Question about driving through Yellowstone in Jan

Me and a few friends are flying into Jackson Hole in Jan, driving to Big Sky to snowboard for a couple days then driving back to Jackson to snowboard for a couple more.

My question is: we’re hoping to drive through and see Yellowstone on the way to Big Sky but I’ve heard some of the roads through Yellowstone can close during the winter due to weather and snow/ice conditions. Does anyone have any insight into whether this will be feasible or not? Also, what would be the best route for this if it is feasible?

Thanks!