In most of the reliability-based design optimization (RBDO) researches, accurate input statistical model has been assumed to concentrate on the variability of random variables; however, only a limited number of data are available to quantify the input statistical model in many practical engineering applications. In other words, irreducible variability and uncertainty due to lack of knowledge exist simultaneously in random design variables, which may result in uncertainty of reliability. Therefore, the uncertainty induced by insufficient data has to be accounted for RBDO to guarantee the confidence of reliability. Using the Bayesian approach, the uncertainty of input distributions is successfully propagated to a cumulative distribution function (CDF) of reliability under reasonable assumptions, but it requires a number of function evaluations in double-loop Monte Carlo simulation (MCS). To tackle this challenge, the reliability measure approach (RMA) in confidence-based design optimization (CBDO) is proposed to handle the uncertainty of reliability following the idea of performance measure approach (PMA) in RBDO. Input distribution parameters are transformed to random variables following the standard normal distribution for the most probable point (MPP) search based on the proposed stochastic sensitivity analysis of reliability. Therefore, the reliability is approximated at MPP with respect to input distribution parameters. The proposed CBDO can treat confidence constraints employing the reliability value at the target confidence level that is approximated by MPP in standard normal space. In conclusion, CBDO can be performed in a probabilistic space of input distribution parameters corresponding to the conventional U-space in RBDO to yield the probability (confidence) that reliability is larger than the target reliability. The proposed method can significantly reduce the number of function evaluations by eliminating outer-loop MCS while maintaining acceptable accuracy. Numerical examples are used to demonstrate the effectiveness of the developed sensitivity analysis and RMA to estimate the confidence of reliability in CBDO.