  We explore the issue of prior dependence in the context of one-sided
constraints on the dark matter-photon and dark matter-neutrino elastic
scattering cross-sections derived from cosmic microwave background (CMB)
anisotropies measurements. Testing in particular the linear flat, Jeffreys, and
logarithmic flat priors, we find that the former two yield upper limits on the
cross-sections that are mutually consistent to within 20%. In contrast, bounds
derived under the assumption of the logarithmic flat prior are strongly
sensitive to the choice of the lower prior boundary. Indeed, surveying the
recent literature, we find that this pathology of the logarithmic prior has
resulted in published constraints that are up to an order of magnitude
artificially tighter than they should objectively be. Our revised `objective'
constraints from the 2015 data of the Planck CMB mission on the present-day
scattering cross-sections are $\sigma_{\rm DM-\gamma} < 1.72 \times 10^{-6} \,
\sigma_{\rm T} \, (m_{\rm DM}/{\rm GeV})$ and $\sigma_{\rm DM-\gamma} < 2.74
\times 10^{-15} \, \sigma_{\rm T} \, (m_{\rm DM}/{\rm GeV})$ for dark
matter-photon interactions scaling as $a^0$ and $a^{-2}$ respectively, where
$a$ is the scale factor, and $\sigma_{\rm T}$ the total Thomson scattering
cross-section. Their dark matter-neutrino counterparts read $\sigma_{\rm
DM-\nu} < 2.14 \times 10^{-6} \, \sigma_{\rm T} \, (m_{\rm DM}/{\rm GeV})$ and
$\sigma_{\rm DM-\nu}< 2.46 \times 10^{-15} \, \sigma_{\rm T} \, (m_{\rm
DM}/{\rm GeV})$. All have been computed assuming the Jeffreys prior.
