We study the Bayesian process to estimate the features of the environment. We focus on two aspects of the Bayesian process: how estimation error depends on the prior distribution of features and how the prior distribution can be learned from experience. The accuracy of the perception is underestimated when each feature of the environment is considered independently because many different features of the environment are usually highly correlated and the estimation error greatly depends on the correlations. The self-consistent learning process renews the prior distribution of correlated features jointly with the estimation of the environment. Here, maximum a posteriori probability (MAP) estimation decreases the effective dimensions of the feature vector. There are critical noise levels in self-consistent learning with MAP estimation, that cause hysteresis behaviors in learning. The self-consistent learning process with stochastic Bayesian estimation (SBE) makes the presumed distribution of environmental features converge to the true distribution for any level of channel noise. However, SBE is less accurate than MAP estimation. We also discuss another stochastic method of estimation, SBE2, which has a smaller estimation error than SBE without hysteresis.

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