occur in the Acc>Dat order. In this order, difficulties can also already appear on the dative object itself.

Following this argumentation, free datives can cause problems with the prediction of the exact verb lexeme in the RSB which cannot easily be solved by a variation in the object order in the FV-VL clauses. A clear prediction for those clauses cannot be given yet. The FV-V2 clauses in (12c) should allow both orders due to the processing advantages of presenting the full verb early.

Annotation of the verb position

The information about whether a clause is FV-V2 or FV-VL is gained using the POS-tags from the corpora and the annotation of the LSB (Ortmann Reference Ortmann2022). If the POS-tag is VVFIN or VVIMP (Schiller et al. Reference Schiller, Teufel, Stöckert and Thielen1999:7), the sentence has a VF-V2 word order, as in (10), and is annotated as such. For all other cases, including empty LSBs, the clause is annotated as FV-VL, as in (11). This includes subordinate clauses.

Surprisal calculation

The surprisal values used in this study were computed as unigram surprisal values calculated on a lemma basis as in Wallenberg and colleagues (Reference Wallenberg, Bailes, Cuskley and Karl Ingason2021). This lemma-based calculation and the removal of all punctuation addresses the variability inherent in historical texts. German lacked an orthographic standardization until 1901 (e.g. Wolf Reference Wolf2022:16). That often resulted in significant discrepancies among words. Furthermore, the amount of historical data is too sparse to get good enough results for even normalized data (see section 3.2 and figure 1). Making data more similar, for example by lemmatization, is one way to work with limited resources (Manning & Schütze Reference Manning and Schütze2005, Lemnitzer & Zinsmeister Reference Lemnitzer and Zinsmeister2015 among others).

A further and more crucial advantage of the lemma-based surprisal calculation is that it neutralizes grammatical information, such as case, number, grammatical gender; see (13).

A sentence like (13) is the input for the surprisal and for the DORM calculation, described below. Case, number and grammatical gender are even in the ENHG period (already) mostly coded on the article. When the form of the article is kept the same, one cannot be certain about the case of any article. Thus, (13) could be derived from both (14a) and (14b).

Example (14a) is the more likely word order, but the surprisal calculation does not know that and due to the word order freedom, (14b) is also possible.

As a basis for the surprisal value calculation, the different corpora were used. The method of calculating surprisal values across the corpora annotated so far, rather than centuries, was selected primarily for the differing lemmatizations between the corporaFootnote ⁹ as the sixteenth- and seventeenth-century data and modern data are built by two different corpora each.

Finally, the surprisal values were calculated in the following way: The probability of each lemma in each corpus was determined. All punctuation was removed. The surprisal value, then, equals the negative log2-probability of that lemma probability (Hale Reference Hale2001). Based on these surprisal values, DORM_calib was calculated as described below.

DORM (Cuskley et al. Reference Cuskley, Bailes and Wallenberg2021)

DORM is a statistical value developed by Cuskley and colleagues (Reference Cuskley, Bailes and Wallenberg2021) and applied by Wallenberg and colleagues (Reference Wallenberg, Bailes, Cuskley and Karl Ingason2021) to describe the change from OV to VO order in English and Icelandic. The DORM value gives information about the distribution of high and low surprisal values in a sentence. The less variance there is, the lower the DORM value is. Accordingly, a low DORM value indicates a uniform information profile, while a high DORM value indicates a less uniform information profile. To calculate DORM, one must “take the arithmetic mean of every adjacent, overlapping pair of information content values (i.e. a rolling mean) … then compute the sample variance of the resulting list of means (an unbiased estimator, using denominator n −1) and arrive at a single number, the DORM” (Cuskley et al. Reference Cuskley, Bailes and Wallenberg2021:4). Thus, the DORM calculation needs two steps. First (eq. 1), the rolling mean must be calculated.Footnote ¹⁰ Second, then the sample variance of these rolling means (eq. 2) is calculated (Ortmann et al. Reference Ortmann, Voigtmann, Dipper, Speyer, Lemke, Schäfer and Reich2024: 70). Table 1 illustrates the calculation of DORM with an example from the corpus.

