# Streamlined Variational Inference for Higher Level Group-Specific Curve Models

## TL;DR

This paper develops a systematic approach to streamline variational inference for complex multi-level group-specific curve models, enabling analysis of hierarchical data such as ultrasound technology measurements.

## Contribution

It introduces a structured method for variational inference in multi-level models, extending previous sparse matrix techniques to higher levels.

## Key findings

- Effective inference for two- and three-level models demonstrated.
- Framework facilitates analysis of hierarchical data structures.
- Pattern generalizes to models with more than three levels.

## Abstract

A two-level group-specific curve model is such that the mean response of each member of a group is a separate smooth function of a predictor of interest. The three-level extension is such that one grouping variable is nested within another one, and higher level extensions are analogous. Streamlined variational inference for higher level group-specific curve models is a challenging problem. We confront it by systematically working through two-level and then three-level cases and making use of the higher level sparse matrix infrastructure laid down in Nolan and Wand (2018). A motivation is analysis of data from ultrasound technology for which three-level group-specific curve models are appropriate. Whilst extension to the number of levels exceeding three is not covered explicitly, the pattern established by our systematic approach sheds light on what is required for even higher level group-specific curve models.