Let $X$ be a metric space and let $f:X\to R$ be a real-valued Lipschitz function. Given a sample $S$ of $X$ and the values $f$ at the points of $S$, we want to know something about the persistent homology of the sublevel set filtration of $f$. Under reasonable assumptions about the space $X$ and the sample $S$, it is possible to give strong guarantees about the bottleneck distance between the ground truth barcode of $f$ and a barcode computed from the sample. This talk will explore the question of what one can say in the absence of any guarantees about the quality (i.e. density) of the sample. I will present a theory of sub-barcodes that provides an alternative to bottleneck distance for theoretical guarantees on barcodes. Then I will show that it is possible to compute a barcode from Lipschitz extensions of $f(S)$ that is guaranteed to be a sub-barcode of the barcode of every Lipschitz function that agrees with $f$ on the points of $S$.