Low-rank adaptation is effective partly because downstream updates lie in a low-dimensional subspace, but the latent rank coordinates of LoRA are not identifiable: any invertible reparameterization of the adapter factors leaves the weight update unchanged. We show that variational inference with a diagonal rank-wise posterior turns this non-identifiability into a useful inductive bias. By breaking LoRA's rotational gauge symmetry, the variational objective selects a preferred basis in rank space, enabling automatic relevance determination over rank directions. This yields Low-Rank Variational Dropout (LRVD), a Bayesian framework that performs inference directly in the low-rank adaptation space rather than the ambient weight space. As an instantiation, BayesLoRA jointly learns effective adapter rank and predictive uncertainty with only $\mathcal{O}(r)$ additional parameters. Empirically, BayesLoRA induces stable rank structure aligned with the dominant singular directions of learned updates, yields compact predictive calibration and matches or exceeds strong low-rank sparsification baselines at comparable training cost.