The board of directors of yahoo has a fiduciary duty to their shareholders to maximize shareholder revenue. From the wikipedia article on fiduciary duty:

"A fiduciary duty is the highest standard of care at either equity or law. A fiduciary is expected to be extremely loyal to the person to whom they owe the duty (the "principal"): they must not put their personal interests before the duty, and must not profit from their position as a fiduciary, unless the principal consents."

The yahoo board legally can't allow their desire to remain independent to color their judgement. The cost of going public is that it's not your company any more, it's your shareholder's company.