Are Language Models Any Good at Density Modeling?

Large Language Models (LLMs) surprised the world with their ability to mimic humans in writing and are starting to be used as simulations of human writers for various kinds of linguistic analysis. However, these analyses rest on the belief that LLMs are good density models, that accurately capture the underlying probability distribution of the language. In this paper, we question this basic assumption and try to evaluate language models on their density modelling capabilities. Since a ground truth does not exist for the probability distribution of any natural language, we come up with a synthetic language made up of decimal numbers written in words in English. We train language models from scratch on various probability distributions over this synthetic language and compare the distributions learned by the models with the original ones. Experiments show that language models can learn underlying probability distributions across a wide range of cases, but they fail when those distributions depend on deep semantic properties of numbers that cannot be inferred from syntactic patterns. Additionally, we observed a strong bias in the models toward numbers that frequently occur as substrings within other numbers. In natural language models, this bias can impact downstream tasks that rely on model-generated probabilities.