An Information-Theoretic Parameter-Free Bayesian Framework for Probing Labeled Dependency Trees from Attention Score
Authors: Hongxu Liu
Jing Ma
Xiaojie Wang
Caixia Yuan
Fangxiang Feng
Venue: ICLR 2026 Poster (2026)
Keywords: probing
attention score
dependency syntax
mutual information
Primary area: interpretability and explainable AI
TL;DR: A method being able to estimate attention score-syntactical dependency MI and reconstruct labeled dependency trees from probabalistic distributions, requiring no trainable networks.

Abstract: Figuring out how neural language models comprehend syntax acts as a key to revealing how they understand languages. 
We systematically analyzed methods for finding syntax structures in models, namely _probing_, and found limitations yet widely exist in previous probing practice. 
We proposed a method capable of estimating mutual information (MI) and extracting dependency trees from attention scores in a mathematical-rigorous way, requiring no additional network training effort.
Compared with previous approaches, it has a much simpler model, while being able to probe more complex dependency trees, also transparent for fine-grained explanation.
We tested our method on several open-source LLMs and demonstrated its effectiveness by systematically comparing it with a great many competitive baselines. Several informative conclusions can be drawn by further analysis of the results, shedding light on our method’s explanatory potential.
Our code is released at https://github.com/ChristLBUPT/IPBP.

Published as a conference paper at ICLR 2026
AN INFORMATION-THEORER PARAMETER-FREE BAYESIAN FRAMEWORK FOR PROBING LABELED DEPENDENCY TREES FROM ATTENTION SCORE
Hongxu Liu'
Jing Ma² Xiaojie Wang" Caixia Yuan²
Fangxiang Feng²
1 Nanyang Technological University, Singapore
BBeijing University of Posts and Telccommunication, Beijing, China hongxu001e .t....tu..ommm edu.sg
(majing, x xjwang, yuancx, fxfeng @bbpt. edu. cn
ABSTRACT
Figuring out how neural language models comprehend syntax acts as a key to revealing how they understand languages. We systematically analyzed methods for finding syntax structures in models, namely probing, and found limitations yet widely exist in previous probing practice. We proposed a method capable of estimating mutual information (MI) and extracting dependency trees from attention scores in a mathematiial--igorour way, requiring no additional network training effort. Compared with previous approaches, it has much simpler model, while being able to probe more complex dependency trees, also transparent for fine- grained explanation. We tested our method on several open-source LLMs and demonstrated its effectiveness by systematically comparing with a great many competitive baselines. Several informative conclusions can be drawn by further analysis of the results, shedding light on our method's 's explanatory potential. Our code is released at https: /tte:://gtttttt /gggttttthh...gg/gggggg..g..h. com/
INTRODUCTION
Recent advancements in Large Language Models (LLMs) have left the world with deep impressions. This process is accompanied by confusion, since LLMs are largely black-box and usually trained on simple next-token- prediction next-token-prediction LM tasks. While the interpretability community recently assigns great
importance to mechanistic (circuit-basee based) model explaining (Elhage et al., 2021 Wang et al.) 2022 Ferrando & Voital 2024), traditional probing methods, which aim at extracting syntax structures from model states, are still worth working on for two reasons: 1. They can provide dataset- wide conclusions, while most circuit-based methods tend to be sample-wise (except for [Elhage al.) (2021), which is a static parameter analysis) 2. Syntax structures are among the most topologically- complicated and all-around concepts about languages. They are also essential humans' language comprehension, verified by various brain studies (Lopopolo et al. 2021 Dotan & Brutmann 2022 Fallon & Pylkkänen 2024
A common practice of syntactical probing is to train supervised classifier network on top of model
states (Hewitt& & Manning] 2019 Pimentel et al.) 2020b] Müller-Eberstein et al.) (022) to predict dependency syntax trees, or directly take some model states as evidence for syntax (Htut et al. 2019). Despite the insights they gave, is obvious that most of the previous probing methods are explaining by unexplainibility. meaning that most of them are introducing external trainable networks to extract
syntax, ranging from simple linear mappings (Liu et al. 2019) to deep MLPs (Hewitt Liang 2019
Voita & Titov 2020 Pimentel et al. 2020b) pseudo attention heads (Pimentel al.) 2022 This results in a trade-off: Linear mappings are simple and explainable, but have limited expressivity. Deeper networks can fit any co-relationships, but deep probing network itself is hard to explain. so it's natural to raise the doubt on whether the extracted syntax structures really come from the probed LM, just the strong probes have learned unconditionally predict them. This gets for modern LLMs that have larger hidden dimensions compared with preitrained models, making trainable probing networks inevitably larger to fit in the dimensionality.
"Correpooniigg Author

---

Published as a conference paper at ICLR 2026
If we dive deeper, we might find some clues about this bitter tradeoff: previous methods put their attention mainly on contextualized hidden states. Since hidden state vectors have rather different structures compared with syntax trees, trainable mapping network is necessary. Using hidden states is also a primary cause for the aforementioned concern of the probe itself learns the task: Contextualized embeddings emhed abundant semantics, so extreme case is that even if the probed LM knows nothing about syntacties, it's still possible that the deep probing model learns it (see Hewitt & Liang| 2019)). A simple thought experiment is that, given two single words eat and breakfast, it is plausible that breakfast acts as the noun object of the verb eat even without giving any context. This exactly the case of this concern.
If hidden states aren't 't good enough, what's the better choice? Maybe we should put attention on attention. Attention is the only component that involves inter-token relationships (MIP and add/norm are applied token-wise), while dependency syntax is also an inter-word relationship. Attention scores are stacked matrices, while dependency trees can also be described as adjacency matrices (See Section 3.1 for details). This means that attention maps are conceptually, also topologically consistent with dependency syntax trees.
Unfortunately, despite those nice consistencies, there are only few methods focusing probing attention for syntax (Clark et al., 2019 Huut et al.) 2019 Vig & Belinkov 2019, Ravishankar et al. 2021), only being able to extract low-quality or incomplete dependency trees. A seemingly good choice yields results that are not desirable. Why is it? This introduces the second limitation of previous methods, over-trusting attention scores: Due to the aforementioned nice consistencies, logether with the fact that normallze)) attention scores form valid probability distributions across tokens, seems natural directly take them as probability distributions of the dependency relationships of words. However, we must admit that attention scores wouldn't 't need to be dedicated to syntax, meaning that filtering out highly syntactical attention heads and explainable transformations on raw scores are necessary.
Based on our analysis, we proposed a method called infformttion-theoreee Parameter-free Bayesian Probing (IPBP): Instead of training supervised networks on hidden states, we chose to directly estimate the multivariate distributions between attention scores and dependency relationships, and integrate on those distributions to calculate mutual information (MI) between each attention head's scores and the existence of a certain dependency relationship in a mathematically closed form. Since it's parameter-free and attention-based it almost denies the possibilities of the probe itself learns the task, as mentioned before. We further designed a well-formed decoding algorithm incorporating the estimated MI and Bayesian posteriors, being able efficiently reconstruct labeled dependency trees, while preventing us from dropping into the trap of using attention scores themselves as dependency probabilities. Comparisons with series of strong baselines indicate that our method has the best head importance estimation and cree-constructing qualities. We further derived informative conclusions on the estimated MI and distributions, including the one obsessing probing researchers for a long time: do model layers accord with tree heights? (A similar question once discussed by Tenney et al. (2019)). In word, our method addresses the two limitations in an elegant way, while offering vast possibilities for the upcoming conclusion-intensivee research thanks to its fine-grained MI and probability functions.
RELATED WORK
Just after the birth of deep contextualized embeddings (Peters al. 2018) and transformer-based pre-trained models (Devlin et al.) 2019). researchers have started to investigate whether or not linguistic properties are embedded in these models (Conneau et al. 2018 Liu et al.) 2019 Tenney et al. 2019b; Hewitt& & Manning 2019). Then arguments began in this area. The frontline of these arguments is about what probing models can we use to prevent them from learning the task itsef. While early practices and preliminary methods suggested on strictly linear probes (Alain &
Bengio 2017 Hewitt& & Manning 2019: Liu et al., 2019), Hewitt& & Liang (201)) instead proposed control tasks that penalizes models being ability to learn the task itself, and had attempts on several Deep MLPs. Furthermore, Pimentel et al. (2020a) admitted this trade-off and took probing as an encuracy-complexity two-goal optimizing problem, and most radically, Pimentel et al. (2020b) insisted that probes should be as deep and complex as possible since they used them as estimators for Informmatio (Xu et al. 2020). Apart from disputes, there are also alternative theories proposed by
2

---

Published as a conference paper at ICLR 2026
the researchers, like the sode-description-lens length theory by Voita & Titov (2020) and the architectural bottleneck principle by Pimentel et al. (2022). These theories can be seen as patches under the
supervised probing context since they re also addressing the complexity accuracy tradcoff.
Apart from supervised probes, there are also parameter- free probes taking attention scores as evidence for dependencies (Clark al.) 2019 Vig & Belinkov [019 Ravishankar et al., 2021), resulting in individual dependency ares or unlabeled dependency trees. If we take a broader view, we'll also find parameter-free activation explanation methods for more general-purpose concepts in deep learning research (Mu & Andreas 2020, Antverg & Belinkov (0022) together with some supervised methods (Radford et al. [019] Lakretz et al. [019 et al. 2019). These methods, also called
neuron analysis methods, were systematically evaluated by a recent work (Fan et al., 2023). We'll
systematically compare our methods with these baselines in
Similarly, there are also methods for extracting syntax rules from sentences without annotations, ranging from probability-based methods (Klein & Manning, 2002) to neural-network approaches (Shen et al. 2018). While achieving different tasks, they are essentially another side of coin within the field of computational syntactics.
IPBP METHODOLOGY
To foster understanding, we'll first break our method into key points in the first section, and then introduce the details.
3.1 KEY ASPECT ANALYSIS
Given a sentence X = T2 2n from dataset DD and an arbitrary token pair (x,, fyomfm that sentence, we define f(ا|ا> as the variable (also an element in the dependency tree adjacency matrix) for which kind of dependency exists from o; to Tj. 2|81bl can be specific dependency type like nsubj, j. or á if there's no dependency. If the sentence is fed into transformer LM, there will be
series of attention scores matrices. An element in a specific matrix is denoted as Gghh which stands
for the attention score from the i-th token to the j-th token in the attention matrix from the h-th
attention head of the b-th transformer block. If we gather observations [|{[ and ai [dbb] for each token pair in the dataset, we'll get two co-occurring dataset-wide variables, L and As, standing for the dependency and attention score of head (b, h) at any token pair. Therefore, the goal of our probing can be divided into two:
MI Estimation:
For any head b, h, estimate the mutual information (MI) between L and As, (denoted by Al,,))
Tree Reconstruction: A method for deriving full dependency tree based on attention scores As,
Specifically, since L is discrete variable and As, is continuous, the joint distribution is mixture distribution, the formula of MI is as follows (slightly different from classical definition):
f(1,a) da Σ [S(,a)log log (())f(a) LEQUS
MI(I; Ao.
Where L stands for the set of all dependency relationships (nsubjj j, dobi, ... and f(l, a). P(I). f(a) is short for the density value of joint distribution fLL, Ap. at L = 1, Ab, = a, density of marginal distribution (A at Ab, a, and scalar probability P(L 1).
Moreover, the second goal can be regarded as a Bayesian inference process taking As,h as the evidence and L as the hypothesis. The posterior distribution (f = l| = a)) is required for tree reconstruction. Therefore, the key to achieving these two goals are those probabilistic distributions.
In the following sections we'll II dive deep into how infer these distributions from the dataset.
3

