What is the reason for self dissociation of water?
A weak acid has a strong conjugate base but this strong base behaves as a weak base in water. This means that this base is not able to dissociate hydrogen from water molecules but pure water undergoes self dissociation. What is the reason of this dissociation?
Pure water will self-dissociate (also called auto-ionization) to form H+ and OH- ions:
H2O <--> H+ + OH-
At constant temperature and pressure, the value for this self-dissociation is a constant value. For water at 25 degrees C, the value is 10^-14 and is called Kw:
Kw = [H+][OH-] = 10^-14
Keep in mind that this is a very tiny amount of protons and hydroxide ions in solution. This puts the concentration of each species at 0.0000001 moles/liter. So in any kind of real, measurable sense it is negligible. This self-dissociation occurs due to the highly polar nature of water and the O-H bond. Oxygen is very electronegative and essentially pulls both of the electrons from their covalent bond to itself to form the hydroxide anion, thus liberating the proton. But this only occurs as a tiny percentage of all water molecules at any given point in time as the equilibrium for the reversible reaction lies highly to the left.
You are correct that a weak acid has a strong conjugate base. But even when we say that a particular strong conjugate base does not deprotonate water, it actually does to a very small extent and probably more so than the self-ionization of water. It's just that this amount is so small that it is essentially negligible for all practical purposes.