SlowGuess/ACL_2020_OCR huggingface dataset:
Meaning and gender:  $q(c \mid \mathbf{v}, g)$ . We trained a simple multilayer perceptron (MLP) classifier to predict the declension class from the WORD2VEC representation. When conditioning on gender, we again embed each gender class, concatenating these embeddings with the WORD2VEC ones before feeding the result into the MLP.
Form, meaning, and gender:  $q(c \mid \mathbf{w}, \mathbf{v}, g)$ . We again trained two LSTM classifiers, but this time, also conditioned on meaning (i.e., WORD2VEC). Before training, we reduce the dimensionality of the WORD2VEC embeddings from 300 to  $k$  dimensions by running PCA on each language's embeddings. We then linearly transformed them to match the hidden size of the LSTMs, and fed them in. To also condition on gender, we followed the same procedures, but used half of each LSTM's initial hidden state for each vector (i.e., WORD2VEC and one-hot gender embeddings).
Optimization. We trained all classifiers using Adam (Kingma and Ba, 2015) and the code was implemented using PyTorch. Hyperparameters—number of training epochs, hidden sizes, PCA compression dimension  $(k)$ , and number of layers—were optimized using Bayesian optimization with a Gaussian process prior (Snoek et al., 2012). We explore a maximum of 50 models for each experiment, maximizing the expected improvement on the validation set.
5.3 An Empirical Lower Bound on MI
With our empirical approximations of the desired entropy measures, we can calculate the desired approximated MI values, e.g.,
$$
\begin{array}{l} \mathrm {M I} (C; W \mid G) \approx \tag {11} \\ \hat {\mathrm {H}} (C \mid G) - \hat {\mathrm {H}} _ {q} (C \mid W, G) \\ \end{array}
$$
where  $\hat{\mathrm{H}} (C\mid G)$  is the plug-in estimation of the entropy. Such an approximation, though, is not ideal, since we do not know if the true MI is approximated by above or below. Since we use a plug-in estimator for  $\hat{\mathrm{H}} (C\mid G)$ , which underestimates entropy, and since  $\mathrm{H}_q(C\mid W,G)$  is estimated with a cross-entropy upperbound, we have
$$
\begin{array}{l} \operatorname {M I} (C; W \mid G) = \operatorname {H} (C \mid G) - \operatorname {H} (C \mid W, G) \\ \underset {\approx} {\gtrsim} \hat {\mathrm {H}} (C \mid G) - \mathrm {H} (C \mid W, G) \\ \underset {\approx} {\gtrsim} \hat {\mathrm {H}} (C \mid G) - \hat {\mathrm {H}} _ {q} (C \mid W, G). \\ \end{array}
$$
We note that these are expected lower bounds, i.e. they are exact when taking an expectation under the true distribution  $p$ . We cannot make a similar statement about tripartite MI, though, since it is computed as the difference of two lower-bound approximations of true mutual information quantities.
Our main experimental results are presented in Table 2. We find that both form and lexical semantics significantly interact with declension class in both Czech and German (each  $p < 0.01$ ).<sup>6</sup> We observe that our estimates of  $\mathrm{MI}(C;W\mid G)$  is larger (0.5-0.8 bits) than our estimates of  $\mathrm{MI}(C;V\mid G)$  (0.3-0.5 bits). We also observe that the MI estimates in Czech are higher than in German. However, we caution that the unnormalized estimates for the two languages are not fully comparable because they hail from models trained on different amounts of data. The tripartite MI estimates between class, form, and meaning, were relatively small (0.2-0.35 bits) for both languages. We interpret this finding as showing that much of the information contributed by form is not redundant with information contributed by meaning—although a substantial amount is.
<sup>6</sup>All results in this section were significant for both languages, according to a Welch (1947)'s  $t$ -test, which yielded  $p < 0.01$  after Benjamini and Hochberg's correction. A Welch (1947)'s  $t$ -test differs from Student (1908)'s  $t$ -test in that the latter assumes equal variances, and the former does not, making it preferable (see Delacre et al. 2017).