---

Published as a conference paper at ICLR 2026
3.2 GETTING THE DISTRIBUTIONS
Initialization. Assume the dataset 'D is already annotated with a series of <sentence, dependency free> pairs, and model with b blocks and 5 attention heads within each block. We first initialize series of attention score sets As, where b E {1 ...... b). E {1.... h and E C Φb. Ab. means all possible attention scores of attention head b, h between token pairs having dependency I.
Gathering attention scores. We iterate over the dataset and for specific sentence X E D, we feed X 21 ....,, Tn into the model, and for any token pair (I,, (i,j E {1...n.) n)), the dataset provides [[[ its dependency relationship 2[4|[)] and the model provides the attention scores dg (vb, h). We add
3| the corresponding attention score set Ab, A After the iteration, all attention score sets bbh
will have all possible attention scores in the dataset.
Getting the distributions. After gathering attention scores, we estimate those required probabilities. The most intuitive one might be P(L. 1), since we can take the empirical probability P(L. = =) = Lirecute) As.h,r Vb, h (the number proportions) on the dataset as the approximate value. The tricky ones are the continuous probabilities. Given we already have abundant attention score samples, and the mixture distribution render requirements for only univariate densities, it's nice to Kernel Density Estimation (KDE) to estimate them. Specifically, for every possible As, we regard it as the observation of the attention variable As, under the circumstance L I. The samples in As, follow the conditional density of [As.h L = 1). Using the Gaussian kernel and taking a specific bandwidth B (See D.1 the kernel density f(Au,,) 1) can be estimated
30
A b.hi!
exp
Ab
22T As.ud
B
where aa is the standard deviation of Ao,ii and A means the i-th value of As,
Again, if we take the view of Bayesian inference, with As, as evidence and L as hypothesis, then the estimated f(As.a))) is essentially the likelihood density. Applying the Bayesian theorem, we'll get the following equation:
f(L[Ab.N) = ((A.A)))P(L) f(Ao,t)
f(Ab.h, L) f(Ao,n)
(3)
This means that given the likelihood f(Ao,u L) and the prior P(L), we can multiply f(Ab, (L) with P(L) to get the joint densities f(Ab, k: L). Moreover, by summing estimate marginal density f(Au.a). and then the posterior probability [(I]AA..) computable. This resolves the computation of all the required probabilities as mentioned in Section 3.1
all possible Ls, we can
3.3 ESTIMATING MI
With all these distributions, we're able to proceed to our two main goals: MI estimation and Tree Reconstruction. However, if we reexamine the MI formulation in Equation [ we'll Il find that the (T; Ap,a) ) in Equation 1 measures how much shared information head (b. h) has about every possible dependencies. However, more probable case might be that head (b, h) is only responsible for certain dependencies. This kind of specialist head also the assumption of previous attention-
analysis research like (Htut al. 2019). Even though such versatile heads exist, MI corresponding to all dependencies still too coarse- -griindr.dddd..... Therefore, it is necessary to tweak Equation to make the MI formulation fit this specialist assumption. The new formulation is follows:
f(l,a) da + f(-1,a) log f(-1,a) da Mibinary (1:iinar(1:444)))) Ab,n) / f(l.a)log (())((a)
(4)
P(-I)f(a)
In that equation, f(-l, a) is short for the density value of Ab, at As, = a, where f(-1, Ag.u) stands for the joint density between all dependencies other than 1 and At,h- In practice, it can be
4

---

Published as a conference paper at ICLR 2026
gained by marginalizing f(A L) over all possible L E (CU {o)) - {1}. P(-I) stands for the possibility of dependencies other than (, which can be estimated using P(I).
3.4 GETTING HIGHLY SYNTACTICAL HEADS
By now, having posterior distributions Ao.)) and Miburry feasible for estimating the independent importance of each dependency type, it's possible to reconstruct the trees. The basic idea of our tree reconstruction is: for every dependency relationship l, we pick out attention heads highly responsible for I, constituting the head set Hi. We then infer the possibilities of dependency ares of jointly decided by the posteriors of heads from Hi by using Miniaay to balance between posteriors of each head from H4, forming the overall possibility for dependency are with relation l. Finally, we use decoding algorithm to build the dependency tree based on these overall possibilities.
To form H4, it's natural to think about setting threshold on Mininary (1; Ab. However different dependency relationships might have different MIsisury magnitudes, so an adaptive threshold condi- tioning on specific relations is necessary. Remind the fact that mutual information is upper-bounded by the individual entropies of each random variable (in our case, H (1)) and H (As,)), where (L) denotes L equals to not). Since As, is continuous, and the entropy analogs of continuous variables (variational entropies) are known as incorrect analogs (and sometimes negative), this makes unable act an upper bound. Therefore, we choose estimate (L(()) as:
(L)) = ((L)) P(-L) log P(-L)
(5)
If MI is divided by the entropy, the resulting proportions ( Mluan((iAn will be normalized into H(((L))
[0, 1], which can act as the adaptive threshold.
3.5 TREE- CEEEERECOOSTTUCIOO ALGORITHM
After getting His, another problem occurs: As mentioned before, previous probing practices mainly aim at building unlabeled trees. Even those supervised dependency parsing methods (Dozat & Manning, 2017, TTin et al., al. (2022) rely on separate networks for predicting arcs and labels. Therefore, these methods are operating on a simple probability space with only probabilities on the existence of dependency arcs, and then individual probability spaces for the label of cach arc. Moreover, their methods only involve one or two networks responsible for predicting probabilities, while our method instead has bunches of posterior probabilities. Therefore, it's necessary that we design decoding algorithm that not only balances each posterior but also constitutes a valid probability space.
We first make assumption that the overall possibility of dependency ares can be individually conditioned on each head in H4. Theoretically, a good model to balance each posterior is to treat the prediction of dependency arcs as voting process: for dependency I, each head (hi, h2) Hi can be seen as a participant with a representation of (I.A.o, The probability of dependency are of I can be seen as the probability of a total representation of approval larger than a proportion (like half or two-thirds). However, due the non-discrete weights, the problem cannot be efficiently dynamically programmed, resulting in search space of O22 which will be terribly inefficient during inference. Instead, we relax this voting problem to an easy-computing yet rational form: We take the geometric mean of the posteriors. Specifically, let GP Hi Izy; () be the geometrically-averr probability of an are of / between tokens 2; and Iy conditioned on heads in H4. In logarithmic space, the geometric mean is:
Σ Exbe, (ba Mininary (1; Abe.ta.) log f(I. [TT==[AApp AlA
log GP Hi (Ziz21fi1)
(6)
E Exbi dem) Jem)) Mininary (1; Abr..tu.. )
This is approximately equivalent to the Logarithmic Opinion Pooling (Heskes 1997) technique widely adopted in Bayesian inference, thus acting as a reasonable approximation when the number of experts (in our case, heads in H₂) is relatively large. Given this good approximation, it still has problem: if we sum over all probabilities of each possible dependency relationship (in our MI case, and -D) it is not guaranteed to be 1, and different kind of dependency ares are decided by different sets of
5

---

Published as a conference paper at ICLR 2026
heads, recalling the problem of valid and homogeneous probability space. To resolve this, we build larger multivariate probability space of (0, 1)14H+1 We take the voting process of the dependency between H2 and v as |C] + independent votes. The I-th ballot votes for the existence of the 1-th dependency from L, using the GPH, (z;, in Equation 6 as the probability of existence, and Zy; 1 GPH, Zy; 1) as the probability of non-existence. The overall probability P(r,, æz; meaning the probability of an are of between tokens 2; and 2- conditioned all highly responsible heads Hi U ....UU
U..UUHI U4a U H., is calculated as follows:
P(11,4011 = GPH, (Zi,zzi)) x II
{1-GHH, (5is);;))
(EC+(0)
This results the decision of all kinds of dependency ares in unified and valid probability space. By now, the two problems introduced by multi-head and multi-label are solved. We're only need decoding algorithm utilizing the overall probabilities to arrive complete trees. Specifically, following previous supervised dependency parsing methods, we re using the Eisner dynamic pro- gramming algorithm (Eisner) 1996) as the decoding algorithm. Readers might refer Appendix D.]] for implementation details of our methods, like hyperparameters and our GPU-optimized KDF and integral methods.
3.6 IS IPBP TRIVIAL?
Since MI estimation is a small hot topic in statistics, it's helpful to perform methodological com- parisons with related methods. We found two methods sharing (minor) principles with our method: The first one (Moon et al.) 1995) is method for estimating MI between two observations within a time series using KDE. They did three separate KDEs, with one multivariate one. While it's a known issue that KDE quickly becomes inferior when there is more than one variable, known as the dimensionality curse, their method is inevitably introducing errors (and also unapplicable to our peveationndependenee mixure distribution setting). We instead dexterously circumvented the dimensionality curse, also making the least number of estimations possible (limited to 1) by exploiting mixed-joint distribution and Bayesian theorems.
Another one also focuses on mixed joint distribution (Gao et al. 2017). However, they use kNN- like algorithm to estimate point-wise mutual information (PMI) and average it over the dataset. Their method didn't it provide any valid probability distributions, thus offering no possibility of tree reconstruction, and also allowing less chance for datasct-level or visualization-bar based explanations.
EXPERIMENTS
In this section, we're going to systematically compare our method with a series of probing as well as neuron analysis baselines.
4.1 EVALUATION STRATEGY
We divide the baseline employed into two sets based on the two sub-tasks introduced in Section 3.1 For the first subtask MI Estimation, introduce a series of head-selection baselines, replace the estimated MI with the importance introduced by these methods, keeping the tree-construction algorithm fixed, and compare the qualities of reconstructed trees. This essentially uses constructed trees as an external "proxy" for the quality of head-selection. Apart from that, we also employed the voting theory-basen intrinsic comparison method introduced by (Fan al. (023). The core idea is that they compare neuron importance methods by letting other methods vote neuron selection method with higher consistency with the rest is considered a preferred one. Since the high-importance head sets like Hi of variable sizes, instead compare the importance rankings of all heads given by two head-selection methods.
method. A
For the second subtask Tree Reconstruction, we compared our MI and posterior- based algorithm with the common practice of previous aatention-based methods. We believe dividing into two groups of baselines is better for illustrating the individual contributions of each submodule of IPBP.

