As a clarification and driving tips:
1. Most roads in Yellowstone are closed to private vehicles in winter. However, you can still access these roads with a snowmobile or snow coach (with guide, group travel etc).
2. Only the road between Gardiner and Silvergate/Cooke City remains open for private vehicles in winter. The road "dead-ends" at Cooke City for wheeled vehicles (snow mobiles only beyond this point).
3. Do get good snow tires; the road is typically snow packed and icy, and there are many slopes and curves. We use studded tires in winter for the van. Sliding would be very bad.
4. Position your vehicle based on where you think the yellow line is. Do not position your vehicle based on where you think the edge of the pavement is. The snow plow tends to plow beyond the edge of the pavement... put your wheels there and you are off the road.
5. Drive slow, and downshift on descends; brake using your gears, minimize the use of your brakes.
6. Take the above very seriously... I think there are perhaps 5-10 cars that go off the road on an average day...
Finally: There are several ski and snowshoe trails. Some are groomed, others are not. These trails are awesome! Stretch your legs, get some fresh air, and you have an opportunity to find real solitude.