Laplace approximation for GP posteriors

pi(f | data) is approximated by N(f-hat, (K^(-1) + W)^(-1)) with W = diag(-d^2 log p(y | f)) evaluated at the mode. For a non-Gaussian likelihood the GP posterior is not available in closed form at all, so some Gaussian surrogate is unavoidable; Laplace is the cheapest, and its covariance is the one place the likelihood curvature enters. Logistic classification here, with W = diag(p(1 - p)).

Usage

Ghosalgplaplace(x = NULL, y = NULL, length = 0.5, seed = 42)

Arguments

x

Covariate values; a small separable design when NULL.

y

Binary responses matching x.

length

Squared-exponential length scale, positive.

seed

Unused; kept for call compatibility.

Value

List with estimate (fitted probability at the last point), mode_probs, laplace_var_site0, separates, method.

Details

Formula: f <- f + 0.3 (K (y - p) - f) to the mode; Var(f_1) = ((K^(-1) + W)^(-1))_{11}.

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 11.7.5.

Examples

Ghosalgplaplace()
#> $estimate
#> [1] 0.6393151
#> 
#> $mode_probs
#> [1] 0.3606849 0.3955306 0.4536524 0.5463476 0.6044694 0.6393151
#> 
#> $laplace_var_site0
#> [1] 0.603688
#> 
#> $separates
#> [1] TRUE
#> 
#> $method
#> [1] "GP Laplace approximation (GvdV 2017 sec. 11.7.5)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"