(1) $${\rm{Rolling \ Mean\!\!:}}\,{\rm{for}}\;i\;{\rm{in}}\;\left( {1 \ldots {\rm{n}} - 1} \right)\!\!:\;RM_{i} = \;{{{s_i} + {s_{i + 1}}} \over 2}$$

(2) $${\rm{DORM\!:}}\ {s^2} = {{\sum\nolimits_{i = 1}^n {{{\left( {R{M_i} - {\overline x}} \right)}^2}} } \over {n - 1}}$$

Table 1.

Example calculation of DORM value for TIGER corpus text 0317_0001, sentence 2

Table 1 Long description

The table presents the calculation of the DORM value for the sentence ‘Bagdad gives UN new information.’ It includes columns for text, lemma, surprisal, rolling mean, DORM, UIDO, and DORM(UIDO). The table has 6 rows and 8 columns. The first row contains the text ‘Bagdad’, ‘übergibt’, ‘UN’, ‘neue’, and ‘Informationen.’ The second row lists the lemma ‘Bagdad’, ‘übergeben’, ‘UN’, ‘neu’, and ‘Information’. The third row shows the surprisal values for each lemma: 15.27, 14.92, 13.92, 9.22, and 12.78. The fourth row presents the rolling mean values: 15.095, 14.42, 11.57, and 11.00. The fifth row displays the DORM value of 4.14. The sixth row lists the UIDO values: 9.22, 14.92, 13.92, 12.78, and 15.27. The seventh row shows the DORM(UIDO) value of 1.06. The final row presents the DORM_calib value of 3.08.

For this sentence from the TIGER corpus (Brants et al. Reference Brants, Dipper, Eisenberg, Silvia Hansen-Schirra, Lezius, Rohrer, Smith and Uszkoreit2004), the unigram lemma-surprisal values of the five words are [15.27, 14.92, 13.92, 9.22, 12.78]. That results in the following four rolling mean values [15.095, 14.42, 11.57, 11.00]. The variance of these values corresponds to the DORM of the whole clause, which is 4.14.

However, DORM values are only directly comparable for sentences with the same number of elements (Cuskley et al. Reference Cuskley, Bailes and Wallenberg2021). To compare the DORM values of different clauses, they must be calibrated using the Uniform Information Density Optimization (UIDO) (Cuskley et al. Reference Cuskley, Bailes and Wallenberg2021). Wallenberg and colleagues (Reference Wallenberg, Bailes, Cuskley and Karl Ingason2021:8) describe the calibration as follows: “For any array of information content values, UIDO systematically reorders them, seeking to minimize the DORM for the whole strings. We output a ‘calibrated’ DORM measure for each sentence, i.e., DORM – DORM(UIDO).”

The application of UIDO is one of main reasons unigram surprisal was chosen, besides the fact that it has been successfully applied in previous studies using DORM (Wallenberg et al. Reference Wallenberg, Bailes, Cuskley and Karl Ingason2021). The UIDO function orders the surprisal value multiple times to check whether this order results in the smallest possible variation between values (Cuskley et al. Reference Cuskley, Bailes and Wallenberg2021). If UIDO were to be calculated with a bigram model, not only would the variance have to be recalculated after each change in the order, but the surprisal value of each word would also have to be recalculated beforehand based on the changing context. This effort exceeds my resources.

