I just lost two major battles! huzzah!

[Bob the Insane]

The AI also gets stat bonuses on Hard, but I think they are small...

This comes from my observations of 1v1 identical unit fighting in custom battles. In Medium (Normal), it is 50-50 split on who wins. On Hard the AI will always win, but only just (it is not just down to Morale as the AI will have always caused a few more casualties). On very Hard (Expert) the AI will kill your troops at a rate of 2:1 and always win pretty quickly.

This is with basic Line Infantry...

I am impressed with anyone taking the AI on with significantly fewer numbers on VH. Especially the Native American factions as they tend to move directly to melee combat.

I like Hard battle difficulty myself because you are forced to be tactical because on an even footing and a simple head to head fight you will tend to lose, or at the very least take very high casualties.