content_image/311327.jpg content_image/311328.jpg content_image/311329.jpg
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$_{2}$ . 

<img src='content_image/311327.jpg'>

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 

<img src='content_image/311328.jpg'>

<img src='content_image/311329.jpg'>

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 