The method of calculating UIDO and DORM(UIDO) is shown in the last three lines of table 1; the UIDO results in this order of surprisal values [9.22, 14.92, 13.92, 12.78, 15.27], which corresponds to the word order: new give UN information Bagdad. The example illustrates that the UIDO does not necessarily (or at all) result in a meaningful or grammatical word order, but this is not the point of the UIDO as it is only used for calibration. For the example sentence, the calibrated DORM is 3.08. Since DORM is calculated on the unigram lemma surprisal values, it is possible to say that this type of DORM calculation can only make statements about the lexical information profile.Footnote ¹¹

Whenever the uniformity of a sentence is discussed in the remainder of the article, it refers to the calibrated DORM (DORM_calib), that is, the DORM of the original sentence minus the DORM of this sentence’s UIDO.Footnote ¹²

DORM_calib

A low DORM_calib indicates a smooth information profile while a high DORM_calib suggests an information profile with peaks and troughs. Smooth information profiles are presumably felicitous for sentence processing.

Table 2 shows some descriptive statistics of DORM_calib of the whole data set. The values for the Dat>Acc order vary between 0 (only two sentences, due to rounding) and 18.83, with a median of 7.54 and a mean of 7.81. The variance is 9.74 and the standard error is only 9.74. In the Acc>Dat order, the DORM_calib varies between 1.03 and 16.16, with a mean of 7.07 and a median of 6.91. The variance is smaller than in the Dat>Acc order with only 8.05, but the standard error is higher with 0.23. When all centuries are combined, the DORM_calib of both orders is normally distributed.

Table 2.

Descriptive values of DORM_calib, all data

Table 2 Long description

The table presents descriptive statistics for DORMcalib in two data orders: Dat>Acc and Acc>Dat. It has 11 rows and 2 columns. The rows are labeled as Minimum, 1st quartile, Median, Mean, 3rd quartile, Maximum, Variance, Standard error, and Normal distribution. The columns are labeled DORMcalib Dat>Acc and DORMcalib Acc>Dat. For DORMcalib Dat>Acc, the values are: Minimum 0.00, 1st quartile 5.71, Median 7.54, Mean 7.81, 3rd quartile 9.86, Maximum 18.83, Variance 9.74, Standard error 0.08, Normal distribution W = 0.99. For DORMcalib Acc>Dat, the values are: Minimum 1.03, 1st quartile 5.22, Median 6.91, Mean 7.07, 3rd quartile 8.62, Maximum 16.16, Variance 8.05, Standard error 0.23, Normal distribution W = 0.98.

Interestingly, the information profile of Acc>Dat is smoother than the information profile of Dat>Acc. However, since the data is not equally distributed across all periods under investigation, with the modern data accounting for 76.23 percent of the data, it is worthwhile to split the data per century (figure 4).

Figure 4.

DORM_calib per century and object order.

Here, the differences between the centuries become visible. DORM_calib is lowest, only between about 2.5 and 5, in the data built by the Anselm and the RIDGES corpora.Footnote ¹³ The Anselm corpus consists of theological texts, which might indicate a smaller vocabulary and more repetition of the words used, especially since one lexeme carried more meanings at that time compared to later centuries. In the sixteenth-century corpora, the Dat>Acc order indicates a smoother information profile ( $${\rm{\overline x}} = {\rm{\;}}3.45$$ ) than the Acc>Dat order ( $${\rm{\overline x}} = {\rm{\;}}4.09$$ ).

In the seventeenth century, the DORM_calib is generally higher than in the sixteenth century. This might be due to a larger variation in the content of the RIDGES and GerManC corpora. However, data from both the sixteenth and seventeenth centuries does not behave like modern data. The median of the Acc>Dat order ( $${\rm{\tilde x}} = $$ 8.21) is above the median of the Dat>Acc order ( $${\rm{\tilde x}} = $$ 7.76). The means of the two orders only differ slightly (Dat>Acc $${\rm{\overline x}} = 8.296$$ vs. Acc>Dat $${\rm{\overline x}} = $$ 8.21). However, data sparsity is also still a problem here (n = 100, 90 percent Dat>Acc). In the eighteenth-century data, the Acc>Dat order often has higher DORM_calib values compared to the Dat>Acc order. Figure 4 shows that the median values are rather similar (Acc>Dat: $${\rm{\tilde x}} = $$ 7.88 vs. Dat>Acc: $${\rm{\tilde x}} = $$ 7.81). The mean DORM_calib of the Acc>Dat order is 8.27 and the mean DORM_calib of the Dat>Acc order is 7.95. In the modern data (twentieth and twenty-first century), the Acc>Dat order sentences have a smoother information profile ( $$\bar x = $$ 6.96, $${\rm{\tilde x}} = $$ 6.58) than the Dat>Acc order sentences ( $$\bar x = $$ 7.94, $${\rm{\tilde x}} = $$ 7.63).

