It depends upon what sort of mud it is to be honest.
There are generally two forms of motion offered by mud-tyre design, the first is intended to propel you forward by displacement (moving as much mud from infront of you to behind you), tyres with heavy sidewall design tend to be for this (note "tend").
The second mechanism is by digging downwards to find something more solid to drive on. This tends to work quite well in clay but isn't especially effective in a bog, as you tend to find the floor with the axles long before you find anything solid.
Essentially it comes down to a lot of complicated mathematics (which I don't even pretend to understand) but a "good" mud-terrain tyre will generally try to reach a compromise between the two anyhow.
There are reasons for a lift of course, but that's mainly with increasing ground clearance to overcome obstacles, whilst this comes into play when stuck in mud it's not the whole story, and hopefully better tyres and a more sensible line would prevent you from getting into difficulty in the first place.
Having said all of that, there is a huge amount of driver ability to throw into the mix, rubber and air can only compensate for so much. That's not to say you're a bad driver (I have no idea whether you are or not) merely that experience and practice can make a much bigger difference to ability than just about anything else.
