Our answer always starts out the same, "Interrupt your dog while it is in the process of the behavior you want to change."

After this point you need to remember some things.  First, we're talking about LGDs here and secondly, the behavior almost always has to do with chasing, biting, chewing or somehow hurting or bullying the stock the dog is supposed to nurture and protect.  We are not talking about training a dog to perform specific actions nor are we talking about getting the dog to DO something.  We are talking about stopping the dog from doing something.

Perhaps the best place to start is when you bring the dog home.  A tremendous number of problems can be totally avoided if you introduce your new dog to its new surroundings and its new companions, canine and caprine, in an appropriate manner.

Before, during and/or after you introduce it to its new surroundings and its new companions gently but firmly declare your alpha status.  You're still working on problem prevention rather than correction.  If a pup, you should be able to pick it up by reaching under its belly and its chest.  If you have an adult you should be able to lift it gently a short distance by holding under its chest.  We pet and fuss over our animals (only in the goat yard - NEVER out of it) and in the process we make sure we straddle the dog. Sometimes we hug it around its neck from this position.  These are superior positions and some dogs will resist letting you maintain that position or they may even try to stop you from getting into the position.  Hold the dog firmly by the collar while talking to it in a soothing tone of voice and petting it.  All but the truly hard core will give in and allow you to do this. We also practice giving and taking food to insure the dog does not become food aggressive toward us.  Simply take a treat and give it to the dog, then immediately remove it speaking soothingly to it, then return the food and let it eat.  Once you've done these things, do them daily until there is no resistance to your claim to alpha and then continue the practice on a regular basis.  If you will do this consistently, you may still have problems along the way but you will definitely have a head start if and when you need to correct them.

Our basic approach is to insure the dog knows two things:

     1) You're the boss, or Alpha, and 

     2) The dog knows exactly what it is doing that is bad.
Being the Alpha of your pack does NOT mean you should be mean or harsh to your Pyr. In fact, that attitude will NOT work with Great Pyrenees. If you do not understand what it means to be Alpha - if you think being the Alpha is achieved from demeaning or bullying the dog, or any other harsh practices, STOP right now and read this article: Alpha Humans: What does it mean to be dominant?

Now that you've set yourself up as the boss, or alpha, you're ready to make corrections to your dog's behavior if necessary.  We add "if necessary" because the next step is to be absolutely sure that your dog is misbehaving.  Remember, LGDs are bred to make their own decisions and you may not understand what they are doing.  For example, many LGDs, including Great Pyrenees, eat dead goat kids or even adults presumably in an effort to reduce the area's attraction to predators.  Because of this trait, if you find your dog eating one of your animals, you cannot just assume that the dog has killed it.  Especially if you are new to LGDs, when you think the dog may be doing something wrong, take the time to find out for sure and to make sure that even if you don't like it, that it is not an acceptable thing for LGDs to do.  An example of this is a new owner did not like her Pyr pup "picking on her house dogs" until the house dogs were afraid to be around the pup.  Keeping other dogs away from stock is what LGDs do!  This was not bad behavior but the new owner did not understand.

When you find your dog misbehaving, interrupt its behavior.  You cannot correct a dog even shortly after the behavior is over, much less hours later and expect any positive results because the dog won't know why you're treating it that way.  If your dog is chasing or biting the stock when you're not around, you cannot afford to leave the dog with the stock until you know the behavior has changed.  Separate the dog from your animals but leave it as close as possible.  An escape proof pen inside the stock yard would be perfect.  Locking the dog in your house or your yard would be truly counter productive.  Just remember, if your stock is in more danger from your LGD than it is from predators, you need to change the situation before you release the dog and leave it unsupervised, even if you can only work with it nights and weekends.

Once you are available, release your dog in the barn yard or other confined area on a long lead and let it interact with the stock.  If it misbehaves, use the lead to interrupt it and speak harshly to it.  Repeat this process as often as necessary until your dog seems to understand and alter its behavior.

Next, release the dog in the same confined area without the lead but while you are right there.  If the dog misbehaves, again interrupt its behavior.  You can do this by roaring, yelling, throwing something distracting at it (a plastic jug with just a few pebbles in it is effective - the objective is not to hurt the dog but to get its attention) or by grabbing it.  Sometimes, an aggressive or fierce roaring/yelling at the dog will be enough to stop it.  Pups are often distracted by the noisy jug landing in their vicinity.  Grabbing the dog, holding it (we do mean holding, NOT shaking it) while looking it in the eye and telling it loudly and firmly "No!" (making sure you try your best to communicate that "NO!" means "This is a bad thing you are doing, don't do it again") is often effective.  When the dog is consistently behaving well, leave the area and observe the dog from a distance.  If the dog continues to behave well over a period of time, release it into the pasture with the stock but stay alert.  If the dog regresses and there are problems, start over again and repeat the earlier steps.

One specific behavior common to Great Pyrenees is demonstrating their joy at seeing you by jumping up on you.  As puppies the results are usually dirty paw prints on your clothing and it is not a major problem when the dog weighs 20 or 30 pounds.  When the dog weighs 100 or so, it can present much more of a problem as well as considerably enlarging the area covered by muddy paw prints.  For this behavior the first step, of course, is to remove the dog from your person, put all 4 feet on the ground and say, "No!" firmly.  If this doesn't work, try stepping on the dog's toes of his hind feet while putting the front feet back on the ground.  Some dogs are so exuberant that nothing that we've mentioned is enough to dampen its enthusiasm.  As goat owners, we purchased a mini hot shot years ago to insure Paula's safety when working around bucks and larger groups of very greedy does.  She has found that a very gentle application of the mini hot shot will calm even the most wildly exuberant dog, or at least prevent it from climbing all over her.  As with any technique that produces a painful sensation, we recommend using the hotshot only in extreme cases when a dog refuses to physically respect your person and then only with the minimum application to change the behavior.

If, for whatever reason, (prudence being an excellent one in some cases) you choose not to correct your consistently misbehaving dog and you are at your wit's end with no one to help you, there is one other alternative available to you: the shock collar.  This is a tool of last resort as the only purpose of it is to cause a brief, painful interruption of the undesired behavior and to teach the dog that this will happen every time it misbehaves in this specific way.  Since many LGDs have a thick coat, the collar must be one with long contacts and you may (definitely on a Pyr) still have to cut the hair on the dog's neck (which can be a whole new adventure in and of itself).  With batteries in the remote and the collar on the dog, raise the setting to the low medium range and wait for the dog to misbehave.  When it does, push the button very briefly.  The dog should react as if it just had a bee sting it on its neck.  Wait again and, if the dog repeats is actions hit the button again briefly.  If the dog does not seem to care about the shock, move the intensity up in small increments and repeat the process.  At some point, if you have good contact and the remote is working, the dog will pay attention and stop the behavior.  It is critical that you start with low intensity and brief exposure to the shock.  Short cutting the step by step method could lead to erratic or aggressive behavior and/or non-performance of its guardian duties.  Extreme and random exposure to the shock can create a psychotic dog.

Hopefully with some attention at the very beginning of your relationship with your dog, you will prevent any major problems from happening and make the minor ones pretty easy to take care of. 

Good luck.