Length difference

The length of both the accusative and dative objects is determined. The dative objects are between 1 and 20 words long with a median of 2 and a mean of 2.92 words. This is expected as most nominal phrases in German have a determiner and their nominal head. The variance of the dative length is 3.8 and the standard error is 0.05. The length of the accusative objects varies between 1 and 36 words with a median of 2 words and a mean of 3.18 words. The variance of the accusative object length is 6.61 and the standard error is 0.06. The sentences with very long objects are sentences in which the accusative or dative object has attributes like adjacent relative clauses.

As described above, I only include the length difference of the two objects in this study. Negative values indicate that the accusative object is longer than the dative object and positive values say that the dative object is longer than the accusative object. Figure 5 shows that the data is normally distributed with some negative outliers. Most values group around 0 indicating that the objects have the same length. The difference values group vary between −34 and 16, the median is 0 and the mean is −0.26. The variance is 10.07 and the standard error is 0.08.

Figure 5.

Length difference of dative and accusative objects.

Additionally, there does not seem to be a difference between two object orders regarding the length difference (figure 6). The Dat>Acc order has a length difference between −34 and 16 with a mean of −0.29 and a median of 0. The variance is 10.3 and the standard error 0.08. The Shapiro-Wilk normality test has a W statistic of 0.86 and a p-value below 0.001. The Acc>Dat order sentences show a length difference between −14 and 8, their median being 0 and the mean 0.05. The variance is 7.57 and the standard error 0.22. The Shapiro-Wilk normality test has a W statistic of 0.91 and a p-value below 0.001. I assume normality despite the outliers.

Figure 6.

Length difference per object order and century.

The Dat>Acc order generally seems to vary more than the Acc>Dat order, especially in modern data. For the oldest texts (sixteenth century), the length difference for Acc>Dat has a median of −1 and a mean of −0.5. For the Dat>Acc order in the sixteenth century, however, most values cluster around 0 with only a few outliers. In the seventeenth century and in the modern data, there is no visible difference between the two object orders. In the eighteenth century, it seems as if the Acc>Dat order is used when the length difference is positive, which is not in line with Behaghel’s law (Reference Behaghel1932).

Full verb position

Besides the DORM_calib, the position of the full verb is the other previously unconsidered variable that might influence the object order in the middle field.

A total of 449 clauses have FV-V2 and 1,284 clauses have FV-VL. Figure 7 combines all centuries. There is a slight decline of Dat>Acc instances in FV-VL (90.65 percent) compared to FV-V2 clauses (92.65 percent). That is contrary to the idea that Dat>Acc should be preferred in FV-VL clauses, where the verb valency is only known at the end of the sentence. However, the difference is small.

Figure 7.

Object order dependent on verb position.

When the data is divided into the four periods (figure 8), some differences can be found. First, the sixteenth-century data behaves like the modern data. Dat>Acc is more frequent in FV-V2 clauses (88.89 percent) compared to FV-VL clauses (73.68 percent). In the modern data, Dat>Acc occurs in 94.79 percent of all FV-V2 clauses but only in 90.85 percent of all FV-VL clauses.

At least descriptively, the expectation that Dat>Acc is less frequent in FV-V2 clauses is met in the seventeenth century. In FV-V2 clauses, one finds 86.96 percent Dat>Acc clauses. In the FV-VL, 90.91 percent of clauses have Dat>Acc. The eighteenth century also aligns with the expectation that Dat>Acc is more frequent in FV-VL clauses (91.19 percent) compared to FV-V2 clauses (85.45 percent).

