I learned a long time ago that leaving anything out on the ice is a bad idea. It's not the dudes out fishing you worry about, it's the passers-by on sleds, trucks or ATV's.
My buddies and I had a shack out on Pewaukee lake in SE Wis, we got hammered one night and got lazy with our crap. Sure enough, broken into when we came back the next day. We lost 2 Vexilars, a few jig rods and a bunch of Beaver Dam flags. Saving grace? Dude left tracks back to the public launch, a public launch with a security camera on the Pay Station roof. We narrowed down the time and the local cops went through the video and found dude's truck plate #. He and two buddies weren't fishermen, they were just cruisers. Cruisers who faced felony burglary charges.