Model The single-index model uses Gaussian process with zero mean and and covariance kernel \(\eta \cdot \text{exp}(-\frac{(t_i-t_j)^2}{l})\) as a link function, where \(t_i, t_j, j = 1, \ldots, n\) is index value.
Index vector should be length 1.
Priors
von Mises–Fisher prior on the index \(\theta\) with direction and concentration.
Covariance kernel: Amplitude, \(\eta\), follows log normal distribution with mean \(a_\eta\) and variance \(b_\eta\).
Length-scale parameter follows gamma distribution with shape parameter \(\alpha_l\) and rate parameter \(\beta_l\).
Inverse-Gamma prior on \(\sigma^2\) with shape parameter \(a_\sigma\) and rate parameter \(b_\sigma\).
Sampling In the fully Bayesian approach, \(\theta\), \(l\), and \(\eta\)
are updated via the Metropolis–Hastings algorithm, while \(f\) and
\(\sigma^2\) are sampled using Gibbs sampling.
In the empirical Bayes approach, \(\theta\), \(l\), \(\eta\),
and \(\sigma^2\) are estimated by maximum a posteriori (MAP), and
\(f\) is sampled from its full conditional posterior distribution.
In the empirical Gibbs sampler, \(\theta\), \(l\), and \(\eta\)
are estimated by MAP, whereas \(f\) and \(\sigma^2\) are sampled
via Gibbs sampling.
For estimation via MAP, effective sample size or potential scale reduction factor is meaningless.
Prior hyper-parameters
These are the prior hyper-parameters set in the function. You can define new values for each parameter in prior_param.
Index vector: Nothing to assign.
Link function:
Length-scale:Gamma distribution is assigned for length-scale parameter, \(l\).
link_lengthscale_shape is shape parameter (default 1/8) and link_lengthscale_rate is rate parameter of lengthscale. (default 1/8)
Amplitude: Log-normal distribution is assigned for amplitude parameter, \(\eta\).
link_amp_a is mean (default -1), and link_amp_b is variance. (default 1)
Error variance (sigma2): inverse gamma prior is assigned to \(\sigma^2\)
where sigma2_shape is shape parameter and sigma2_rate is rate parameter of inverse gamma distribution.
(default sigma2_shape = 1, sigma2_rate = 1)
Initial values
These are the initial values set in the function. You can define new values for each initial value in init_param.
Index vector (index): Initial unit index vector. By default, vector is randomly drawn from normal distribution and standardized.
Link function: link_lengthscale is initial scalar length-scale parameter. (default: 0.1)
link_amp is initial scalar amplitude parameter. (default: 1)
Error variance (sigma2): Initial scalar error variance. (default: 1)