Item response theory models are commonly used in political science to estimate latent traits. We propose a new methodology that considerably generalizes existing research in two respects: First, we define the idea of a "structured" item response theory model as a generalization of the traditional "hierarchical" item response theory model where we allow an individual's latent trait to be a function of (multiple) hierarchical terms as well as modelling the item parameters (e.g., the discrimination parameter) hierarchically. In our empirical example using data from the European Parliament, this allows us to model the MEP's ideal point as depending on their party grouping and country as well as modelling the bill parameters as a function of the topic of the bill, the type of question under consideration, etc. This allows a more nuanced understanding of how different factors shape the ideology of MEPs.

Methodologically, we provide a new framework for fast estimation on large datasets using variational inference. We derive a closed form algorithm for estimating these ideal points even when they are multidimensional as well as some novel parameter-expansion techniques to accelerate convergence. All of the methods are implemented in open-source statistical software.