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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2016/07/15/the-curious-incident-of-the-inverse-of-the-mean/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="http://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong>A</strong> s I figured out while working with astronomer colleagues <a target="_blank" href="https://xianblog.wordpress.com/2016/07/05/a-lobservatoire-de-paris/">last week</a>, a strange if understandable difficulty proceeds from the simplest and most studied statistical model, namely the Normal model</p>
<p>x~N(θ,1)</p>
<p>Indeed, if one reparametrises this model as x~N(υ⁻¹,1) with υ>0, a <em>single</em> 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.</p>
<p>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:</p>
<p><a target="_blank" href="http://i1.wp.com/xianblog.files.wordpress.com/2016/07/invorm2.jpg?ssl=1"><img alt="invorm" class="wp-image-31066" src="http://i1.wp.com/xianblog.files.wordpress.com/2016/07/invorm2.jpg?resize=300%2C300&ssl=1" data-recalc-dims="1"></a>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:</p>
<p><a target="_blank" href="http://i1.wp.com/xianblog.files.wordpress.com/2016/07/invarm.jpg?ssl=1"><img alt="invarm" class="wp-image-31067" src="http://i1.wp.com/xianblog.files.wordpress.com/2016/07/invarm.jpg?resize=300%2C300&ssl=1" data-recalc-dims="1"></a>There is simply not enough information contained in one datapoint (or even several datapoints for all that matters) to infer about υ.</p>
<p>Filed under: <a target="_blank" href="https://xianblog.wordpress.com/category/statistics/r-statistics/">R</a>, <a target="_blank" href="https://xianblog.wordpress.com/category/statistics/">Statistics</a>, <a target="_blank" href="https://xianblog.wordpress.com/category/university-life/">University life</a> Tagged: <a target="_blank" href="https://xianblog.wordpress.com/tag/astronomy/">astronomy</a>, <a target="_blank" href="https://xianblog.wordpress.com/tag/bayesian-inference/">Bayesian inference</a>, <a target="_blank" href="https://xianblog.wordpress.com/tag/inverse-problems/">inverse problems</a>, <a target="_blank" href="https://xianblog.wordpress.com/tag/parallaxes/">parallaxes</a> <a target="_blank" rel="nofollow" href="http://feeds.wordpress.com/1.0/gocomments/xianblog.wordpress.com/31062/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/xianblog.wordpress.com/31062/" border="0"></a> <img alt="" src="http://i1.wp.com/pixel.wp.com/b.gif?resize=1%2C1&ssl=1" border="0" data-recalc-dims="1"></p>
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