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Form & Decension Class (LSTM)	Meaning & Decension Class (MLP)
H(C | G)	H_Q(C | W, G)	MI(C; W | G)	U(C | W, G)	H(C | G)	H_Q(C | V, G)	MI(C; V | G)	U(C | V, G)
Czech	1.35	0.56	0.79	58.8%	1.35	0.82	0.53	39.4%
German	2.17	1.60	0.57	26.4%	2.17	1.88	0.29	13.6%
Both (Form and Meaning) & Decension Class	Tripartite MI (LSTM)
H(C | G)	H_Q(C | W, V, G)	MI(C; W, V | G)	U(C | W, V, G)	MI(C; W | G)	MI(C; W | V, G)	MI(C; W; V | G)	U(C | W; V, G)
Czech	1.35	0.37	0.98	72.6%	0.79	0.44	0.35	25.9%
German	2.17	1.50	0.67	30.8%	0.57	0.37	0.20	9.2%
Table 2: MI between form and class (top-left), meaning and class (top-right), both form and meaning and class (bottom-left), and tripartite MI (bottom-right). All values are calculated given gender, and bold if significant.
H(C)	H(C | G)	MI(C;G)	U(C | G)
Czech	2.75	1.35	1.40	50.8%
German	2.88	2.17	0.71	24.6%
Table 3: MI between class and gender  $\mathrm{MI}(C;G)$ :  $\mathrm{H}(C)$  is class entropy,  $\mathrm{H}(C\mid G)$  is class entropy given gender,  $U(C\mid G)$  is the uncertainty coefficient.
As a final sanity check, we measure mutual information between class and gender  $\mathrm{MI}(C;G)$  (see Table 3). For both languages, the mutual information between declension class and gender is significant. Our MI estimates range from approximately  $3/4$  of a bit in German up to 1.4 bits in Czech, which respectively amount to nearly  $25\%$  and nearly  $51\%$  of the remaining unconditional entropy. Like the quantities discussed in §4, this MI was estimated using simple plug-in estimation. Remember, if class were entirely reducible to gender, conditional entropy of class given gender would be zero. This is not the case: Although the conditional entropy of class given gender is lower for Czech (1.35 bits) than for German (2.17 bits), in neither case is declension class informationally equivalent to the language's grammatical gender system.
Next, we ask whether individual declension classes differ in how idiosyncratic they are, e.g., does any one German declension class share less information with form than the others? To address this, we qualitatively inspect per-class half-pointwise mutual information in Figure 2a-2b. See Table 5 in the Appendix for the five highest and lowest surprisal examples per model. Several qualitative trends were observed: (i) classes show a decent amount of variability, (ii) unconditional entropy for each class is inversely proportional to the class' size, (iii) half-pointwise MI is higher on average for Czech than German, and (iv) classes that
have high  $\mathrm{MI}(C = c;V\mid G)$  usually have high  $\mathrm{MI}(C = c;W\mid G)$  (with a few notable exceptions we discuss below).
Czech. In general, declension classes associated with masculine nouns ( $g = \mathrm{MSC}$ ) have smaller  $\mathrm{MI}(C = c; W \mid G)$  than classes associated with feminine ( $g = \mathrm{FEM}$ ) and neuter ( $g = \mathrm{NEU}$ ) ones of a comparable size—the exception being 'special, masculine, plural -ata'. This class ends exclusively in -e or -ě, which might contribute to that class' higher  $\mathrm{MI}(C = c; W \mid G)$ . That  $\mathrm{MI}(C = c; W \mid G)$  is high for feminine and neuter classes suggests that the overall  $\mathrm{MI}(C; W \mid G)$  results might be largely driven by these classes, which predominantly end in vowels. We also note that the high  $\mathrm{MI}(C = c; W \mid G)$  for feminine 'plural -e', might be driven by the many Latin or Greek loanwords present in this class.
With respect to meaning, masculine declension classes can reflect degrees of animacy: 'animate1' contains nouns referring mostly to humans and a few animals (kocour 'tomcat', colek 'newt'), 'animate2' contains nouns referring mostly to animals and a few humans (syn 'son', k'rest'an 'Christian'), 'inanimate1' contains many plants, staple foods (chléb 'bread', ocet 'vinegar') and meaningful places (domov 'home', kostel 'church'), and 'inanimate2' contains many basic inanimate nouns (kámen 'stone'). Of these masculine classes, 'inanimate1' has a lower  $\mathrm{MI}(C = c; V \mid G)$  than its class size alone might lead us to predict. Feminine and neuter classes show no clear pattern, although neuter classes '-eni' and '-o' have comparatively high  $\mathrm{MI}(C = c; V \mid G)$ .
