Nonparametric Poisson regression with a Gaussian process prior

Y | x ~ Poisson(exp f(x)) with f ~ GP. The log link keeps the intensity positive without constraining f, and as in the binary case the price is the loss of conjugacy, so the posterior mode is found by damped Newton on the log-posterior.

Usage

Ghosalnppoissonreg(x, y, length = 0.7, var = 1)

Arguments

x

Design points.

y

Non-negative counts matching x.

length

Squared-exponential length scale, positive.

var

Kernel variance, positive.

Value

List with estimate (mean intensity), intensity, f, method.

Details

Formula: solve (K^(-1) + diag(lambda)) df = (y - lambda) - K^(-1) f and step by df/2, with lambda = exp(min(f, 30)).

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 2.6.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Ghosalnppoissonreg(V, V)
#> $estimate
#> [1] 3.789453
#> 
#> $intensity
#> [1] 1.065621 1.666354 2.476025 3.307161 4.188774 5.036555 6.102965 6.472169
#> 
#> $f
#> [1] 0.06355737 0.51063775 0.90665441 1.19609002 1.43240800 1.61672238 1.80877467
#> [8] 1.86751134
#> 
#> $method
#> [1] "log-link GP Poisson regression MAP (GvdV 2017 sec. 2.6)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"