A good friend of mine recently got the fatbike bug, largely so that he could ride on snow, and wanted to better understand appropriate tire pressures for that surface.
He's ridden bikes for decades and skied for even longer, but the correct range of pressures for 5" tires on soft snow is not as auto-intuitive as you might think.
I spent a few minutes writing a detailed response, and after sending it to him it occurred that many here might benefit from it as well, if only as a resource for their new-to-fat friends.
Without further ado...
It takes some time to wrap your head around appropriate PSI for snow--it's probably gonna be a lot less than you think. The standard credo for tire pressure when snow riding is 'when in doubt, let air out'.
Best way to be sure is to take a little low-pressure (0-15psi) gauge with you for the first month or so, and check pressure frequently with both the gauge and your hand to get your hand calibrated. The idea being to learn what works by feel, so that you can ditch the gauge sooner than later.
This is a rough guideline. The absolute number is irrelevant, finding a pressure that works, and then being able to both recognize the conditions and duplicate the appropriate pressure is what matters.
10psi and up = pavement or glare-ice pressure.
6-8psi = *very* hardpacked snow.
4-5psi = softer or less consistently packed snow.
2-3psi = deeper snow, when more flotation is needed. If you need this kind of
pressure, you'd probably be having more fun with skis on! But
sometimes you start a ride on hardpack and have an ambitious
objective, then it snows or blows and you have to dump air to keep
0-2psi = what I most often ride at, due to lots of light, dry snow and very little traffic.
As temperatures and conditions change the appropriate pressure for the surface can fluctuate pretty dramatically. 1psi makes a big difference. My way of staying safe (avoiding flats or rim damage) is to lean all my body weight on the saddle, while looking down at the rear tire. Any wrinkles in the sidewall? Add psi until the wrinkles go away. That's your baseline for hardpack. The flipside of that process is that for the softest, least-packed snow (the kind where you should have chosen to ride lifts with skis on that day!) you can go as low as four or five wrinkles in the sidewall as long as you're being delicate. More than five wrinkles and you're generally just adding resistance without increasing float or traction. That said, conditions in my neck of the woods often require 5+ wrinkles just to keep pedaling, and since pedaling beats walking...
One last bit of editorial: Not many people understand how far you can go in a short time on one of these steeds when conditions are good, but how absolutely hosed you can be if it's nuking or blowing or both on the return. Like 7-8mph when it's good, and hours per mile when it's bad. I don't take a sleeping bag with me on every ride, but I *always* have a puffy, firestarter and lighter, and some food on winter rides. Seems like about every other year I get antsy to do something epic, and conditions change halfway through the ride, leaving me out for the night and into the next day (or til a sledneck comes along and packs the track back in).