Path 1: p ( ⃗ x | θ, φ ) π$_{F1}$ ( θ, φ ) } ✲ π ( θ, φ | ⃗ x ) ✲ ∫ π ( θ, φ | ⃗ x ) dφ = π ( θ | ⃗ x ) ≡ π$_{1}$ ( θ | t ) Path 2: p ( t | θ, φ ) ≡ p ( t | θ ) π$_{2}$ ( θ ) } ✲ π$_{2}$ ( θ | t ) ❅ ❅ ❘ ✘ ✘ ✘ ✿ =? FIG. 1: Let ⃗ x be a dataset modeled by the probability density p ( ⃗ x | θ, φ ), where θ and φ are unknown parameters, and consider the following two paths to a posterior density for θ . In path 1, we use a formal prior π F1 ( θ, φ ) to construct the joint posterior for θ and φ , and then integrate out φ . Suppose that the result of this operation only depends on the data ⃗ x through the statistic t ; this gives us π 1 ( θ | t ). For path 2, assume further that the sampling distribution of t only depends on θ . We can then directly construct a posterior for θ , say π 2 ( θ | t ). A marginalization paradox occurs if π 1 ( θ | t ) = π 2 ( θ | t ) regardless of the choice of prior π 2 ( θ ) in path 2. Marginalization paradoxes [15] arise in multiparameter problems when a posterior density can be calculated in different ways that ought to give the same answer but do not (see Fig. 1). This incoherence does not happen with subjective or evidence-based priors because these priors are always proper. With formal priors however, it can only be avoided by allowing the joint prior for all the parameters in a given statistical model to depend on the quantity of interest. This is in fact what the reference prior construction does. For a simple illustration, consider n measurements x$_{i}$ from the normal model with unknown mean µ and standard deviation σ . The likelihood function is: i =1 $^{√$_{2}$$_{π σ}$}$) where ¯ x = ∑ n i x$_{i}$/n and s 2 = ∑ n i ( x$_{i}$ − ¯ x ) $^{2}$/ ( n − 1). When µ is the quantity of interest, the reference prior derived from this likelihood is 1 /σ . Restricting the remaining calculations to the case n = 2 for convenience, the joint reference posterior for µ and σ is then: Integrating out σ yields the marginal µ -posterior, which is a Cauchy distribution with lo- cation parameter ¯ x and scale parameter $_{s/}$√$_{2. Suppose however that our interest lies in}$ π ( µ, σ | ⃗ x ) = √$_{2}$$_{s}$ π σ 3 e $_{−}$1 2 ( s σ ) 2 − ( µ − ¯ x σ ) 2 . (2) p ( ⃗ x | µ, σ ) = n ∏ i =1 e $_{−}$1 2 ( x i − µ σ ) 2 √$_{2}$$_{π σ}$ = e $_{−}$n − 1 2 ( s σ ) $^{2}$$_{−}$n 2 ( ¯ x − µ σ ) 2 n , (1) 5