The depth depends somewhat on the type of soil. Sandy soil is not a structurally sound as clay, so it would require a deeper post hole. Also, the diameter of the hole is critical, unless you fill the hole to the cut edge with concrete, and not use any soil fill. The whole fence is only as good as its foundation, the base of the posts.
Regardless of the depth, all wood will shrink after cut. This shrinkage will cause the post to diminish in size from 4" by 4" to something less with the passage of time. The concrete will not expand to fill this shrinkage void, so a gap will form between the post and the concrete.
This gap will allow moisture to find its way between the wood and concrete and pond there, causing pre-mature wood/water rot, and eventually the base of the post will become loose, weak, and will fail.
Either find another way to secure the base of the wood post above the moisture line, or use metal posts.