Expectation propagation for Gaussian process classification

q(f) = N(mu, Sigma) with Sigma = (K^(-1) + sum_i Lambda_i)^(-1): each likelihood factor is replaced by a Gaussian site, and the sites are refined to a fixed point. Unlike Laplace, which linearises once at the mode, EP matches moments site by site, so its site precisions carry information from the whole factor rather than only its curvature at a single point. The simplified version here takes the site precisions from the curvature at the current mean and damps the mean update.

Usage

Ghosalepgp(x = NULL, y = NULL, length = 0.5)

Arguments

x

Covariate values; a small separable design when NULL.

y

Binary responses matching x.

length

Squared-exponential length scale, positive.

Value

List with estimate (fitted probability at the last point), site_precisions, ep_var_site0, separates, method.

Details

Formula: Lambda_i = max(p_i (1 - p_i), 1e-4); mu <- 0.7 mu + 0.3 (mu + K (y - p)); Sigma_{11} = ((K^(-1) + diag(Lambda))^(-1))_{11}.

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 11.7.4.

Examples

Ghosalepgp()
#> $estimate
#> [1] 0.9664528
#> 
#> $site_precisions
#> [1] 0.03310167 0.10051171 0.10051171 0.03310167
#> 
#> $ep_var_site0
#> [1] 0.8779674
#> 
#> $separates
#> [1] TRUE
#> 
#> $method
#> [1] "EP for GP classification (GvdV 2017 sec. 11.7.4)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"