Marking/Peeing in the house

The first thing I'd mention -- I have a friend who does bichon rescue and she's often mentioned to me that bichons can be remarkably difficult to completely housetrain -- you may honestly just need to go back to Housetraining 101 and RE-train him -- and allow him **zero** opportunity to screw up.  Like Jackie says -- a crate when you can't watch him, or during high risk times, leashing him to you so you know when he's looking FOR a place. 

dgerber4

The problem is that he pees on the floor throughout the house to mark his territory

Here I have to ask -- is he neutered?  He shouldn't be "marking territory" in his own home.  Neutering isn't a complete solution, but it will decrease the testosterone (which honestly helps prolong his life unless he's a show dog with awards out the ying yang being used actively for stud). 

It usually has way more to do with not being fully and reliably house-trained than just "marking">

The other thing you should check is have the vet make sure he doesn't have a UTI/crystals, etc.  But otherwise it looks like you really need to go back to beginning housetraining.  You may think a dog "should know" and they really don't.  They may be housetrained when it's convenient but if he's successfully learned he can go "in there" when no one pays attention otherwise, or isn't awake?  Then as far as he's concerned it's "ok".  And when you "find" it later he's confused. 

bottom line you have to prevent him from screwing up -- keep him contained or leashed to you so you can deal with it immediately. 

training, training, training -- sometimes it seems bizarre that a dog you've had THAT long is regressing in training, but with some breeds it happens easily and with some dogs, it also just plain happens easily.