the curious incident of the inverse of the mean

A s I figured out while working with astronomer colleagues last week, a strange if understandable difficulty proceeds from the simplest and most studied statistical model, namely the Normal model

x~N(θ,1)

Indeed, if one reparametrises this model as x~N(υ⁻¹,1) with υ>0, a single observation x brings very little information about υ! (This is not a toy problem as it corresponds to estimating distances from observations of parallaxes.) If x gets large, υ is very likely to be small, but if x is small or negative, υ is certainly large, with no power to discriminate between highly different values. For instance, Fisher’s information for this model and parametrisation is υ⁻² and thus collapses at zero.

While one can always hope for Bayesian miracles, they do not automatically occur. For instance, working with a Gamma prior Ga(3,10³) on υ [as informed by a large astronomy dataset] leads to a posterior expectation hardly impacted by the value of the observation x:

And using an alternative estimate like the harmonic posterior mean that is associated with the relative squared error loss does not see much more impact from the observation:

There is simply not enough information contained in one datapoint (or even several datapoints for all that matters) to infer about υ.