---

Published as a conference paper at ICLR 2026
4.2 BASELINE METHODS
For head-selection baselines, we'll start from several strong neuron analysis methods evaluated by recent paper (Fan et al. 2023):
Probeless (Antverg & Belinkov 2022): This is a parameter-free method, which gets the correlation scores by calculating mean activation values corresponding having/not having a concept alongside the dataset. In our situation, we use the following instead of Milinaar...
Ab,)) =
Σ
Mo - Av,harl
(8)
(EC+(O)-11)
Where A.,.; denotes the mean value of a specific attention set. Note that despite its simplicity, this method is evaluated as the method being most consistent (thus most preferred) with others by Fan et al. (2023).
IoU (Mu & Andreas 2020 This method uses Jaccard Similarity as a correlation criterion. In our implementation, we use the following form:
Ab, n [T, +oo)|
loU(1; As,h) = Ab,hal + Σrec+ |Ab,,e O [T, +oo)|
(9)
Where 7 is a threshold serving as selecting a salient score. Following the original authors, we set it to the top 99.5% value among values in As,
The Linear Feedforward Family: This method refers to a series of methods performing correlation ranking by training supervised linear network Wg. Specifically, the equation below gives uniform formulation of these methods:
We arg min
Σ
log Po = M+
MD |a b.1......))
(10)
Woe XED XX
BDJ Where We is matrix of shape bh x (1C) + 1), and b.b denotes the concatenated vector of [}]] [+[+)
attention scores between and x; for all attention heads, and Pa ( (...b...,bb stands for the probability of the ground-truth label estimated by the network. When \ 1, A2 O, this equation becomes Lasso (Radford et al. 2019), when A 2019), and Å 1, A2 corresponds to ElasticNet (Dalvi et al., 2019). We use ElasticNet representative. After gaining the trained Wa. we use the weight entry mapping attention score of head (b. h) the probability of relation I the correlation value, b, h).
X 1, it becomes Ridge (Lakretz al.,
V-Information: Xu et al. (2020) proposed to use a trainable network as an approximation of conditional probabilities, and use the mean logarithm probabilities approximations of conditional entropies based on the law of large number. This the ofatc-of-the-art mathematical--iignnn estimation algorithm, used by previous methods also taking information-theorest perspectives (Pimentel et al.) 2020b 2022). Specifically, in our case, we use max Hy..A.,.) (1 Ag,x) as replacement of MIbinary (1; Ao, and as the equation shows:
entropy
Σ Σ Hv(1)An,))) ExED XX
XED (E.,))E
(11)
L(y (llll) log MLPo, (bl) + I lc+ (1 (ibll log log 1 MLP aa (4
Where MLPp,kt. (.) are deep MLPs individually trained using head (b, h) to predict label
Under each head-selection setting, for fair comparison, set a limit of the total number of syntactical heads Eiecu(o) [Pd of 2000 for every head-selection method.
For the conerrsssttttion alternative, we use raw attention score: Under this setting, we still use MI the head importance criteria, but for a specific head (b. h) Hi, we use the attention score
7

---

Published as a conference paper at ICLR 2026
HE instead of the posterior ((L)A in the reconstruction algorithm, with all other formulations (balancing heads and forming valid probability space) unchanged. This simple intuition is the common underlying principle of previous methods attention probing methods (Clark et al. 2019
4b.h
Vig & Belinkov 2019 Ravishankar et al.) (021). We found that due to the absence of our estimated posteriors, if > (E Hal reaches 2000, the scores of all heads will be rather noisy. Therefore, we
choose to select top-k heads based on MI for each label. We did a grid search and found the top-8 setting has ideal performance.
Moreover, we also introduced two simple baselines, left/right branching assuming the dependency head of each word always the previous/latter word (this results in unlabeled dependency trees), and random model that applies IPBP on a model with the same structure but with its weights randomly initialized. These methods are mainly used to rule out the worst situations where our methods (also the baselines) fall behind the simplest baselines.
4.3 MODEL, DATASET AND METRICS
We applied our methods on open_ 11ama 7b[Geng & Liu Aicuna-7b(Chinn
2023).
Deta-LLama-3-Bal Mistral-7R(iiag al., 2023). These models are decoder-based LLMs consisting of a great many layers and heads. Compared with pre-trained models like BERT (Devlin et al., 2019) or XLNet which are widely used in previous probing research, these LLMs' attention heads might have rather varied functionalities, offering more insights under the contemporary LLM research context.
Meta
2024
et al., 2023), and
With respect to collecting attention scores, we cache the Q/K/V of each attention head and use them to re-calculate the unmasked attention scores. This is mainly because, as sentence lengths vary, the from sentences of differennt lengths falling in different intervals, which undermines the density estimation process. (This is especially true for masked attention matrices since they have rows of varying numbers of unmasked elements, even attention scores within the same sentence vary in magnitudes.) Given that softmax is not bijective, using cached QK to reconstruct the unnormalized scores is inevitable. Since directly using the reconstructed attention matrix might involve using masked attention scores for estimation, we talked about two structural alternatives of our method in Section 4.4 to discuss potential effects. To deal with subword tokenization, we use the attention scores on the last sub-token of each word.
attention scores also vary in magnitude, resulting in attention scores
For datasets, we use the English, French, and Spanish partitions of Universal Dependencies (UD) 2.9 (Zeman et al. 2021 as our dataset. Dataset statistics are shown in Appendix [.1
For evaluation metrics on reconstructed trees to evaluate the quality of constructed trees, like supervised dependency parsing methods, we also employ labeled attachment scores (LAS), and unlabeled attachment scores (UAS) (different from previous probing methods, which only used undirected unlabled attachment scores (UUAS)). For intrinsic comparison, since each head-selection method gives a [[C], x h) matrix, for any two head-selection methods, we calculate the Spearman-R with respect to the corresponding rows for each dependency label, and average them across labels.
Note that, unlike IPBP, some baselines are rather computationally prohibitive. Given the observations that most models have similar IPBP tree-extraction performance (Section 4.5) and similar perfor- mances when applied on our multilingual dataset (Appendix [B), also considering the vast number of lagguagge-baseliine compositions, experiments of baseline comparison mainly conducted on the English dataset using model open_ aama_ _7b (it seemed most "familiar" with our dataset as shown in Appendix B acting good representative)
4.4 IPBP STRUCTURAL ALTERNATIVES
Apart from comparing with previous methods, we're re also curious about our model's designs. There- fore, we propose several alternative structures:
We noticed that the attention score samples exhibit a long-tail characteristic: most samples come from Ag. since most pairs of words don't have dependency in between. Ao might be noisy, consisting of various non-syntactic inter-token relationships, and MI estimations based on samples in Ag might be affected by this long tail noisy distribution. Other score sets A Ai,...AA
Positive MI:
8

---

Published as a conference paper at ICLR 2026
are having approximately the same magnitudes and their corresponding token pairs guaranteed to have any dependency relationship. Therefore, we also calculated a more syntactical MI, namely pas with the following formulation:
al) Spma(l, Pp
MIpss (L; Ab,h) =
Σ IEC
(12)
In that equation, Ppos( (.), fpos(::)) actually stand for conditional possibilities when * Φ, estimated by [Ab,aall pos (L - Erecc [As.aar and Spee (L, Ao, f(Ab,, IL) prs (L). During implementation, we'll use balance factor and calculate the mixed MI Mlmix (=: = aMMiiooryy (++ + pos
Are First: Unlike previous methods, we're directly obtaining labeled dependency trees, bypassing the process of dependency are predicting. We're curious about whether it's good choice. Under this setting, instead of estimating (Ab,A [)). we'll directly estimate the unlabeled likelihoods f(As,) E f) and fA [. Φ), and calculate the corresponding multivariate probabilities together with corresponding MI values. We'll compare UAS to check the quality of reconstructed unlabeled trees.
Sometimes, we're unsure whether the attended token acts as a dependency head, or a dependant. Another thing to consider is that lower-triangular entries of the (reconstructed) attention matrix might correspond to dependencies that ought to correspond to the
Transposed:
"useless" (masked) attention scores. So we let [|l]s) correspond to 6. and repeat the whole IPBP process under this setting.
[i]
Undirected: This is more radical leap towards eliminating attention score uselessness. Under this alternative, we relax the problem to probing undirected (but labeled) dependency trees through "flipping" the up-triangular area of the dependency adjacency matrix the lower-triangular part (i.e., [|]b being unable to decide dependency directions, as a tradeoff for using attention scores that are actually in-use during generation. Under this setting, undirected trees reconstructed using the maximum spanning tree algorithm.
(ill)) ), and perform the whole IPBP process. This results in (1]1)) corresponds to oi if > j else 06,h
What's 's more, we've also done ablations regarding KDE bandwidth and posterior pooling methods. See Appendix E for details.
4.5 RESULT AND ANALYSIS
Results are shown in Table I and 3 We can see that our method applies to multiple models and languages. We also observe that it gives better sets of selected heads (using trees as proxies and also intrinsically evaluated), and produces more faithful trees given the same set of heads, compared with all competitive baselines. To be noticed, we can find that supervised methods like LFF or V-Information may still fall behind our statistical method. Considering methods like y-Information require much larger computational budgets, and we have to our attention probing settings (see Appendix D.2 for details), we argue that supervised deep networks might not be the panecea especially when the data long-tailed For tree reconstruction alternatives, the well-used tree-reconstruction baseline raw-score, despite being specially treated, still has a great performance gap compared with our posterior-based method, further justifying its necessity. For structure alternatives, we notice that incorporating MI will give performance benefits, inspiring further improvements. The transposed setting will still capture a relatively smaller portion of dependencies, with undirected setting capturing better trees (ignoring slightly different metrics for simplicity) than it. For more intriguing conclusions regarding how models capture dependencies of different directions, please see Section 4.6 For those simple and random baselines, we find that they fall far behind most of our methods and baselines, indicating that IPBP and the baselines are producing non- trivial results. Finally, by comparing with our arc- based baseline, we'll 'Il find that we'rr re actually at a triangular balance: we probed for more accurate, labeled trees using more straightforward method: no need for individual are probing.
several tricks to adapt V-Information
low-dimensional.

