BAT is a method that uses systematic desensitization (DS). DS involves gradually getting the dog/person/whatever used to the trigger by repeated subthreshold exposure and relaxation. DS is a form of classical conditioning (extinction in the classical sense, not the behavioral sense, i.e., subthreshold form of trigger ->; no scary event, with relaxation). DS can be used with the open bar/closed bar version of classical counterconditioning (CC), but it doesn't necessarily need to be paired with CC. For example, DS is an essential part of BAT training. In fact, one could even go so far as to say that BAT is a way to do DS with animals without using classical counterconditioning.
In order for the dog to be below threshold without the food present, DS in BAT is usually done further from the trigger than DS with classical CC or with other operant procedures using treats, like LAT (Look at That).
BAT is also Differential Reinforcement of Other behavior (DRO) using rewards determined by a functional analysis. We increase or decrease exposure to the trigger (walk toward or away), depending on what reinforced the original problem behavior. For example, if a fear aggressive dog offers a replacement behavior (usually a polite cut-off signal), we mark with yes, praise, and walk about 15-20 feet away. If they bark and lung, we call them away and move only as far as the dog needs to be below threshold, which may be like 5-10 feet. Then we see a replacement behavior and mark that with yes, praise, and walk away about 15-20 feet further away. So the problem behaviors are only rewarded with a short distance and the replacement behaviors are rewarded with a greater distance, a conditioned reinforcer, and social reinforcement from the handler.
Walking away is negative reinforcement (R-), praise is positive reinforcement (R+) and if you give a bonus reward, that is also R+. So BAT is DRO with functional R- and nonfunctional R+. The functional piece is what makes it so powerful, by the way. That makes it so that even though it is R-, the level of aversion to the stimulus can be very low, with exposure to stresses that are even lower than with DS.
Because we are working with triggers that the dogs are curious about, we do not expose the dog to the trigger, but rather allow them to engage with the trigger themselves. Our role, as handlers, is to prevent the dog from going so close that they go over threshold.