In-context learning (ICL) empowers large language models (LLMs) to tackle complex tasks without explicit training. But why does simply providing examples work, and how can we predict its effectiveness? New research explores the surprising connection between ICL and Bayesian learning, revealing a family of "Bayesian scaling laws" that shed light on this mysterious process. These laws not only predict ICL accuracy better than existing methods, but they also offer valuable insights into how LLMs understand and represent knowledge. Imagine the brain encountering a new concept. It doesn't require retraining on all prior knowledge; rather, it integrates the new information into its existing understanding, much like Bayesian inference. This research suggests LLMs operate similarly during ICL. By treating ICL as a Bayesian process, researchers derived equations that link the number of examples to the model's prediction accuracy. Experiments with different-sized GPT-2 models demonstrated that these Bayesian scaling laws accurately predict ICL behavior, often outperforming existing scaling laws. Interestingly, the laws have interpretable parameters that represent the model’s prior beliefs about tasks, how efficiently it learns from examples, and the probabilities of different outcomes for each task. This interpretability is crucial for understanding how LLMs learn and represent knowledge. Researchers further explored how post-training techniques like supervised fine-tuning (SFT) and direct preference optimization (DPO) affect ICL. Results on synthetic data suggest SFT primarily alters the model’s prior beliefs, while DPO impacts its deeper knowledge about tasks. Intriguingly, larger models seem more resistant to these changes, suggesting that post-training is less effective at larger scales. Finally, the researchers tested their Bayesian scaling laws on real-world LLMs, using both capability and safety benchmarks. The results confirmed that the Bayesian approach remains competitive, accurately modeling ICL behavior across various tasks. Comparing base and instruction-tuned LLMs revealed that instruction tuning primarily affects the prior probabilities of safe/unsafe behavior, but doesn't prevent jailbreaking in the long run. These findings have significant implications for real-world LLM applications. Predicting ICL effectiveness could streamline development by guiding decisions about the number of examples and the need for fine-tuning. Moreover, the interpretability of Bayesian scaling laws offers a glimpse into the black box of LLMs, aiding in safety and alignment research. While the research doesn't definitively prove LLMs are Bayesian reasoners, it presents compelling evidence that they operate similarly. This Bayesian perspective opens up exciting avenues for understanding and improving ICL, paving the way for more powerful and reliable LLMs.