Just want to make sure I'm playing this correctly. Say I'm playing Bara vs Greyjoy and I have a Bay of Ice out. I throw a 6 initiative plot and my opponent throws a 4. He then says he's going to play Ahead of the Tide. Am I correct in assuming that since Bay of Ice has a passive effect and Ahead of the Tide is a response even though the event changes the actual winner of Initiative the kneeling and draw from Bay of Ice would still happen first yes?

I'm afraid that you are mistaken. Bay of Ice let's the player who won Initiative draw a card. To determine who wins Initiative there is an Initiative count. You're Initiative count was 7 (plot + Bay of Ice), while your opponent's was 4. Normally you would win, but after the Initiative count there is save/cancel opportunity against the Initiative Count. This is where Ahead of the Tide comes into play. Ahead of the Tide cancels your Initiative count and let's the player that played it win Initiative instead. So while you would have won Initiative, you never actually did win it. Since the Greyjoy player won Initiative, he or she would kneel a Bay of Ice and draw a card.

As an extra bit of information. Plots actually have to resolve first before the player that won Initiative can kneel a Bay of Ice. Yes, Bay of Ice is passive, but it doesn't get the opportunity to happen until plots are resolved. For more information on the timing sequence related to your question and this, check out this very good thread on the same topic:

http://www.fantasyflightgames.com/edge_foros_discusion.asp?efid=18&efcid=4&efidt=314317

Correct me if I'm wrong but Ahead of the Tide is not a regular Response: it's a cancel Response. Therefore, Ahead of the Tide affects the initiative count before it resolves and before pasives for winning initiative are triggered.

Yep that was the question if it cancel/saved or was a response and where that fit. I hadn't seen that other post, just didn't look back far enough I guess.

I wonder what would happen with Host of the Bear when your opponent cancels your iniciative. Would it stay in play because Ahead of theTide just cancels the count or would it be discarded since at some point my initiavide dropped under 3.

Well, remember that your total initiative didn't actually fail. It is still the total of your plot and your modifiers. The count and comparison was canceled, but that's not the same thing. It's kind of like the difference between STR and printed-STR. Ahead of the Tide doesn't interact with Host of the Bear.