Setting the mean module#
Prior mean \(\mu(x)\) can be directly set to the prior distribution \(\mathcal{N}(\mu(x), k(x, x'))\).
This method is convenient to implement, but it can be inappropriate when latent GP \(g_j(x)\) does not directly form quantiles \(Q_{\tau_i}(x)\). When indirect representation or correlation structure is involved, special care is needed. Refer to Basic Usage for more details on representation and correlation structure.
Direct representation#
If quantile functions \(Q_{\tau_i}(x)\) are directly represented and independent, prior means \(\mu_i(x)\) can be individually set for each latent GP \(g_i(x)\).
If quantile functions \(Q_{\tau_i}(x)\) are directly represented but correlated, their prior means cannot be directly set. However, if the desired prior means of \(Q_{\tau_i}(x)\) are members of a family of functions that is closed to linear combination, that family of functions can be used as prior means for \(g_j(x)\).
Center-gap representation#
Directly setting prior mean for each \(Q_{\tau_i}(x)\) is impossible for center-gap representation.
However, setting informative prior mean only for the central quantile \(Q_{\tau_0}(x)\) is often enough, as the information can be propagated to other quantiles through the additive structure.
A special mean module CenterGapMean is provided to support this approach.
If \(Q_{\tau_0}(x)\) and \(\Delta Q_{\tau_i}(x)\) are correlated by LMC structure, segregating prior means for central quantile and gaps is generally impossible.
This problem can be circumvented by introducing a special LMC structure that assumes no correlation between \(Q_{\tau_0}(x)\) and \(\Delta Q_{\tau_i}(x)\).
A special variational strategy CenterGapLMCVariationalStrategy is provided to support this approach.