---

Published as a conference paper at ICLR 2026
Table 2: Tree reconstruction results of our IPBP and different baselines. For undirected settings, results* are UUAS and ULAS instead.
Method
UAS
LAS
Table 1: Results across models and languages
Random Model
12.4
0.8
L. Branching
16.6
Model/Language
UAS
LAS
R. Branching
26.4
Meta-Llama-3-88.
39.9
26.6
Probeless
34.8
20.9
Mistral-7B
50.4
30,9
IoU
38.3
26.6
vicuna-7h
41.5
25.8
ElasticNet
41.9
31.3
Open_llama 7b
49.1
30.6
V-Information
41.3
20.9
open_llaman 7b+fr
44.7
25.6
Raw Score
32.3
16.6
open_llama_700
44.8
21.9
IPBP
49.1
30.6
IPBP (transposed)
42.6
28.0
IPBP (undirected)
45.3*
28.4"
IPBP + Mllos
49.9
34.8
IPBP (arc only)
36.5
N/A
4.6 FURTHER ANALYSIS
Like previous probing methods, we'll also do fine-grained analysis of our reconstructed trees and estimated MI values. Instead of listing up MI values and doing trivial analyses. we decided to provide several intriguing and informative conclusions, giving inspiration to those upcoming research.
The first conclusion that decoder models adaptively capture left/right dependencies: Since the masked decoder attention can only attend to previous tokens, it's reasonable that left dependencies (dependencies pointing front words) can be well captured. What makes it more intriguing that right dependencies (pointing to rear words) might also be captured in "attend-bace draw this conclusion by comparing the top- most well-reconstrueted labels between original IPBP and the transposed alternative. We find that there's re more look-ahead dependencies (5 of 10) under the transposed setting compared with the original setting (3 10). The second conclusion is that model layers correspond to tree layers some extent: lower layers are for local/phrasal dependencies, dependencies. This corresponds to intuitions but was
manner. We
while higher layers are for global/sentence-we never systematically justified before due the lack of MI-like criteria and overfocusing unlabeled trees. Thanks to the -raiied MI, similar to the Prenter-of-Graving introduced by (Teneyy
ct al. 2019a), we can calculate the MI-weighted layer indices Σhe(t...) Ene(t...)) MIpu(Ar.a3)) Σne(1...)) Loe( Ab.s()) for each label, where smaller weighted indices indicate dependencies having more lower-layer heads responsible for it. Among these top- 10 labels, we calculated the Pearson correlation coefficient p between the weighted layer index and average depth (maximum distance to leaf nodes) of each dependency label, getting result of 0.69 with 0.03 for a null hypothesis of no correlation. Apart from these, we also drew some other informative conclusions by series of further MI/distribution analysis methods, including visualization, mechanistic analysis, and unbiased (not specifying "things to probe is dependency" in advance). These results further ensure the faithfulness of IPBP and give intriguing conclusions and insights. Please refer to Appendix A for details.
5 CONCLUSION
We proposed method that can estimate MI and reconstruct labeled dependency trees without introducing any trainable networks. Indeed, our method is achieving an "impossible triangle": it has simpler architectures requiring small computation budgets, while producing more complex and high-quality trees, and also transparent for explanation, meaning that researchers can get fine-grained head-level MI estimation, and a bunch intuitive probability functions, without worrying about did my network furtively learn the task? Through comparing with a series of competitive baselines, ensured its effectiveness, and then made several informative conclusions based on our estimated MI and reconstructed trees. The number of conclusions is limited due to the content limit. Since method provides an analytical backbone, we strongly encourage future fine-grained research based on the fine-grained MI values and distributions of our method.
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
This research is supported by National Key R&D Program of China (No. 2024YFF0907033 and Beijing Natural Science Foundation (L247710). We'd also thank the anonymous reviewers for their efforts and valuable suggestions.
REFERENCES
AI@ Meta. Llama model card. 2024, URL wtt.///tss///gttab...oo/o//////e 11ama/ Lama33
http//aainMMMDELL CARD md
Guillaume Alain and Yoshua Bengio. Understanding intermediatee layers using linear classifier probes, 2017.
Omer Antverg and Yonatan Belinkov. On the pitfalls of analyzing individual neurons in language models. In International Conference on Learning Representations, 2022. URL https: //
openreview wetenreviewnet//ooreeeeeeeeeeeee www/teetoooroee - www.://wwww..../////0/0
Wei-Lin Chiang, Zhuohan Li, Zi Lin, Ying Sheng. Zhanghao Wu, Hao Zhang, Lianmin Zheng, Siyuan Zhuang, Yonghao Zhuang, Joseph E. Gonzalez, lon Stoica, and Eric P. Xing. Vicuna: An open-source chatbot impressing gpt-4 with 90% * chatgpt quality, March 2023. https
///mmys..rrg/blo////00 -3--0-3---0000000 30 111-000vvieee vvicuna
Kevin Clark, Urvashi Khandelwal, Omer Levy, and Christopher D. Manning. What does BERT look at? an analysis of BERT's attention. In Tal Linzen, Grzegorz Chrupata, Yonatan Belinkov, and Dieuwke Hupkes (eds.), Proceedings of the 2019 ACL Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP. pp. 286, Florence, Italy, August 2019. Association for Computational Linguistics. doi: 10.18653/1/W119-4000 URL (tthstts:: httttps://aclanttoo...
org/W19-4822
Alexis Conneau, German Kruszewski, Guillaume Lample, Loic Barrault, and Marco Baroni. What you cram into single $&.#* vector: Probing sentence embeddings for linguistic properties. In Iryna Gurevych and Yusuke Miyao (eds.), Proceedings of the 56th Anmual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 2126 2126-2136. Melbourne, Australia, July 2018. Association for Computational Linguistics. doi: 10. 10.18653/1//188-1199
URL httptttp///cccc antho ogy.orgg/1881111
Fahim Dalvi, Nadir Durrani, Hassan Sajjad, Yonatan Belinkov, Anthony Bau, and James Glass. What is one grain of sand in the desert? analyzing individual neurons in deep nlp models. Proceedings of the AAAI Conference on Artificial Intelligence, 33(01):6309 6317, Jul. 2019. doi: wtw6609aaai.v3300 133001693,,,,99 URL Inttps:///oo/ aaai org/ index. php/AAAT/
article/view/4599 wtt.cle//viee//....
Jacob Devlin, Ming-Wes Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Jill Burstein, Christy Doran, and Thamar Solorio (eds.), Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume I (Long and Short Papers), pp. 4171-41 Minneapolis, Minnesota, June 2019. Association for Computational Linguisties. doi: 10. 10.18653/v/NN19-14222 1423.
URL htttps://aclatthooogr
org/N19 N0g/N19-220 1423/
Dror Dotan and Nadin Brutmann. Syntactic chunking reveals a core syntactic representation of multi-digit numbers, which is generative and automatic. Cognitive Research: Principles and Implications, 7(1):88, 2022. doi: 10. 11186/641255-022-004222
Timothy Dozat and Christopher D. Manning. Deep biaffine attention for neural dependency parsing. In International Conference on Learning Representations, 2017. URL htttps://operree
net / Eorum? 2 id- HK95PK911
Jason M. Eisner. Three
probabilistic models for dependency parsing: An exploration. In COLING 1996 Volume 1: The 16th International Conference on Computational Linguistics, 1996.
URL https: //ccl antho www.olannoooooooooooooooooo wtwpo///oooo///0000000003

---

Published as a conference paper at ICLR 2026
Nelson Elhage, Neel Nanda, Catherine Olsson, Tom Henighan, Nicholas Joseph, Ben Mann, Amanda Askell, Yuntao Bai. Anna Chen, Tom Conerly, et al. A mathematical framework for transformer circuits. Transformer Circuits Thread, 1(1):12, 2021. URL https://
trranformmr-ctt ci rcu rcuits. pttpppp/////aeeeeeeeeee
Jacqueline Fallon and Liina Pylkkänen. Language at glance: How our brains grasp linguistic structure from parallel visual input. Science Advances, 10(43)xeadr9951, 2024. doi: 10.1 126/
sciadv.adr9951.
Yimin Fan, Fahim Dalvi, Nadir Durrani, and Hassan Sajjad. Evaluating neuron interpretation methods of nip models. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (eds.), Advances in Neural Information Processing Systems, volume 36, pp. 75644-75668. Curran Asso- fieepppperreeeeppppprerpp
reeri ciates, Inc., 2023. 000.100eeee0000112222010011000-0
pdf
Javier Ferrando and Elena Voita. Information flow routes: Automatically interpreting language models at scale. In Yaser Al-Onaizan, Mohit Bansal, and Yun-Nung Chen (eds.), Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pp. 17432-17445, Miami, Florida, USA. November 2024. Association for Computational Linguistics. doi: 10.18653/v I/ /ttp///////.///////////////// 2024. emn smmnl-ma manmmipmmnn...... 965/ /tt.aaattttooooooooooooooooeeeee
2024.emnlp-main.66. URL https:
Weihao Gao, Sreeram Kannan, Sewoong Oh. and Pramod Viswanath. Estimating mu- tual information for discrete-continuous mixtures. In Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Ad- vances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017. URL https::
wwwrips.cc/paaeeee. /i.es//ppper///17 fi F1le/4r//a33900060055e3e22b....2 Pappoea.aa02PPee.ppp..
Xinyang Geng and Hao Liu. Openllama: An open reproduction of Ilama, May 2023. URL https
7/github. com cowp//ennmmmmmmmmmmmm httm:/pppn.mmeeearreeeeeeaeeeee wwweearchoopen_leeeeee.
Tom Heskes. Selecting weighting factors in logarithmic opinion pools. In M. Jordan, M. Kearns, and S. Solla (eds.), Advances in Neural Information Processing Systems, volume 10. MIT Press, 1997. URL /tPLitttpp//pppoeeeenggggg
nwwr.iss......u................ cc/paper files/paper/ 2tt::oocc2PPPPaepeeee Paper. pdf
John Hewitt and Percy Liang. Designing and interpreting probes with control tasks, In Kentaro Inui, Jing Jiang, Vincent Ng, and Xiaojun Wan (eds.), Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLL) pp. 2733 2743, Hong Kong, China, November 2019, Association for Computational Linguistics. doi: 10. 10.18653/v1/D19 1/D19///11D1111/777 URL https 7/aclanthe woaerhooogyy..oo ottp/19-//1119
John Hewitt and Christopher D. Manning. A structural probe for finding syntax in word repre- sentations. In Jill Burstein, Christy Doran, and Thamar Solorio (eds.), Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Shori Papers), pp. 4129-4138, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics. doi: 10.18653/v1/N19-111 URL https:
1419/
Phu Mon Htut, Jason Phang, Shikha Bordia, and Samuel R Bowman. Do attention heads in bert track syntactic dependencies? arXiv preprint arXiv:1911.1224 2019.
Albert Q. Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, Lélio Renard Lavaud, Marie-Anne Lachaux. Pierre Stock, Teven Le Scao, Thibaut Lavril, Thomas Wang, Timothée Lacroix, and William El Sayed. Mistral 7b, 2023. URL https: //arxivv
org/abs 2.31 06825
Dan Klein and Christopher D. Manning. A generative constituent-conee model for improved grammar induction. In Pierre Isabelle, Eugene Charniak, and Dekang Lin (eds.), Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics, pp. 128 135,
12

