  We study the Bayesian approach to thermometry with no prior knowledge about
the expected temperature scale, through the example of energy measurements on
fully or partially thermalized qubit probes. We show that the most common
Bayesian estimators, namely the mean and the median, lead to high-temperature
divergences when used for uninformed thermometry. To circumvent this and
achieve better overall accuracy, we propose two new estimators based on an
optimization of relative deviations. Their global temperature-averaged behavior
matches a modified van Trees bound, which complements the Cram\'er-Rao bound
for smaller probe numbers and unrestricted temperature ranges. Furthermore, we
show that, using partially thermalized probes, one can increase the range of
temperatures to which the thermometer is sensitive at the cost of the local
accuracy.
