Bayesianism has some faults some of which involve the problem of old evidence and the issue of new theories. Are these two problems linked to Quine’s underdetermination? Or are they contrasting it? What is the relationship between these issues and Quine’s underdetermination?
Here's a brief definition of Quine's undetermination that I found online: "Underdetermination is a thesis explaining that for any scientifically based theory there will always be at least one rival theory that is also supported by the evidence given, and that that theory can also be logically maintained in the face of any new evidence."
I think my favorite current example of this would be cosmological theories about Dark Matter. DM is supposed to explain certain gravitational anomalies both on a galactic scale and on much larger scales. Based on traditional (including relativistic) gravitational theories, galaxies should fly apart because of insufficient internal gravitational attraction. And, on a much larger scale, the motions of galactic clusters do not seem to follow strict gravitational "rules" either. So cosmologists have posited DM as the source of the extra mass to explain the extra gravitational force required. However, DM has never been directly detected (despite numerous efforts). This led Israeli physicist Mordehai Milgrom to question its existence and posit a different explanation for these gravitational anomalies called MOND (Modified Newtonian Dynamics). Interestingly, MOND actually explains galactic-level gravitational coherence more accurately than DM but doesn't work as well for the larger scale anomalies where DM proves more accurate. Maybe this doesn't quite fit Quine's "equally well" criterion but that all depends on what you mean by the term. Another example might be neo-Darwinian evolutionary theory vs. current epigenetic research and theories (a form of neo-Lamarkianism). In both cases, implications of the actual evidence do indeed seem "underdetermined" in Quine's sense and in both cases, one has to wonder whether the evidence (given its nature) ever will be sufficient to be considered determinative.