Thanks to all of you for posting your opinions and thoughts. I decided today that I will practice moderation. My best bet is to buy the kids some things that I am not tempted by. I got one package of cookies for the whole family (which comes to about 5 Oreo's each) and each of them a treat or two. I'm going to make faux fried ice cream for the family (and I will have some of this). And I made sure everyone heard me say, "Once the snacks are gone, they are gone and you're not getting any more for the rest of the week. So eat wisely."
After reading your thoughts, I decided that "none" is probably too extreme and may not be the best policy because it could lead to binging.
And I got home just in time from the grocery store. My daughter almost starved to death.