

\subsection*{Related Works}









\paragraph{Uncertainty Quantification in Supervised Learning.} Existing work on UQ mostly focused on supervised learning settings.
For example, Bayesian inference quantifies uncertainty by placing a prior distribution over model parameters, updating this prior distribution with observed data to obtain a posterior distribution, and examining the inconsistency of predictions derived from the posterior distribution \citep{neal1996bayesian, mackay1992practical,kendall2017uncertainties,depeweg2018decomposition}. Since the posterior distribution may not have an analytical form, many approximating approaches have been introduced, including Monte Carlo dropout \citep{gal2016dropout}, deep ensembles \citep{osband2018randomized, lakshminarayanan2017simple, wen2020batchensemble}, and Laplace approximation \citep{daxberger2021laplace,sharma2021sketching}. In this paper, we focus on quantifying the uncertainty of representations and prove that standard supervised-learning frameworks cannot be directly applied to investigate representation uncertainty (see Section~\ref{sec:consistency} for more details). 







\paragraph{Novelty Detection and Representation Reliability.}  Self-supervised learning is increasingly used for novelty/OOD detection. These approaches train self-supervised models and then compute an OOD score for a new test point based on its distance from the training data in the representation space \citep{lee2018simple, van2020uncertainty, tack2020csi, mirzae2022fake}.
However, OOD detection and our representation reliability are different concepts. 
\rev{The former identifies whether a test point belongs to the same distribution as the (pre-)training data, while the latter evaluates the possibility that a test point can receive accurate predictions when the self-supervised learning model is adapted to various downstream tasks (see Section~\ref{sec:defin} for more details).}

To compare with this line of work, we conduct comprehensive numerical experiments (Section~\ref{sec::exp}). 
The results suggest that our approach more robustly captures the representation reliability compared with state-of-the-art OOD detection measures and the empirical metrics proposed in \citet{ardeshir2022uncertainty}.
Finally, our representation reliability extends the notion of probe as in \citet{haochen2021provable} to multiple downstream tasks. We introduce an algorithm for estimating the representation reliability without prior knowledge of the specific downstream tasks.

\paragraph{Uncertainty-Aware Representation Learning.} 
There is a growing body of research aimed at training robust self-supervised models that map input points to a distribution in the representation space, rather than to a single point \citep{vilnis2014word,neelakantan2015efficient,karaletsos2015bayesian,bojchevski2017deep,oh2018modeling,chen2020simple,wu2020simple,zhang2021temperature,almecija2022uncertaintyaware}. 
They rely on special neural network architectures and/or introduce alternative training schemes. 
For example, the approach by \citet{zhang2021temperature} requires an additional output (i.e., a temperature parameter) and the approach by \citet{oh2018modeling} requires the network to output means and variances of a mixture of Gaussian distributions.
In contrast, we avoid making any assumptions about the training process of the embedding functions, while only needing black-box access to them. 
Furthermore, we provide a theoretical analysis of our method and explore the impact of the representation reliability on the performance of downstream tasks.
We provide a more in-depth discussion about related works in Appendix~\ref{app:summary}.

















































      




