We investigated the performance of distribution estimation of hyperparameters in Markov random field models proposed by Nakanishi-Ohno et al., J. Phys. A 47, 045001 (2014) when used to evaluate the confidence of data. We analytically calculated the configurational average, with respect to data, of the negative logarithm of the posterior distribution, which is called free energy based on an analogy with statistical mechanics. This configurational average of free energy shrinks as the amount of data increases. Our results theoretically confirm the numerical results from that previous study.
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