As we mentioned in the previous section, it would be possible to proceed by linking the parameters θ1, θ2 in a Bayes linear structure. However there are advantages in transforming the parameters first. The transformed parameters η1, η2 are then linked in a Bayes linear structure. The reasons for using the transformation are as follows.
Firstly, the range of θi is bounded to 0 < θi < 1 in the binomial case and 0 < θi < ∞ in the Poisson case. The combination of linear updates with bounded parameter spaces seems undesirable both in terms of first and second moments. If information leads to adjustment of the expectation for a quantity towards a boundary, it seems clear that this adjustment should not continue to be linear as the boundary is approached.
It is to be expected that variances will be affected by the proximity of a boundary and beliefs, when the mean is close to a boundary, will no longer be symmetric in the sense that deviations from the mean in either direction would be regarded in the same way. Similarly there are difficulties with covariances in bounded spaces where the tendency would be to imagine rather nonlinear relationships between unknowns close to boundaries. So it is desirable to transform the parameters onto unbounded spaces.
Secondly it is possible for the variances of the untransformed parameters θi to increase when data are observed. For example, in the beta-binomial case above, when ai = 7, bi = 1, ni = 4 and xi = 2. In the gamma-Poisson case the posterior variance can be greater than the prior variance if xi is sufficiently large. While Goldstein &
Shaw (2004) (Theorem 5) give conditions for the existence of unique Bayes linear kinematic updates which allow some such variance increase, the transformations have the effect of making reductions in variance of the transformed parameters occur when observations are made, at least in most circumstances, and therefore allow use of the simpler sufficient condition given in Corollary 3 of Goldstein & Shaw (2004).
Bayes linear kinematics, without transformation, gives a rule for adjusting beliefs about
θ1, θ2 by Bayes linear updates. Similarly Bayes linear kinematics, with the transfor-mation, gives a Bayes linear rule for updating beliefs about η1, η2, where there is a 1 - 1 relationship between ηi and θi. Any further use of conjugate Bayesian updating of beliefs about θj, given observation of Xj, after already adjusting by observation of Xi, relies on the idea that θj still has a distribution of the required conjugate form, whether or not a transformation is used. Similarly evaluating predictive distributions for new observations or credible intervals for θ1, θ2depends on such an idea. Additionally, when a transformation is used, this preserved conjugate form is required in order to convert back from the adjusted moments of ηj to the new distribution for θj.
Clearly, if adjustments were only ever made in one direction, eg. of beliefs about θj by observing Xi, and this was never reversed to adjust beliefs about θi by observing Xj, then it could simply be declared that the conditional distribution was the required conjugate distribution. Such one-way belief adjustment might be appropriate, for ex-ample, in a time-series forecasting context, as in West et al. (1985). Even in this case, however, we would be saying that the conjugate distribution holds both when we make the update and for forecasts to time t + k.
When commutativity, in the strong sense that conjugate updates of the marginal dis-tributions of θ1, θ2 are always appropriate, is required then this might be regarded as a pragmatic approximation which does not correspond exactly to a full Bayesian con-ditioning analysis. With no transformation, this assumption is made directly on the distributions of θ1, θ2 under Bayes linear kinematic updates. With transformation, the assumption applies to the corresponding distributions of η1, η2, in the same way.
3.7.1 The transformed approach
Having decided on the use of transformations of the binomial and Poisson parameters we represent them using the function g(), where
ηi = g(θi),
for i = 1, 2. The transformation g() is such that for either 0 < θi < 1 in the beta-binomial case or θi> 0 in the gamma-Poisson model then ηi ∈ (−∞, ∞). We then link η1, η2, rather than θ1, θ2, in a Bayes linear structure.
In order to perform Bayes linear kinematics we shall need the prior means and variances
of η1, η2. In the beta-binomial model they are
where f0i(θi) is the prior beta density for θi. After the conjugate updates the expres-sions for the mean and variance, E1(ηi) and Var1(ηi), remain of the same form but with ai and bi replaced by ai+ xi and bi+ ni− xi respectively.
In the gamma-Poisson model the prior means and variances of each ηi are E0(ηi) =
We can propagate these changes in belief through to the other group via the Bayes linear kinematic updating equations.
for i 6= j. We wish to know when a unique, commutative Bayes linear kinematic solution exists. A sufficient condition for uniqueness, using Equation 3.8 is
Var1(ηi) < Var0(ηi) (3.17) for i = 1 or 2 or both. If this condition holds then the Bayes linear kinematic adjusted expectation and variance of ηi are given by
E(2)(ηi; xi, xj) = Var(2)(ηi; xi, xj)[Var−11 (ηi)E1(ηi)
+ Var−11 (ηi; xj)E1(ηi; xj) − Var−10 (ηi)E0(ηi)], (3.18) and
Var(2)(ηi; xi, xj) =Var−11 (ηi) + Var−11 (ηi; xj) − Var−10 (ηi)−1
. (3.19)
In the notation above E(2)(ηi; xi, xj) and Var(2)(ηi; xi, xj) represent the Bayes linear kinematic commutative expectation and variance (Equations 3.9 and 3.10) of ηi having made 2 observations (given in brackets in the subscript). The quantities after the semi-colon indicate that these are the adjusted expectation and variance having observed xi and xj.
3.7.2 Predictive distributions
Suppose now we have p ≥ 2 groups X1, . . . , Xp where each is a binomial or Poisson count. Imagine we have performed a Bayes linear kinematic update of the form in the previous section on X1, . . . , Xp and obtained adjusted expectations and variances for η= (η1, . . . , ηp)′.
Now suppose we imagine updating by X1, . . . , Xp−1. The ηp, Xp structure has to be such that we would get the “correct” update by X1, . . . , Xp. This means that it has to be the conjugate beta-binomial or gamma-Poisson structure.
Therefore, to be consistent with potential future updates, predictive distributions are calculated on the basis of the same structure.