I agree with Arrows completely. Positive reinforcement training does not mean they get food every time, nor does it mean a dog is rewarded for simply not biting. I have a dog who is reactive towards other dogs on leash. When we see a dog approaching, I get his attention BEFORE he fixates on the other dog and ask for behaviors he does well and that give him the opportunity to be successful at ignoring the other dog (primarily “look” and focus on me as we walk). He is rewarded for doing THAT when asked, and the behavior I chose to ask for “eyes on me” means that if he is listening he is not barking or running towards the other dogs. I am not rewarding his lack of action - I am consciously choosing to create a behavior I DO want in place of one I don’t want.
The reward can be food, his ball or just a verbal praise. If he does make a mistake, then yes, I have to physically stop him from going towards the other dog for safety reasons. However, at that point he’s excited, stressed and not focused or learning. Me physically stopping him does not decrease the unwanted behavior. Having him attentive and focused on me and not the distractions around him creates a calmer, happier dog who is more apt to offer the correct behavior going forward. Can I say we accomplish this perfectly? No. However there is no question there is a huge improvement in my dog when I take the time to work with him on giving the correct behavior as opposed repeatedly punishing him for incorrect behavior.
For a food aggressive dog, that means creating the drop command over a series of steps to keep you safe. It’s great to say I’d never need to use mind games (trading) to teach correct behavior - but the reality is while punishment in that scenario may have worked for one dog many people who try pure punishment in a guarding situation end up bitten with the dog worse than when they started. I had an Aussie who guarded as a child. I only wish I knew then what I do now. Maybe it would have given us a better alternative than prying bones away from him with a broom while he was growing when he accidentally got ahold of a high value object we normally did not give him due to his instinct to guard them.