---

Published as a conference paper at ICLR 2026
Philadelphia, Pennsylvania, USA, July 2002. Association for Computational Linguistics. doi: http://00.00...1.........../a//. http://:///aaaaatoooooo...oo.// 01111///17730831003166 URL
Yair Lakretz, German Kruszewski, Theo Desbordes, Dieuwke Hupkes, Stanislas Dehaene, and Marco Baroni. The emergence of number and syntax units in LSTM language models. In Jill Burstein, Christy Doran, and Thamar Solorio (eds.), Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume (Long and Short Papers), Pp. 11-20. Minncapolis, Minnesota, June 2019. Association for Computational Linguistics. doi: 10. 105186///111119100022 URI. https
//aclantho ogy. www.o/rg////OO//
Nelson F. Liu, Matt Gardner, Yonatan Belinkov, Matthew E. Peters, and Noah A. Smith. Linguistic knowledge and transferability of contextual representations. In Jill Burstein, Christy Doran, and Thamar Solorio (eds.), Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume / (Long and Short Papers), pp. 1073-1094 Minneapolis, Minnesota, June 2019. Association for Computational Linguistics. doi: 10. 10.18633/1//119-11222 URL
Drg/N19-1112
Alessandro Lopopolo, Antal Van den Bosch, Karl-Magnus Petersson, and Roel M Willems. Distin- guishing syntactic operations in the brain: Dependency and phrase-structum parsing. Neurobiology of Language, 2(1): 2021. doi: 10.1162/nol 00029.
Young-Il II Moon, Balaji Rajagopalan, and Upmanu Lall. Estimation of mutual information using kernel density estimators. Phys. Rev. E, 18-22211-2233 Sep 1995. doi: 10.1 1..1003/ URL httpttt::///tit
aps wwtp:oogggooooooooooooo/ooo/oott http:///.....11111111111 http://...../////./.e........ 52.2311..
Jesse Mu and Jacob Andreas. Compositional explanations of neurons. In H. Larochelle, M. Ranzato. R. Hadsell, M.F. Balcan, and H. Lin (eds.). Advances in Neural In- formation Processing Systems, volume 33, pp. 17153-17163. Curran Associates, Inc., 2020. URL htttps://proceeediiee file/c74956f (077555ff.Baaaeeeeee777766622221
cc/paper /t1le//pappr/2220/ pdf
Max Müller-Fberstein, Rob van der Goot, and Barbara Plank. Probing for labeled dependency trees.
In Smaranda Muresan, Preslav Nakov, and Aline Villavicencio (eds.), Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 7711-7726, Dublin, Ireland, May 2022. Association for Computational Linguistics. doi: 10. 1865// v1/2022.acl-long.55 532. htthttps: /tw.aaaaatttte htt///acaaaattoooo..aoeeo org. acl- 532/
Matthew E. Peters, Mark Neumann, Mohit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee, and Luke Zettlemoyer. Deep contextualized word representations. In Marilyn Walker, Heng Ji, and Amanda Stent (eds.), Proceedings of the 2018 Conference of the North American Chapter of the As- sociation for Computational Linguistics: Human Language Technologies, Volume (Long Papers), pp. 2227-2237, New Orleans, Louisiana, June 2018. Association for Computational Linguistics. doi: 18.18653/v 1016655/1//188... 1202. URL pnttps://ede antho wog.. wwwooogg////... 1202/
Tiago Pimentel, Naomi Saphra, Adina Williams, and Ryan Cotterell. Pareto probing: Trading off accuracy for complexity. In Bonnie Webber, Trevor Cohn, Yulan He, and Yang Liu (eds.), Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 31 3138-3153, Online, November 2020a. Association for Computational Linguis- URL httptttt////c httpsss//acaaaathoho ogy org.y//rg/220
tics. doi: 10. http:///1/2220000/mmmmmaaa
emn p-ma 1 n, 2.577
Tiago Pimentel, Josef Valvoda, Rowan Hall Maudslay, Ran Zmigrod, Adina Williams, and Ryan Cotterell. Information-theoore probing for linguistic structure. In Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel Tetreault (eds.), Proceedings of the 58th Annual Meeting of the Association for Computational Linguisties, pp. 4609-4622, Online, July 2020b. Asso- URL https
ciation for Computational Linguistics. doi: 10. 33653/v1/20200 //aclanthologys org/2020 acl-main..mm....mmmmm...m......m 4207]
Tiago Pimentel, Josef Valvoda, Niklas Stochr, and Ryan Cotterell. The architectural bottleneck principle. In Yoav Goldberg, Zornitsa Kozareva, and Yue Zhang (eds.), Proceedings of the
13

---

Published as a conference paper at ICLR 2026
2022 Conference on Empirical Methods in Natural Language Processing, pp. 11459-11472 Abu Dhabi, United Arab Emirates, December 2022. Association for Computational Linguis- tics. doi: emn p- i n.. 7
URL httppp://acaaaaaooooooooooo
Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, llya Sutskever, et al. Language models are unsupervised multitask learners. OpenAl blog, 1(8);9 2019.
Vinit Ravishankar, Artur Kulmizev, Mostafa Abdou, Anders Søgaard, and Joakim Nivre. Attention can reflect syntactic structure (if you let it). In Paola Merlo, Jorg Tiedemann, and Reut Tsarfaty (eds.), Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume. pp. 3031-3045, Online, April 2021. Association for Computational Linguistics. doi: 10.18653/v /ttp:/2222eeellmmeieeeee 264. URL https: http:::ss/aaaaaatoooogooo /rg/ 2021 eec.acllllo. ma i n 2647
Yikang Shen, Zhouhan Lin, Chin wei Huang, and Aaron Courville. Neural language modeling by jointly learning syntax and lexicon. In International Conference on Learning Representations, 2018. URLtttpss: http::/::////eeeee.eeee//ee net for.mmidd-rgglllbbb OW
Ian Tenney, Dipanjan Das, and Ellie Pavlick. BERT rediscovers the classical NLP pipeline. In Anna Korhonen, David Traum, and Lluís Mårquez (eds.), Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pp. 4593-4601, Florence, Italy, July 2019a. Association for Computational Linguistics. doi: 10. 18653//11P19-1452 URL https: http////aaaantttoooo//o/
htggPP9-14455
Ian Tenney, Patrick Xia, Berlin Chen, Alex Wang, Adam Poliak, R Thomas McCoy, Najoung Kim, Benjamin Van Durme, Sam Bowman, Dipanjan Das, and Ellie Pavlick. What do you learn from context? probing for sentence structure in contextualized word representations. In International Conference on Learning Representutions, 2019b. URL htttps://oppeneeee
seet/forum2id=SDsSgniee
Yuanhe Tian, Yan Song, and Fei Xia. Enhancing structure-aware encoder with extremely limited data for graph-based dependency parsing. In Nicoletta Calzolari, Chu-Ren Huang, Hansaem Kim, James Pustejovsky, Leo Wanner, Key-Sun Choi, Pum-Mo Ryu, Hsin-Hss Chen, Lucia Donatelli, Heng Ji, Sadao Kurohashi, Patrizia Paggio, Nianwen Xue, Seokhwan Kim, Younggyun Hahm, Zhong He, Tony Kyungil Lee, Enrico Santus, Francis Bond, and Seung-Hoon Na (eds.), Proceedings of the 29th International Conference on Computational Linguistics, pp. 5438-5449, Gyeongju, Republic of Korea, October 2022. International Committee on Computational Linguistics. URL atttss:/accaatthot
org/ 2022. coling 1.iiingg.483))
Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cris- tian Canton Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel Kloumann, Artem Korenev, Punit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael Smith, Ranjan Subramanian, Xiaoqing Ellen Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang Kuan, Puxin Xu, Zheng Yan, lliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom. Ilama 2: Open foundation and fine-tuned chat models, 2023. URL httptttts//aaaat
org. Nbrggabbs/3000 /2007.2307..0 09288
Jesse Vig and Yonatan Belinkov. Analyzing the structure of attention a transformer language model. In Tal Linzen, Grzegorz Chrupata, Yonatan Belinkov, and Dieuwke Hupkes (eds.), Proceedings of the 2019 ACL Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP, pp. 63-76, Florence, Italy, August 2019. Association for Computational Linguistics. doi: 10.18653/ v1/W19-4808. URL httttppsss/acccatcool..... wtt.:r//////-100//////
9--199-4
Elena Voita and Ivan Titov. information-theercee probing with minimum description length. In Bonnie Webber, Trevor Cohn, Yulan He, and Yang Liu (eds.), Proceedings of the 2020 Conference
14

