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This paper reframes self-attention as a Bayesian posterior and argues that fixed positional encodings are a poor fit for time series whose dependency structure shifts across input windows. Its proposed Bayesian Time-aware Attention uses a mixture-of-experts router to generate input-conditioned positional biases. The reported gains are consistent across Transformer backbones with modest overhead, but they come from standard LTSF benchmarks, so readers should check whether they hold on their own non-stationary data.
Beyond Static Positional Encoding: Bayesian Time-aware Attention for Non-Stationary Long-term Time Series Forecasting
arXiv:2610.19603v1 Announce Type: new Abstract: Transformers have demonstrated strong performance in long-term time series forecasting (LTSF), yet modeling non-stationary temporal dependencies remains challenging. In this paper, we interpret self-attention as a Bayesian posterior over attended positions, combining a content-based prior induced by scaled dot-product similarity with a temporal-evidence likelihood induced by positional encoding. This formulation provides a probabilistic interpretation of how positional encoding shapes the temporal dependencies captured by self-attention. Building on this perspective, we derive a population-level evidence lower bound (ELBO) that characterizes the role of positional encoding in forecasting weakly stationary time series. Our analysis further highlights a limitation of static positional encoding: when temporal dependence structures vary across input windows, a fixed temporal bias must accommodate heterogeneous dynamics and can therefore be suboptimal for non-stationary forecasting. Motivated by this analysis, we propose Bayesian Time-aware Attention (BTA), which employs a mixture-of-experts (MoE) router to adaptively combine temporal kernels and generate input-conditioned, head-specific positional biases. Extensive experiments on standard LTSF benchmarks demonstrate that BTA consistently improves forecasting performance across multiple Transformer backbones and achieves competitive overall performance with modest computational overhead and stable training.
Abstract:Transformers have demonstrated strong performance in long-term time series forecasting (LTSF), yet modeling non-stationary temporal dependencies remains challenging. In this paper, we interpret self-attention as a Bayesian posterior over attended positions, combining a content-based prior induced by scaled dot-product similarity with a temporal-evidence likelihood induced by positional encoding. This formulation provides a probabilistic interpretation of how positional encoding shapes the temporal dependencies captured by self-attention. Building on this perspective, we derive a population-level evidence lower bound (ELBO) that characterizes the role of positional encoding in forecasting weakly stationary time series. Our analysis further highlights a limitation of static positional encoding: when temporal dependence structures vary across input windows, a fixed temporal bias must accommodate heterogeneous dynamics and can therefore be suboptimal for non-stationary forecasting. Motivated by this analysis, we propose Bayesian Time-aware Attention (BTA), which employs a mixture-of-experts (MoE) router to adaptively combine temporal kernels and generate input-conditioned, head-specific positional biases. Extensive experiments on standard LTSF benchmarks demonstrate that BTA consistently improves forecasting performance across multiple Transformer backbones and achieves competitive overall performance with modest computational overhead and stable training.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2610.19603 [cs.LG]
(or arXiv:2610.19603v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.19603 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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