= Absolute magnitude–Hubble constant degeneracy
In <Type Ia supernova cosmology>, the predicted <apparent magnitude> depends on the common <absolute magnitude> $M_0$ and the <Hubble constant> only through $\beta=M_0-5\log_{10}(H_0/H_{\rm ref})$, after choosing a fixed reference $H_{\rm ref}>0$. A flat <improper prior> on $M_0$ makes the likelihood integrated over $M_0$ independent of $H_0$, by translation of the integration variable. If the other prior factors are independent and the posterior is proper, the marginal posterior of $H_0$ equals its prior. An external calibration of $M_0$ can break this lack of <identifiability>.
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