Kelly Sizing with an Unknown Win Probability

A coin's win probability $\theta$ is unknown, with prior $\theta\sim\text{Beta}(2,2)$. You observe $7$ wins and $3$ losses in calibration trials, then must place one even-money bet on the next flip, choosing a fraction $f$ of wealth to maximize the expected log-wealth after that bet. What fraction should you bet, and why is the posterior mean (rather than, say, the posterior mode) the right quantity to plug into the Kelly formula?