A recent study examined mitochondrial DNA (mtDNA) taken from the bones and teeth of the extinct Irish bear and were compared to all other bears. What they found was really interesting, 10 of these Irish bears shared the same maternal lineage that gave rise to modern day polar bears.
mtDNA is only passed on down the maternal line, you only get it from your mother. It also lasts longer in fossilized bones, such as those from these Irish bears that may have died up to 43 000 years ago. On the other hand Nuclear DNA, which you receive from both parents, doesn't share the same shelf life.
The study hypothesizes the reason polar bears have Irish heritage after a period of hybridisation with Irish bears during the last glacial maximum (Ice Age) at ~22 000 years ago. There would've been much greater ice sheet coverage of the arctic that would have allowed polar bears to move much further south, as well as into north east Europe. There they would have encountered Irish bears, a close relative or 'sister species', and interbred with them, thus receiving their mtDNA. By chance through genetic drift, the Irish bear's mtDNA went on to replace all other mtDNA (assumably from American brown bears) in the polar bear.
My only criticism of the study is that it's a bit of a big call if firstly, the study only compared a single marker, one section of DNA, and it was only 100-170bp long. Secondly the fact that the marker used was so short. Using another one or two markers of greater lengths would definitely solidify these claims, though I can imagine it's easier said than done when using ancient DNA from bones and teeth.
Journal reference: Current Biology, DOI: 10.1016/j.cub.2011.05.058