Predictive distribution of new responses — likelihood

Integrates a likelihood against a Gaussian distribution of the latent value, $\int p(y_* \mid f_*) N(f_* \mid \mu, \sigma^2) df_*$: the predictive distribution of a new response given the posterior of the latent function at its input.

Usage

likelihood_predictive(
  likelihood,
  mean,
  variance,
  observed = NULL,
  exposure = NULL,
  quadrature_order = 40L
)

Arguments

likelihood: A likelihood specification (gp_likelihoods).
mean, variance: Mean and variance of the latent value, one per prediction.
observed: Optional responses whose log predictive density is returned.
exposure: Optional positive exposure for Poisson likelihoods: one value or one per response.
quadrature_order: Number of Gauss-Hermite nodes for integrals without a closed form.

Value

A list with the predictive mean and variance of $y_*$, probability, $\Pr(y_* = 1)$, for Bernoulli likelihoods, and log_density of observed when it is given.

Details

Closed forms are used where they exist: the Gaussian likelihood adds its variance, the probit link gives $\Pr(y_* = 1) = \Phi(\mu / \sqrt{1 + \sigma^2})$, and the Poisson mean and variance are those of a Poisson with a lognormal rate, $E e^{\mu + \sigma^2 / 2}$ and $\mathrm{E}[y_*] + E^2 e^{2\mu + \sigma^2}(e^{\sigma^2} - 1)$. The logistic probability and the Poisson and logistic predictive densities use adaptive Gauss-Hermite quadrature (Liu and Pierce, 1994). With $\hat f$ the mode of $\psi(f) = \log p(y \mid f) + \log N(f \mid \mu, \sigma^2)$, found by Newton's method, and $s = (-\psi''(\hat f))^{-1/2}$, $$\int p(y \mid f) N(f \mid \mu, \sigma^2) df \approx \sqrt{2} s \sum_j w_j e^{t_j^2} e^{\psi(\hat f + \sqrt{2} s t_j)},$$ computed in logs. Centring the nodes on the integrand rather than on $\mu$ matters when the observation lies in the tail of the latent distribution, as for large counts.

Quadrature converges more slowly as $\sigma^2$ grows, because the logistic function has poles at $\pm i\pi$. With the default 40 nodes, the logistic probability is accurate to $10^{-10}$ for $\sigma^2$ up to 3; increase quadrature_order for wider latent distributions.

Stability

Experimental: this interface may change in a minor release, and every change is listed in NEWS. See gaussianprocesses-package for the policy.

References

Liu, Q. and Pierce, D. A. (1994). A note on Gauss-Hermite quadrature. Biometrika, 81(3), 624–629.

Examples

likelihood_predictive(bernoulli_likelihood("probit"), mean = c(-1, 0, 2),
  variance = c(0.5, 1, 4))$probability
#> [1] 0.2071081 0.5000000 0.8144533

likelihood_predictive(poisson_likelihood(), mean = log(3), variance = 0.2,
  observed = 4, exposure = 2)
#> $mean
#> [1] 6.631026
#> 
#> $variance
#> [1] 16.36622
#> 
#> $log_density
#> [1] -2.168429
#>