Why do people have such a hard time admitting they've made a mistake? We are all humans, we all make mistakes, we regret many of them later. But, geez, say so. I won't think less of you, on the contrary.
I find it especially important in my role as a parent to acknowledge a mistake and apologize for it - yes, to my child. Otherwise I am modeling behavior I don't want to see perpetuated. We have this underlying belief that we are perceived as weak when we say "I was wrong," "I made a mistake," "I'm sorry," or"I wished I'd never done this." I build up a huffing and puffing resistance and fight response to a lame "wasn't me" or "I would never have done this;" when, on the other hand, someone says right away "oops, so sorry, my mistake," that feeling implodes and my reaction softens immediately.
How cool would it be if a politician or CEO honestly said "I made the wrong decision at the time, but I understand the issue better now and take full responsibility, and I promise to correct the situation." Wouldn't you appreciate the humanness in that person? I guess this behavior all starts with you and me, though.