---

Published as a conference paper at ICLR 2026
Empirical Methods in Natural Language Processing (EMNLP), pp. 183-196, Online, November 2020. Association for Computational Linguistics. doi: 10.18653/w1/2020.com emnlp- www.:mmain.nmmnnmmmmm.mm.. main.14. URL
bttps::/aclaattoooogyy wtwp:/hhooooyor///22000 wrg/2020 www.:eeammmmmmmmm.m.mm....mm
Kevin Ro Wang, Alexandre Variengien, Arthur Conmy, Buck Shlegeris, and Jacob Steinhardt. Interpretability in the wild: circuit for indirect object identification in gpt-2 small. In The Eleventh International Conference Learning Representations, 2022.
Yilun Xu, Shengjia Zhao, Jiaming Song, Russell Stewart, and Stefano Ermon. A theory of usable infor- mation under computational constraints. In International Conference on Learning Representations. 2020. BtLnttps: /tttps::/ppenreviieeee.e net. wttp://wwwwwetrrrrrra-.--..-.---
Daniel Zeman, Joakim Nivre, Mitchell Abrams, Elia Ackermann, and et al. Aepli. Universal dependen- cies 2.9, 2021. URLhttp: /thp:://hl //dll.... handle digital library at the Institute of Formal and Applied Linguistics (UFAL).
LINDAT/CLARIAH
A VARIFYING THE FAITHFULNESS OF IPBP
While IPBP produces probability distributions and MI between a great many attention head- dependency label compositions, it is hardly ever possible (also inefficient) to list or visualize all MIs/distributions.
far, we mainly discussed tree extraction results, which naturally raises the doubt that is the high MI expert heads just "happened" have high MI? We'll investigate this question by showcasing other fine- -raiggrinnd (also page-consuming) conclusions that may otherwise be exhibited after Section 4.6
A.1 VISUALIZATION OF ESTIMATED PROBABILITY DISTRIBUTIONS
The most thrilling one we find might be the apparent dual-peak pattern among the é-joint densities of heads in H4: Figure | displays the joint distributions (A ()) of the attention heads whose M Ibinary rank top-5 and last-5, for E [sobj.nsubj] (these are two well-captured dependencies). From that figure we might notice an apparent distributional pattern of these expert heads: the ø-joint densities f(-1, Ab,h) of these heads tend to have an apparent sub-peak, and these sub-peaks overlap with the main peaks of these Í(, As,x) distributions, while those of non-expert heads don't (expect for head 29, 2 and 1 15, 29 which are expert heads for other dependencies by coincidence). While the non-expert heads exhibit approximately Gaussian-like densities, one seemingly plausible explanation of this "unexpected sub-peak" might be that, the high-MI expert heads do have the bahaviour of attending words having that dependency, while the sub-peaks correspond to the cases where the head thought the token pair was of that dependency, but doesn't. 't. The "errors" that expert heads made instead become the evidence for their behaviors.
A.2 MECHANISM-BASE CASE STUDIES
Another conclusion is derived via a mechanistic interpretability perspective. We ask the following questions: If an expert attention head filtered by MI expected be highly responsible for certain dependency, in real forward passes, will that attention head give human-understandable and interesting effects model behavior?
Table 3: Intrinsic evaluation (label-averaged Spearman-R) between each pair of head-selection methods. Higher value means higher consistency with other methods. Values are averaged for each method in the last row.
Probeless
IoU
Elasticnet
IPBP
Probeless loU ElasticNet
1
0.198
0.276
0.327
0.198
0.224
0.438
0.276
0.224
0.427
IPBP
0.327
0.438
0.427
1
Column Average
0.267
0.286
0.309
0.398
(w.o. diagonal)
15

---

Published as a conference paper at ICLR 2026
Eat vertrct mi
Figure 1: Estimated joint densities (f(1, Ab, f(-1, Ab,a)) of heads with top-5/last-5 MI, when I (nsubj, pobi]. The upper half for dependency pobj, while the bottom half is for dependency nsubj. For each dependency, the first ten plots are joint densities of attention heads with top-5 MI, with the first five plots for fl,, Ab. and last five plots for f( A.. Correspondingly. the last ten plots are joint densities of attention heads with last-5 MI.
To investigate this, we proposed "head ablation" technique: for a specific expert attention head, during the forward pass, we the value vectors of MHSA to zero. Since attention heads operate the residual stream by taking the weighted sum of value vectors, zeroing out the value states equal totally removing one attention head. This is also equal cutting off the "OV-cireuit" (proposed by (Elhage et al.) 2021)). ). It also share some principles with the "activation patching" technique of many mechanism interpretability practices (Wang et 2022). The most observable model behavior (also well-used by previous mechanistic research) is the next-token logit, so we compare the logits after ablating the top-5 expert heads of a specific dependency with the original logits. Moreover, order to exclude possible effects of the ablation itself, we also performed a series of random ablations as the control group (we ablated several times, each time five random heads were ablated, and we took their average logits).
The noticeable result is that, for a specific dependency label I, its expert heads particularly add to the logits of words that require the model to understand l. One typical example is shown as follows: for the incomplete sentence "In order to protect the environment, eco-friendly industries were", ", for the tokens having top- 10 logits among the predicted next tokens, the original logits logits after ablating amod's t transpose (amod a right dependency) expert heads, and the average random ablation logits are shown in Table 4
16

---

Published as a conference paper at ICLR 2026
Table 4: (Unnormalized) next-token logits for sentence In order to protect the environment, eco- friendly industries were before/after top-5 head ablation and random head ablation. means the logits has decreased, and vice versa.
Avg. Logits
Next Token
Original Logits
Logits w/o Top-5 Expert Heads
After Random Ablation
established developed
5.75
5.69
5.881
5.72
5.69
5.751
introduced
5.00
4.94
4.97
encouraged
4.88
4.72
5.061
set started
4.81
4.72
4.971
4.47
4.47
4.44
promoted
4.38
4.28
4,471
required
4.13
4.03
4.221
born
4.03
4.134
3.97
formed
4.00
4.00
4.031
We might notice an interesting phenomenon: these top-10 words basically express positive actions on eco-friendly industries. After ablating several amod expert heads, most of the logits decreased, while logits didn't exhibit unified increase/decrease behavior after random ablations. The difference shows that the decrease of logits is dedicated to ablating amod expert heads. This can be explained intuitively: there weren't the adjective modifier sccoffriendly,, considering the context "to protect the environment", common-purpose "industries" (probably polluting) should be restricted or contained. Only the sco-friendly' industries should be "established" or "promoted". Therefore, the appearance of such words is highly dependent understanding that "eco-friendly" modifies "industries", i.e., the amod dependency.
A.3 UNBIASED DEPENDENCY SYNTAX DISCOVERY
Sometimes people may argue, "probing assumes syntactic structure (like the dependency syntax trees) in advance, it is doubtful whether specific syntactical relationships really aside in model's internal mechanism, the researchers are fabricating Texas sharpshooter fallacy.
We think a good solution to this doubt is to design an unbiased strategy: strategy that does not assume any concepts in advance and make its assumptions only based the model structure itself. We take the unbiased assumption of "maximum attention weight": for each token position, the token that an attention head gives most attention weight can be considered as a token forming a relationship that the attention head favors. We designed the "maximum attention label recall", that is. for each dependency relationship, we calculate the count that an attention head assign the maximum attention weight to the dependency head/dependant (depending on which token comes earlier, since we adopt the first conclusion Section S.6). divided by the total number of dependency token pairs of that relation over the dataset. Note that we use the term and concept "recall" since dependencies are sparse over the dataset and on tokens not having such dependency the attention head might achive something else just attend to a "no-op" position just like the first token. For each dependency relationship, we' 'Il find an attention head having most maximum attention label recall. We then analyse how well the results from this unbiased strategy align with the results of IPBP by calculating the MI rank of head having most recall among all heads for each dependency. Results shown follows in Table 5
17

---

Published as a conference paper at ICLR 2026
Table 5: Maximum attention-based unbiased attention head analysis results for dependency relation- ships having top-10 10 largest maximum attention label recalls.
Recall Rank
Label Name
Maximum Atin. Label Recall
MI Rank
#1
root csubjpass iobj expl possessive
100.00
#2
100.00
329
#3
95.24
#4
93.94
#5
92.63
#6
prt mwe
92.24
53
#7
91.07
10
#8
auxpass
86.12
#9
pobj det
85.55
#10
82.70
N/A
Average of all Labels
57.25
66
We can draw two exciting conclusions from the results. First, attention heads do capture de- pendency syntax even under simple unbiased assumptions. We can find that the top- 10 recalls are rather high, with the average maximum recall still more than half. Many of the top- 10-recall dependencies are frequent and common dependencies, meaning that there are for most of the attention pairs in real- world data, there'll be heads that covering by assigning maximum attention score. Second, maximum attention score aligns well with IPBP MI. For the top- 10-recall dependencies, the attention head that contribute the maximum label recall are mostly heads that are highly-ranked by MI (except for nsub jpass which is rare in the dataset). The average rank is also among the top 10% (open's A_ Imaaama \_7b 7b has 1024 attention heads in total). The unbiased probing results and comparisons towards methods reveal the Texas sharpshooter concern to a large extent.
B LLM ABILITIES COMPARISONS AND ANALYSIS
Since experiments conducted multiple models and languages, it is necessary to investigate the (possibly multilingual) performance differences of the models we' ve tested.
B. METHODS
common intuition is that there open benchmarks D for LLMs' generic abilities and each LLM's research paper also reported relevant performances. However, the UD dataset is syntactically (also lexically) complex compared with most common common-purpose LLM benchmarks. Therefore, is required that we test the LLMs' abilities on our dataset.
While treebanks are sentences, direct way of testing LLM capabilities on such datasets to use perplexity (PPL). However, we indicate that PPL can be used to compare language model abilities for the same model, but might not yield fair results when comparing across models: simple example is that for sentence X IIn for each position i [L...n]. if two models (- 01) and p2(-I...: .... 02) are both predicting x; as the most probable lable, but p2 gives more probabilities to other words (like performing label smoothing). PPL will give p1 (-1 ....... (0) better performance, even though p2(-1.... -.. ;22) exhibit better generalizability. costing no performance loss.
Therefore, instead of perplexity, propose to average token rank to compare across models: for a sentence Ti..... for each token z, given the LLM-predicted conditional probability distribution p(2.(21....11). which is vector P (p1, pP))) where [V] is the vocab size, we sort the probabilities and get the rank x; in it, and average across sentences and datasets. This solves the aforementioned fairness issue some extent (p) (-I... 0,) and pa (-)... :2) are, at least, assigned equal performance scores).
https://huggiin co/ http//aaccs/pppeee 1m 1ede-meababbbbree//pene Teaderboarót/
18

