for d$_{1}$, d$_{2}$ ∈ { 1 , . . . , D − 1 } . Alternatively, it tells us that in this case I$_{dl}$ = ln/ (2 σ 2 $_{e}$), for d = 1 , . . . , D − 1. Moreover, using the above along with Equation (2.1), in conjunction with part (2) of Theorem 2.1, we have that if f$_{p}$ ≤ Z$_{d}$$_{1}$$_{p}$ < e$_{p}$ for p = 0 , . . . , l$_{1}$ − 1 (i.e. if treatment d$_{1}$ is present up to stage l$_{1}$ ) and f$_{q}$ ≤ Z$_{d}$$_{2}$$_{q}$ < e$_{q}$ for q = 0 , . . . , l$_{2}$ − 1 (i.e. if treatment d$_{2}$ is present up to stage l$_{2}$ ), with l$_{1}$ ≤ l$_{2}$ , l$_{1}$, l$_{2}$ ∈ { 1 , . . . , L } and d$_{1}$, d$_{2}$ ∈ { 1 , . . . , D − 1 } (taking f$_{0}$ = −∞ and e$_{0}$ = ∞ ) then for a ≤ b , a = 1 , . . . , l$_{1}$ , and b = 1 , . . . , l$_{2}$ . For further clarity, as an example, consider the case D = 3, L = 2, and the associated value of pr ( ω$_{R}$ = ω , ψ$_{R}$ = ψ | τ ) when ω = (2 , 1) T and ψ = (1 , 0) $^{T}$. Using the above we know the following elements of the matrix Λ$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ and vector I$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ = ( I T 1 , ( ω , ψ $_{)}$, I T 2 , ( ω , ψ $_{)}$) T where we have used • to signify an element we do not know the value of. Now consider our computation of pr ( ω$_{R}$ = ω , ψ$_{R}$ = ψ | τ ). We have As we have seen we do not know the values of the final row and column of the matrix Λ$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ , or the final element of the vector I$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ . But, the fact mentioned in Theorem 2.2 becomes clear: this does not matter as the limits of integration corresponding to this variable are ( −∞ , ∞ ). Indeed, by the marginal distribution properties of the multivariate normal distribution, we need only as stated consider one matrix Λ$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ , and one set of vectors I$_{l,}$$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ ; exactly those given by the case ω = ( L, . . . , L ) $^{T}$. We denote these 17 I$_{d}$$_{1}$$_{a}$ = an 2 σ 2 e , I$_{d}$$_{2}$$_{b}$ = bn 2 σ 2 e , cov( Z$_{d}$$_{1}$$_{a}$, Z$_{d}$$_{2}$$_{b}$ | ω , ψ ) = I 1 / 2 d$_{1}$$_{a}$cov(ˆ τ$_{d}$$_{1}$$_{b}$, ˆ τ$_{d}$$_{2}$$_{b}$ | L$_{b}$$_{1}$, . . . , L$_{bD}$ ) I 1 / 2 d$_{2}$b = ( an 2 σ 2 e ) 1 / 2 σ 2 e bn (1 + δ$_{d}$$_{1}$$_{d}$$_{2}$ )$^{(}$ bn 2 σ 2 e ) 1 / 2 , = 1 2 ( a b$^{)}$$^{1}$$^{/}$$^{2}$ (1 + δ$_{d}$$_{1}$$_{d}$$_{2}$ ) , Λ$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ =        1 1 2 ( 1 2 ) 1 / 2 • 1 2 1 1 2 ( 1 2 ) 1 / 2 • ( 1 2 ) 1 / 2 1 2 ( 1 2 ) 1 / 2 1 • • • • •$^{}$       , I$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ = ( n 2 σ 2 e , n 2 σ 2 e , 2 n 2 σ 2 e , •$^{)}$$^{T}$ , pr ( ω$_{R}$ = ω , ψ$_{R}$ = ψ | τ ) = ∫ e$_{1}$ f$_{1}$ ∫ f$_{1}$ −∞ ∫ ∞ e$_{2}$ ∫ ∞ −∞ φ $^{{}$x , r ( τ , 2) ◦ I$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ , Λ$_{(}$$_{ω}$$_{,}$$_{ψ}$$_{)}$ $^{}}$d x$_{22}$ d x$_{12}$ d x$_{21}$ d x$_{11}$.