The first mistake is often cheaper than everything done to avoid admitting it.

Admitting an error early is usually cheap. What gets expensive is everything built to avoid the admission, the cover, the elaborate justification, the second decision made to protect the first. That machinery costs more than the mistake ever did and keeps costing until someone stops paying.

If you are currently defending a position you privately think is wrong, price the defence. Count the time, the strain, the other things you cannot do. Then compare it with the size of simply saying you got it wrong.