For  $\mathrm{MI}(C = c;V;W\mid G)$  , we observe that 'masculine, inanimate1' is the smallest quantity, followed by most other masculine classes (e.g., masculine animate classes with -ove or -i plurals) for which  $\mathrm{MI}(C = c;W\mid G)$  was also low. Among non-masculine classes, we observe that feminine
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(a)Czech
(b) German
Figure 2: Pointwise MI for declension classes. MI for each random variable  $\mathcal{X} \in \{V, W, \{V, W\}, \{V; W\}\}$  are plotted for classes increasing in size (towards the right):  $\mathrm{MI}(C = c; V|G)$  (bottom),  $\mathrm{MI}(C = c; W|G)$  (bottom middle),  $\mathrm{MI}(C = c; V, W|G)$  (top middle), and tripartite  $\mathrm{MI}(C = c; V; W|G)$  (top).
'pl -i' and the neuter classes -o and -eni show higher tripartite MI. The latter two classes have relatively high MI across the board.
German.  $\mathrm{MI}(C = c;W\mid G)$  for classes containing words with umlautable vowels (i.e., S3/P1u, S1/P1u) or loan words (i.e., S3/loan) tends to be high; in the prior case, our models seem able to separate umlautable from non-umlautable vowels, and in the latter case, loan word orthography from native orthography.  $\mathrm{MI}(C = c;V\mid G)$  quantities are roughly equivalent across classes of different size, with the exception of three classes: S1/P4, S3/P1, and S1/P3. S1/P4 consists of highly semantically variable nouns, ranging from relational noun lexemes (e.g., Glied 'member', Weib 'wife', Bild picture') to masses (e.g., Reis 'rice'), which perhaps explains its relatively high  $\mathrm{MI}(C = c;V\mid G)$ . For S1/P3 and S3/P1,  $\mathrm{MI}(C = c;V\mid G)$  is low, and we observe that both declension classes idiosyncratically group clusters of semantically similar nouns: S1/P3 contains "exotic" birds (Papagei
'parrot',  $P_{fau}$  'peacock'), but also nouns ending in -or, (Traktor 'tractor', Pastor 'pastor'), whereas S3/P1 contains very few nouns, such as names of months (März, 'March', Mai 'May') and names of mythological beasts (e.g., *Sphinx*, Alp).
Tripartite MI is fairly idiosyncratic in German: The lowest quantity comes from the smallest class, S1/P2u. S1/P3, a class with low  $\mathrm{MI}(C = c; V \mid G)$  from above, also has low tripartite MI. We speculate that S1/P3 could be a sort of "catch-all" class with no clear regularities. The highest tripartite MI comes from S1/P4, which also had high  $\mathrm{MI}(C = c; V \mid G)$ . The existence of significant tripartite MI results suggests that submorphic meaning bearing units, or phonaesthemes, might be present. Taking inspiration from Pimentel et al. 2019, which aims to automatically discover such units, we observe that many words in S1/P4 contain letters  $\{d, e, g, i, l\}$ , often in identically ordered orthographic sequences, such as Bild, Biest, Feld, Geld, Glied, Kind, Leib, Lied, Schild, Viech, Weib,
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etc. While these letters are common in German orthography, their noticeable presence suggests that further elucidation of declension classes in the context of phonaestheses could be warranted.
We adduce new evidence that declension class membership is not wholly idiosyncratic nor fully deterministic based on form or meaning in Czech and German. We quantify mutual information and find estimates which range from 0.2 bits to nearly one bit. Despite their relatively small magnitudes, our estimates of mutual information between class and form accounted for between  $25\%$  and  $60\%$  of the class' entropy, even after relevant controls, and MI between class and meaning accounted for between  $13\%$  and nearly  $40\%$ . We analyze results per-class, and find that classes vary in how much information they share with meaning and form. We also observe that classes that have high  $\mathrm{MI}(C = c; V \mid G)$  often have high  $\mathrm{MI}(C = c; W \mid G)$ , with a few noted exceptions that have specific orthographic (e.g., German unlauted plurals), or semantic (e.g., Czech masculine animacy) properties. In sum, this paper has proposed a new information-theoretic method for quantifying the strength of morphological relationships, and applied it to declension class. We verify and build on existing linguistic findings, by showing that the mutual information quantities between declension class, orthographic form, and lexical semantics are statistically significant.
Thanks to Guy Tabachnik for discussions on Czech phonology, to Jacob Eisenstein for useful questions about irregularity, and to Andrea Sims and Jeff Parker for advice on citation forms. Thanks to Ana Paula Seraphim for helping beautify Figure 1.
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