---

Published as a conference paper at ICLR 2026
Moreover, since different models have different vocabulary sizes, also calculate the average token rank proportions, that is, the average token rank of each model divided by its vocabulary size.
Table 6: Average token ranks (and average token rank proportions) on UD-2.9 EnglisshFFrenchSSpanihh train/dev set
Language
Model Name
Avg. Rank
Avg. Rank Prop.
English
Meta-Llama-3-888
330.48
0.26
English
Mistral-7B
175.10
0.54
English
ppen_llama_a __
282.89
0.88
English
vicuna-7b-yl..
1177.39
3.68
French
Meta-Llama-3-88
275.13
0.21
French
Mistral-7B
212.42
0.65
French
open_llama_ 7b
123.21
0.39
French
vicuna7bb-vl.5-1
957.52
2.99
Spanish
Meta-Llama-3-8B
273.54
0.21
Spanish
Mistral-7B
223.06
0.68
Spanish
[ien_lama_71e
119.66
0.37
Spanish
vicuna-7b-vl.5-hff
919.78
2.87
B.2 RESULTS
Results are shown in Table 6 We can see that except for vicuna-7b, other models have similar average token rankings and average token ranking proportions. That might be because vicuna-7b is a instruction fine-tuned model and its attention pattern might be corrupted when not given correct input templates. Among the other three models, open _lal_llaaa__ 7b seems to have the best average token rank performances, while Meta-Llamaa3--88 is better in terms of token rank proportions. Note that the actual token usage of LLM given real-world data is actually long-tail distribution (the most frequently-used tokens constitute the majority of token usage, and most of the tokens are rare ones). Therefore, take the average token rank better criterion of LLMs' "familiarity" of our dataset, and choose open laama_ 7b as the representative.
C MI VISUALIZATION
Our dataset has 45 dependency labels, and open_ Ilama_ 7b has 1024 attention heads, it is feasible to list up all the MI(-,-) values in a table. This means that it's important to choose good visualization technique, which not only shows as much information as possible. but also intuitive, easy for us to observe patterns and rules. Therefore, we choose to draw the stack plot of MI values of all heads, grouped by layers, with respect to all dependency labels. Specifically, for dependency I L. we gather all MIninary (I; As,a) values for each head (b, h) [1...b] x [1...b]. forming label MI vector MI, E Bbxb We then group MI, by layers and draw the stack plot, with the x-axis corresponding to model layer index (let's say bg), and the y-axis corresponding to the sum of all MI values En Misnnny (4; Abg,k)) ). To be noticed, considering the first conclusion gain in 4.6 if it right dependency P then we'll choose the Milimary values under transposed setting.
In summary, the MI stack plot can: 1. Show the overall trend of MI distribution between layers. 2. Show the MI distribution of heads within specific layer. 3. By unifying the y-axis, it can also show the MI distribution between different labels. 4. It encapsulates all possible (b, h, I) e
[1...b] .6) b) x L.
The MI stack plots are shown in Through analysing the visualized MI, we draw the following conclusions:
Label occurrence frequency, tree reconstruction accuracy and MI value are highly related. Dependencies that well-captured (like prep,det, www..ooo.nnnnj....o..
also have higher accuracy in
"The definition is the same as 4.6 if dependency relation has more ares where the head is before the dependant, then it is left dependency. On the other hands, right dependencies has more ares with heads after the dependants
19

---

Published as a conference paper at ICLR 2026
Figure MI stack plot with respect all labels and heads. The x-axis corresponds to layer index and the y-axis stacks all MI values for all heads within that layer. Different plots corresponds different dependencies. Depenency root has far more MI compared with other dependencies due to its relative simple pattern, divide it by 2 to avoid letting all other dependencies shrink vertically.
20

---

Published as a conference paper at ICLR 2026
reconstructed trees. They's re also more frequent in the dataset and real-world language, meaning that dependencies are possibly a emergent ability model learns from data. 2. Expert heads tend to obey "principle of locality": if head has high MI at a specific layer, it's plausible that the head's MI is also relatively high in adjacent layers. In the figure, there are many occurrences of this pattern: In layer head h holds large portion of MI, and for adjacent layers - 1, b b 2, 2, ..... head also has relatively larger portion of MI. This might indicate that attention Q/K/V or the residual stream not rotation invariant. Among those dependencies that exhibit relatively apparent layer-wise MI differences, lower(local) dependencies have more heads in lower layers, which further justifies the conclusion model layer correspond to tree layer in 4.6 typical example that dependency det, which is an extremely low-level dependency, has an obvious peak at the first few layers, indicating that the processing of such low-level dependencies might complete within the lower layers, and later layers might just happen to have similar pattern due to the information passed from the first layers either by residual shortcut or by normal forward pass.
D IMPLEMENTATION DETAILS
D.1 IMPLEMENTATION DETAILS OF IPBP
In this section, we'll briefly introduce the implementation details, like the hyperparameters and key algorithms we to implement IPBP.
As shown by the source code, we use the PyTorch framework to implement the whole IPBP pro- cess. We're not relying on off-the-shelf packages that have KDE functionalities like SciPy g and
Seikit-Learr" since their KDE implementations are CPU-based and thus too in-efficient under our experiment settings.
Specifically, take samples in Ab, whole long tensor a R As. We calculate the minimum and maximum values As, and build tensor of real numbers X = { 22, In, }- ensuring that 2 < min Ar, In >> max As, and Φ1 22 < In. These discrete values serve as the points to calculate densities. Next, we calculate the mutual differences between each point
|As.hal times
in X and each element in A, by repeating T for Ab. times, getting matrix Th
X
of
shape n2 AAb, and repeating as,,, for n. times, also getting matrix (ai,h. lai.,,...... T
a.., of shape
T
nr Ab. The absolute differences of the two matrices
X
.T)
a b, a
are the mutual differences, let's say D(X a6,1.) E RR..X(A,.u Then we caleulate the standard
deviation of Ab, Le,, As and take rule-of-thumb value
for the bandwidth
B, with all weight w;s equal to 1. We then calculate element-wise to get the kernel values, following the following equation:
D(X as.ha) 2
exp
(13)
B /2x As.ua
B
After getting the kernel values in shape Nhe x Ab, we calculate the row-wise mean to get the final kernel density values in shape Thz- Since all operations of this process are element-wise matrix operations, this is casily parallel-optimizable by PyTorch. As result, the computation timi for extracting all attention score sets (A....) and performing all kernel density estimations on the training set is within hour using single RTX 4090 GPU. Since attention score allocating and distribution estimations are only required to be done once, our method is extremely time-saving compared to most of the supervised probing methods.
https://scipy.org/
https://scikit-lee
21

---

Published as a conference paper at ICLR 2026
Table 7: Number of train/dev samples of different language partitions of our dataset
Language
Split
Number of Samples
English
train
39832
English
dev
1700
French train
14442
French dev
1476
Spanish train
14182
Spanish dev
1399
For inferring on estimated probabilities (like inferring on posteriors ((]]A),.) Section 3.5 we
take the estimated posteriors as set of nt discrete points, and an attention score in range [Ti, Vi+ i] will get its corresponding posterior value by interpolating between x; and T;+1- The interpolation- based inferring. together with other processes mentioned in Section 3.5 like head selection and score-weighted averaging, are all parallel-optimizer resulting in being able to run inference within minutes on all baseline settings on 4090 GPU.
One thing to note that the Eisner algorithm is originally designed for unlabeled tree extraction, while method gives probabilities for each label (P(x,, 1), YI E L (o)). For each token pair (I,, kz)), we calculate the maximum labeled overall probability maxtec P(z;, xz;() (not including the situation of I = Φ) as the arc probability inputs of the Eisner algorithm.
For calculating integrals, specifically, the MI values like Minaary. MI we use the trapezoid method estimate the integral value: as mentioned in the section before, the kernel densities are described by nh points, we take the nh points as the integral limits and for every interval between 2; and T4+1, we calculate the trapezoid areas and add them up get the integral values. During tree reconstruction, we empirically set the total number of heads, i.e., 2iecu(o) [Ha to given proportion of total number of heads (EEECUE))
|Ha] 2000 when the model has 1024 heads).
We use the English, French and Spanish partitions of the Universal Dependencies 2.9 (Zeman jet al. (022)) dataset. The train/dev sample numbers are shown in Table UD 2.9 is dependency treebank covering texts from multiple languages and various sources like literature, news articles, spoken languages, elc., with diverse morphological and grammatical features. That dataset is publicly available, using BYSA 4.0 license allowing free redistributions upon notifications. The Universal Dependencies (UD) dataset is designed to provide a standardized framework of grammatical identifications for NLP researchers, so we're following intended usage.
We applied IPBP on four publicly available LLMs: open 11ama 7b Geng & Liu 2023), --ta--lama--ae BB ABAAI@MMM Metal 2024), vicuna-7b(Chian et al.) (023)) and
Mistral 7B BJing et al. 2023). We obtained each model's checkpoint from the Hugging Face M
@aar__laaa______ and Mi stral 77B are distributed under the Apache 2.0 license" M
agreeing on their respective license agreement. Specifically,
Liu 2023
is released
under the Meta LLaMA Community License" which allows research use subject to attribution and other conditions. Vicuna-77 which was derived from LLaMA-2 (TTuvron al. (023). follows the LLaMA 2 Community License Agreemen D All experiments reported in this paper are conducted strictly within the scope of these licenses' permissions.
D.2 IMPLEMENTATION DETAILS OF BASELINES
For the V-Information MLPs, found that training on all datasets will result in network always predicting ó for all possible attention scores, due to the long-tail essential discussed Section 4.4 This will result in many infinite Y-Informnatonn values, since there will be many estimated probabilities (for label in L other than Φ) rather close to zero. Therefore, we apply a sample balancing technique,
https:/www.apache..rrzz
https://www.i http://www.1laa.ccommm wttp://1aaaa..oooooo http://11m333a/cceeee
https://ai. http:///aa.mmtta.oo...
22