Givenness status

Figure 9 shows the relative frequency of the Dat>Acc order (y-axis) when the accusative object is given (n = 146), inferable (n = 99) or new (n = 1,488) (x-axis). The relative number of Dat>Acc sentences is smallest when the accusative object is given (58.9 percent) and largest when the accusative object is new (94.69 percent). When the accusative object is given, Acc>Dat is more frequent (41.1 percent) than when the accusative object is inferable (14.14 percent) or new (5.31 percent).

Figure 8.

Changes in the object order distribution dependent on the position of the lexical verb.

Figure 9.

Share of Dat>Acc and Acc>Dat, dependent on the information status of the accusative object.

The influence of the information status of the accusative object is stable across time (figure 10). The Dat>Acc order is always more frequent when the accusative is new compared to the instances of given accusative objects and the relative frequency of Acc>Dat is highest when the accusative object is given.

Figure 10.

Share of Dat>Acc and Acc>Dat, dependent on the information status of the accusative object per century.

Figure 11 shows the relative frequency of the Dat>Acc order (y-axis) when the dative object is given (n = 609), inferable (n = 138), or new (n = 833) (x-axis). The relative number of Dat>Acc sentences is smallest when the dative object is inferable (85.71 percent) and largest when the dative object is given (95.01 percent). When the dative object is new, Dat>Acc occurs in 89.47 percent of sentences.

Figure 11.

Share of Dat>Acc and Acc>Dat, dependent on the information status of the dative object.

Compared to the influence of the accusative object’s information status on the object order, the dative object’s givenness annotation does not provide a clear tendency: A given dative precedes the accusative object less often than a new dative in the sixteenth and eighteenth centuries (figure 12). Given datives first holds in the seventeenth and twentieth centuries, but the differences are small. Dat>Acc is least frequent when the dative object is inferable in the modern data, but in the historical data, Dat>Acc occurs always when the dative object is inferable.

Figure 12.

Share of Dat>Acc and Acc>Dat, dependent on the information status of the dative object per century.

Figure 12 Long description

Panel A: A bar graph showing the share of object order dependent on the status of the dative object in the 16th century. The horizontal axis represents the information status of the dative object with categories given, inferable, and new. The vertical axis represents the percent. The graph shows two sets of bars for each category, one for Dat>Acc and one for Acc>Dat. The values are 76.19 percent and 23.81 percent for given, 100 percent and 0 percent for inferable, and 87.8 percent and 12.2 percent for new. Panel B: A bar graph showing the share of object order dependent on the status of the dative object in the 17th century. The horizontal axis represents the information status of the dative object with categories given, inferable, and new. The vertical axis represents the percent. The graph shows two sets of bars for each category, one for Dat>Acc and one for Acc>Dat. The values are 94.12 percent and 5.88 percent for given, 100 percent and 0 percent for inferable, and 87.1 percent and 12.9 percent for new. Panel C: A bar graph showing the share of object order dependent on the status of the dative object in the 18th century. The horizontal axis represents the information status of the dative object with categories given, inferable, and new. The vertical axis represents the percent. The graph shows two sets of bars for each category, one for Dat>Acc and one for Acc>Dat. The values are 89.39 percent and 10.61 percent for given, 100 percent and 0 percent for inferable, and 89.89 percent and 10.11 percent for new. Panel D: A bar graph showing the share of object order dependent on the status of the dative object in the 20th and 21st centuries. The horizontal axis represents the information status of the dative object with categories given, inferable, and new. The vertical axis represents the percent. The graph shows two sets of bars for each category, one for Dat>Acc and one for Acc>Dat. The values are 96.54 percent and 3.46 percent for given, 84.77 percent and 15.23 percent for inferable, and 89.69 percent and 10.31 percent for new.