Measurements are inseparable from inference, where the estimation of signals of interest from other observations is called an indirect measurement. While a variety of measurement limits have been defined by the physical constraint on each setup, the fundamental limit of an indirect measurement is essentially the limit of inference. Here, we propose the concept of statistical limits on indirect measurement: the bounds of distinction between signals and noise and between a signal and another signal. By developing the asymptotic theory of Bayesian regression, we investigate the phenomenology of a typical indirect measurement and demonstrate the existence of these limits. Based on the connection between inference and statistical physics, we also provide a unified interpretation in which these limits emerge from phase transitions of inference. Our results could pave the way for novel experimental design, enabling assess to the required quality of observations according to the assumed ground truth before the concerned indirect measurement is actually performed.
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