Pearl's causal hierarchy shows that observational, interventional, and counterfactual queries are qualitatively distinct. We ask a quantitative version of this question: how many additional bits are needed to specify higher-rung causal answers once lower-rung answers are known? We formalize this via query-class description length, the Kolmogorov complexity of the answer oracle induced by an SCM for a class of queries. Our main construction gives binary acyclic SCMs whose observational distribution has constant description length, while the single-variable interventional answer oracle has description length $Θ(n^2)$. A degree-sensitive upper bound shows that finite-gate-schema SCMs of indegree $d$ have observational-interventional gap at most $O(nd \log(en/d) + n \log n)$, making the quadratic construction order-optimal in the dense regime and a rooted-tree construction order-optimal for bounded indegree. The quadratic separation persists under $\varepsilon$-accurate total-variation descriptions for every fixed $\varepsilon < 1/4$. At the next rung, the full hard-do interventional oracle can still leave a $Θ(n)$ counterfactual description gap. A general ambiguity-to-bits theorem and Shannon analogue show that these gaps equal the logarithm of residual higher-rung ambiguity up to lower-order terms.

Neural Information Processing Systems http://nips.cc/

In this paper, we introduce the forecast reconciliation packages FoReco and FoRecoML for R (RCore Team 2026). Forecast reconciliation adjusts forecasts for linearly constrained multiple time series (such as hierarchical or grouped series, or series observed at different temporal frequencies) so that they are coherent with respect to the underlying constraints, improving both accuracy and consistency for informed decision making. The contributions of the packages are threefold. First, FoReco and FoRecoML are the first to offer functionality for forecast reconciliation methods across cross-sectional, temporal and cross-temporal frameworks. Second, the packages provide a comprehensive set of forecast reconciliation approaches, including classical (e.g., top-down, bottom-up and middle-out) and regression based reconciliation methods - in FoReco - as well as non-linear reconciliation methods using machine learning - in FoRecoML. A third key contribution is their unified design, which enables easy-to-use forecast reconciliation functions built on the same philosophy, regardless of the reconciliation framework or method.

In training every agent we use a distributed framework for simulation and training. For simulation, we run 6400 Hanabi environments in parallel and the trajectories are batched together for efficient GPU computation. This is done efficiently as every thread can hold many environments in which many agents interact. Every agent chooses actions based on neural network calls, which are more intensive and done by GPUs. By doing these calls asynchronously it allows a thread to support multiple environments while waiting for prior agents' actions to be computed.

We organize our supplementary document as follows: 1. Results on additional dataset 2. Results for limited data settings on YFCC26k and GWS15k datasets 3. Additional Ablations (a) Gallery Size (b) Queue Length (c) ση for Batch GPS noise (d) ση for Queue GPS noise (e) σ for Random Fourier Features (f) Number of hierarchies (M) 4. Different selection choices for GPSGallery Construction (a) Evenly Spaced GPSCoordinates (b) Test Set GPSCoordinates 5. Analysis of Runtime and Memory Footprint 6. Motivations for using Pretrained CLIP as Image encoder Backbone 7. Qualitative Demonstration (a) Hierarchical learning in our location encoder L () (b) GeoCLIP with Image Query (c) Distribution of correct predictions of GeoCLIP on different datasets (d) GeoCLIP with Text Query 8. Discussion on Ethical Issues and Possible Mitigation In section 4.1 of the main paper, we demonstrated the performance of our GeoCLIP method on Im2GPS3k [2] and GWS15k [1] datasets and compared them with the state-of-the-art methods. Here, we perform experiments on another dataset YFCC26k [6]. The results are provided in Table 1. This result highlights that GeoCLIP performs well across datasets, being useful across different data distributions. GeoCLIP achieves decent performance across datasets even when the training data is significantly reduced. 2 We show the efficacy of GeoCLIP on limited training samples of Im2GPS3k in section 4.2 of the main paper. Now, we further investigate the performance of GeoCLIP for limited data settings on other datasets (YFCC26k and GWS15k).

This paper presents a topological learning-theoretic perspective on causal inference by introducing a series of topologies defined on general spaces of structural causal models (SCMs). As an illustration of the framework we prove a topological causal hierarchy theorem, showing that substantive assumption-free causal inference is possible only in a meager set of SCMs. Thanks to a known correspondence between open sets in the weak topology and statistically verifiable hypotheses, our results show that inductive assumptions sufficient to license valid causal inferences are statistically unverifiable in principle. Similar to no-free-lunch theorems for statistical inference, the present results clarify the inevitability of substantial assumptions for causal inference. An additional benefit of our topological approach is that it easily accommodates SCMs with infinitely many variables. We finally suggest that the framework may be helpful for the positive project of exploring and assessing alternative causal-inductive assumptions.