  In the present paper, using a replica analysis, we examine the portfolio
optimization problem handled in previous work and discuss the minimization of
investment risk under constraints of budget and expected return for the case
that the distribution of the hyperparameters of the mean and variance of the
return rate of each asset are not limited to a specific probability family.
Findings derived using our proposed method are compared with those in previous
work to verify the effectiveness of our proposed method. Further, we derive a
Pythagorean theorem of the Sharpe ratio and macroscopic relations of
opportunity loss. Using numerical experiments, the effectiveness of our
proposed method is demonstrated for a specific situation.