---

Published as a conference paper at ICLR 2026
truncating A.,; make their numbers of samples the same as the total number of samples in other score sets A., A... A.c. What's more, we also did a search on several network sizes, and found that if MI.P(-) Wa W (-))), where W in shape achieves better fitting. This aligns with (Pimentel et al. 2020b) to some extent. We also use PyTorch to implement the baselines. Specifically, for ElasticNet that requires additional training, we use Adam W optimizer, le 5 for both A and X2, and 12 epochs. For the V-Inforrmatinn MLP, since we need ( + 1)individual networks for predicting the alternatives of binary MI, initialize bh x (14) 1) sets of matrices, each constituting the weights of specific network WT, W Wajyerss with W, having a dimension of and W,. layers concatenate all attention scores
and W2 in shape
constant learning rate of le 3, training for
having dimension of L 1. During training and inferring,
alls] Lb,h for any E {1...b) b) and h E (1 {1...h) b} into a tensor of shape bh, and use torch. bmm to map each element of that tensor to bl x (I7.) + probabilities (standing for the probabilities of each label conditioned on each attention head's attention score, estimated by the variational family). Using torch. bmm will avoid training bh x (IL) + 1) networks separately, which is disaster computation loads, and can exploit GPU's parallel processing abilities. We use leaky_ jeluu between hidden layers and use sigmoid to form the final probabilities. We use le-2 learning rate with exponential decay (0.8 at each epoch), together with an additional warmup epoch at the beginning. The hyperparameters differ for V-Information since otherwise the variational family network will be more poorly trained. We also trained for 12 epochs.
E OTHER STRUCTURAL ABLATIONS
Apart from the formulation of MI and dealing with different dependency directions, we've also done ablations regarding minor design choices.
The first one is for the bandwidth of KDE, Apart from the rule-of-thumb value (Bo
), we' ve also done experiments varying the bandwidth by 0.5 x Bg. 2 x Bo.
AAAAAA wo and Bg-
Table Ablation results on different bandwidths.
Bandwidth Multiplier
UAS
LAS
0.5x
49.2
30.8
1x
49.9
34.8
2x
49
29.8
5x
34.8
21
Results are shown in Table 8 We can see that the rule-of-thumi bandwidth might yield a (locally) optimal result. It can also act as great influence factor it is not properly set (such as 5 Bo).
Another ablation is for posterior pooling methods, we proposed to use logarithmic opinion pooling (Heskes) 1997). For comparison, also used simple arithmetic mean and weighted arithmetic mean. Both of them correspond to the notation in Equation 14, while the former one uses for Whb..h)) and the latter one uses MIninary (1: Ao.,)a
E Eeb Wabb. log f(L Abb
ArithMean, (z;
(14)
Σ Extma W (ba.))r))
23

---

Published as a conference paper at ICLR 2026
Table 9: Ablation results different posterior pooling methods.
Pooling Method
UAS
LAS
Logarithmic Pooling
49.9
34.8
Simple Arithmetic Mean
36.2
18.4
Weighted Arithmetic Mean
36.3
18.9
Results are shown in Table 9 We can observe an obvious performance drop when using arithmetic mean methods. We think the reason might be because MI, which characterizes shared bits, vary exponentially (for example, a 2-bit event has 4x numbers outcomes compared with a 1-bit one), and geometric means naturally captures this. On the other hand, when applying arithmetic means, heads with slightly lower MI (which actually shares much lower information) have similar weights. This may "dilute" and add noise to the pooled probability.
F THE USAGE OF LLMS
We used ChatGPT to aid our paper retrieval and literature reviews on fields that we're not familiar with.
24


Reviews:
[{"review_id": "pFe8x2OZ5z", "reviewer": "Reviewer_nQmi", "summary": "This paper takes a step to address how LLMs comprehend syntax by proposing a novel method to extract labeled dependency trees directly from attention scores without training any additional neural networks. They aim to solve two problems: Directed Tree Extraction and Mutual Information Estimation for LLMs in a parameter-free Bayesian framework. Experiments are conducted comparing against several baselines, including probing and V-information comparing on labeled attachment scores (LAS), and unlabeled attachment scores (UAS) , where they show superiority of their method.", "questions": "Refer to the Weakness", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I found the paper overall easy to read, and the proposed Bayesian approach is sound, and I think the baselines considered are sufficient.", "weaknesses": "I am not an expert in this specific area, so my comments focus on high-level suggestions that could help improve the paper overall.\n\n1. How does this work compare with Wu et al. (2020)’s *Perturbed Masking*, which is mentioned as a parameter-free approach? It would be valuable to include a direct comparison on the same datasets and metrics to better position the proposed method.\n\n2. Could the authors consider a simpler baseline by using averaged attention scores directly as edge weights for Eisner or Chu-Liu-Edmonds decoding? This would help clarify how much improvement comes from the proposed components.\n\n3. For ablations, it would be helpful to vary the bandwidth parameter for KDE across a range (for example, 0.5x, 1x, 2x, and 5x of the current setting) and report UAS and LAS results to evaluate sensitivity.\n\n4. Another useful ablation would be to compare logarithmic opinion pooling with simpler aggregation methods such as the arithmetic mean or the standard geometric mean to understand whether the more complex approach provides a meaningful benefit.\n\n5. The motivation and implications section could be made stronger. At present, the introduction seems to overstate the practical and scientific significance given the modest LAS scores of around 30 to 49\\%. It would help if the authors discussed more candidly when attention-based syntax extraction is sufficient, when hidden states are preferable, and what these results reveal about whether large language models actually encode syntactic structure.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ytZPFRCHie", "reviewer": "Reviewer_adce", "summary": "This paper proposes a parameter-free probing method for detecting how well attention maps in decoder-only transformers encode dependency tree information. This approach relies on estimating the mutual information between the attention scores of a pair of tokens and the dependency label, and no added learning of probe weights. This approach is evaluated for a number of models and compared against various baselines on UAS and LAS. Results show that this approach is revealing more syntactic structure than previous methods.", "questions": "-", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Parameter-free probes generally seem like a good idea and this approach generates MI estimates that can be useful for more fine-grained analysis.  \nThe paper is thoughtful about considering a number of different baselines in their experiments.", "weaknesses": "There is a mismatch between the attention scores being used for this analysis (bidirectional) and the actual scores being used by the model during generation (causal). It seems a bit strange to make claims about the syntax structure encoded in the model that is never actually used for generation. This point makes the conclusions and implications of the findings a bit hard to reason about.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xrnWkDLlL7", "reviewer": "Reviewer_Y12b", "summary": "The paper tackles the problem of probing labelled dependency trees from attention score when a transformers-based LM processes a sentence. Different from several work using deep neural networks (which, according to the paper, is \"explaining by unexplainability\", the paper proposes to use mutual information between dependency labels and attention scores. Tree reconstruction is done by (i) using the MI to estimate probability that a pair of tokens are linked by a labelled dependency arc, and (ii) using Eisner algorithm. \n\nThe paper demonstrates that proposed method can reconstruct (un)labelled dependency tries with higher UAS/LAS scores than the baselines. Interesting, the paper shows that (i) LM decoders capture left/right dependency adaptively, and (ii) different layers capture different types of dependency arcs (local vs global).", "questions": "* How do left/right-branching baselines perform in terms of UAS?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper is very interesting, especially without any learning. The idea of using mutual information is novel, though using Eisner algorithm in this case is pretty straightforward. \n\n* The real strength of the method lies in its simplicity and explainability. \n\n* Findings in Section 4.5. are particularly interesting, show-casing the strength of the method over learning-based ones.", "weaknesses": "* The use of $MI(l,A)$ to compute $P(x_i,x_j;l)$ doesn't seem information-theoretically motivated. How close is it to estimate $P(x_i,x_j;l|A)$?\n\n* It's unclear whether the tree reconstructed by the Eisner algorithm faithfully expresses the syntax structure from the LM. For instance, thanks to the constraints (and biases) of the algorithm, eventhough $P(x_i,x_j;l)$ > $P(x_u,x_v;l)$, the former pair may not form a dependency arc but the latter may. In this counterexample, it's hard to say that the reconstructed tree faithfully expresses the syntax structure of the LLMs.\n\n* The head-selection experiment doesn't capture intrinsic head-selection performance because it is evaluated via the down-stream task (tree reconstruction) using Eisner algorithm. Is there a way to evaluate head-selection performance in an intrinsic manner? \n\n*  The tree-reconstruction evaluation comparison doesn't include results in the literature.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7FRLWlzRiG", "reviewer": "Reviewer_7oCt", "summary": "The paper aims to reconstruct dependency trees from the attention scores of LLMs using the IPBP method. Precisely, IPBP consists on estimating the Mutual Information between dependencies in a labeled dataset and attention scores. With mutual information, tree reconstruction is performed and evaluated with UAS and LAS metrics. Overall the method proposes a parameter-free way to predict syntax from LLMs. Results show that IPBP is has superior performance to baselines and it works for 7 different LLMs.", "questions": "How are the multi-token words treated? Are attention values averaged?\nHow do the maximum/mean layer-wise MI values look like? Within and across dependency types?\nWhat is the baseline MI when using a randomly initialised LLM?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* The paper spots a limitation of probing techniques and the given solution is simple and elegant.\n* The method is well described and mathematically rigorous, both for MI estimation and tree reconstruction.\n* The results gather several LLMs and well-thought baselines.", "weaknesses": "There are several relevant works which are not cited\n\n* Labeled Dependency probing: \n     - https://arxiv.org/abs/2412.05571\n* Probing attention values: Attention scores probing: \n     - https://www.pnas.org/doi/10.1073/pnas.1907367117\n     - https://arxiv.org/abs/1906.02715\n* Probing ambiguous sentences with causal analysis:\n     - https://arxiv.org/abs/2211.09748\n* Syntactic probes are biased towards linearity\n     - https://arxiv.org/abs/2508.03211\n\nThe results presented fall short at the current version of the manuscript, only different LLMs and baselines are evaluated.\nThere are no insights about LLM depth and tree depth and how syntactic parsing happens inside the model.\n\nAblations are based on a single and arbitrary sentence “In order to protect the environment, ecofriendly industries were”, this analysis should be done per dependency type and using specific tokens for the logits and not the average. For example, long-range dependencies like subject-verb agreement in PP sentences are good candidates to assess syntactic capability.\n\nAlso, the LAS values are far from SOTA, which might be justified by the fact that it is parameter-free. I think the paper could benefit from a syntactic dataset and thus a more controlled setting.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)