Nonparametric binary regression with a Gaussian process prior

P(Y = 1 | x) = Phi(f(x)) with f ~ GP. The probit link destroys conjugacy, so the posterior is approximated at its mode: damped Newton on the log-posterior, which is the Laplace route the chapter's computation sections use throughout.

Usage

Ghosalnpbinaryreg(x, y, length = 0.7, var = 2)

Arguments

x

Design points.

y

Binary responses matching x.

length

Squared-exponential length scale, positive.

var

Kernel variance, positive.

Value

List with estimate (mean fitted probability), prob, f, method.

Details

Formula: with s_i = phi(f_i)/Phi(f_i) for y_i = 1 and -phi(f_i)/(1 - Phi(f_i)) otherwise, and W_i = s_i (s_i + f_i), solve (K^(-1) + W) df = s - K^(-1) f and step by df/2.

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 2.5.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Ghosalnpbinaryreg(V, V)
#> $estimate
#> [1] 0.8352044
#> 
#> $prob
#> [1] 0.8106383 0.8458119 0.8420112 0.8423563 0.8423563 0.8420112 0.8458119
#> [8] 0.8106383
#> 
#> $f
#> [1] 0.8802511 1.0186352 1.0027581 1.0041890 1.0041890 1.0027581 1.0186352
#> [8] 0.8802511
#> 
#> $method
#> [1] "probit-GP binary regression, Laplace MAP (GvdV 2017 sec. 2.5)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"