In single-molecule protein experiments, the observable variables are restricted within a small fraction of the entire degrees of freedom. Therefore, to investigate the physical nature of proteins in detail, we always need to estimate the hidden internal structure referring only to the accessible degrees of freedom. We formulate this problem on the basis of Bayesian inference, which can be applied to various complex systems. In the ideal case, we find that in general the framework actually works. Although careful numerical studies confirm that our method outperforms the conventional method by up to two orders of magnitude, we find a striking phenomenon: a loss-of-precision transition occurs abruptly when the design of the observation system is inappropriate. The basic features of the proposed method are illustrated using a simple but nontrivial model.
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