Occam's Razor is Only as Sharp as Your ELBO

The marginal likelihood, also known as the evidence, is regarded as a mathematical embodiment of Occam's razor, enabling hyperparameter selection that avoids overfitting. The evidence lower bound (ELBO) objective from variational inference has also been used for similar purposes. In this paper, we define a signal-to-noise ratio (SNR) for Bayesian linear regression in terms of key hyperparameters controlling signal and noise variance. We then study how ELBO-based hyperparameter learning can under or overestimate SNR depending on the assumed covariance matrix structure in a Gaussian approximate posterior. We prove that assuming a diagonal or low-rank plus diagonal covariance matrix underestimates the SNR, while a low-rank covariance matrix overestimates the SNR. Underestimating the SNR leads to higher than optimal mean squared error on the training set (underfitting), while overestimating SNR leads to lower error (overfitting). Bayesian practitioners hoping to scale to large models should be cautious about how covariance assumptions needed for tractability may impact model selection.