Someone who corrects themselves publicly becomes easier to trust privately.

Correcting yourself in public looks costly and pays unusually well. It shows that you track accuracy over standing, which is exactly the quality people are trying to assess when they decide whether to rely on you. After watching someone do that once, other people find it much easier to believe them when they are not correcting anything.

When you find you were wrong about something you said out loud, say so in the same place you said it. Do not bury it or let it quietly expire. One visible correction buys more trust than a long run of being right.