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the standardized mean θ = µ/σ . In a non-reference approach one would perform the trans- formation ( µ, σ ) → ( θ, σ ) in Eq. (2) and integrate out σ in order to obtain the marginal θ -posterior. The latter only depends on the data through the statistic t ≡ √$_{2 ¯}$ x/s : 2 

where erf is the error function. Furthermore, the sampling distribution of t turns out to depend on θ only and is a noncentral Student’s t distribution for one degree of freedom and with noncentrality parameter θ : 

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It is clear that there exists no prior (no function of θ only) that, multiplied by the likeli- hood (4), leads to the posterior (3). Hence the marginalization paradox: someone who is only given the data value of the statistic t will be able to make inferences about the pa- rameter θ , but these inferences are guaranteed to disagree with those previously made by the Bayesian who had access to the full dataset. Resolution of this paradox hinges on the realization that lack of information about µ is not the same as lack of information about θ . Therefore, the choice of which quantity is of interest must be done before calculating the prior. Since reference priors are derived from the likelihood function, the latter must first be expressed in terms of the relevant parameters, θ and σ : 

$^{√$_{2}$$_{π σ}$}$) n Applying the reference algorithm to this likelihood while treating θ as the quantity of in- terest yields the prior π ( θ, σ ) = 1 / ( σ √ 1 + θ $^{2}$/ 2), which is very different from the prior 1 /σ obtained by treating µ as the quantity of interest. The resulting reference posterior suffers no marginalization problems. Further details about this example can be found in Refs. [8, 12]. 

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Our discussion of marginalization also helps to clarify the behavior of reference posteriors under transformations in the multiparameter setting. Reference posteriors are invariant under one-to-one transformations of the parameter of interest, but not under transformations that redefine the parameter of interest by mixing in one or more nuisance parameters. However, redefining the nuisance parameters is permitted. Suppose for example that ϕ is the parameter of interest, ν the nuisance parameter(s), and consider an invertible transformation 

