Prior Distributions For Tgp Mcmc

I am trying to understand the interface for tgp method, but much of it is not well documented (yes, despite reading through the package R manual, two nice vignettes, and a the PhD Thesis describing it.)

Just to get some consistent notation down: consider the case of a Gaussian process with a Gaussian/radial basis function kernel, parameterized as:

\[K(X, X') = \sigma^2 e^{\frac{(X-X')^2}{d}}\]

And observations from \(Z = f(X) + \varepsilon\), for which we seek to approximate \(f\) as a (multivariate) Gaussian process, \(Z | X ~ N(\mu, C)\). The covariance matrix \(C\) is given by our kernel \(K\), conditioned on our observations; the mean \(\mu\) is given by a constant or linear model, likewise conditioned on our observations. If the multivariate normal of observed and predicted points is,

\[\begin{align}\begin{pmatrix} y_{\textrm{obs}} \ y_{\textrm{pred}} \end{pmatrix} \sim \mathcal{N}\left( \mathbf{0}, \begin{bmatrix} cov(X_o,X_o) & cov(X_o, X_p) \ cov(X_p,X_o) & cov(X_p, X_p) \end{bmatrix} \right)\end{align}\]

the conditional probability is

\[x|y \sim \mathcal{N}(E,C)\] \[E = \operatorname{cov}(X_p, X_o) (\operatorname{cov}(X_o,X_o) + \varepsilon \mathbb{I}) ^{-1} y\] \[C= \operatorname{cov}(X_p, X_p) - \operatorname{cov}(X_p, X_o) (\operatorname{cov}(X_o,X_o)+ \varepsilon \mathbb{I} )^{-1} \operatorname{cov}(X_o, X_p)\]

Our hyperparameters are \(d\), \(\sigma^2\), and \(\varepsilon\), for which we need priors. We’ll use the inverse gamma,

\[P(x; a, g) = \frac{g^a}{\Gamma(a)} x^{-a - 1}\exp\left(-\frac{g}{x}\right)\]

The distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution if an uninformative prior is used; and as a useful conjugate prior if a less uninformative prior is preferred. However, it is common among Bayesians to consider an alternative parametrization of the normal distribution in terms of the precision, defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior.

  • \(X\) is X, \(Z\) is Z

  • \(\sigma^2\) is the parameter s2 in the tgp package, coming from inverse gamma prior with parameters \(a = a_0\) and \(g = g_0\), which are set by s2.p = c(a_0, g_0). Optionally, \(a_0\) and \(g_0\) can be taken from an exponential prior \(\lambda e^{-\lambda x}\), in which the value of \(\lambda\) is given by s2.lam = c(lambda_a, lambda_g). Setting s2.lam = "fixed" “turns off” the use of a hyperprior.

  • \(d\) I believe is d, which Grammarcy calls the range parameter (other terms include the “lengthscale”). I am not sure what either d or s2 correspond to if the covariance function corr is something other than the radial basis function. I’m also not clear how to set the correlation function to a strictly Gaussian covariance – corr is set to the exp family, which leaves the power as a parameter p_0:

\[K(X, X') = e^{\frac{-(X-X')^p_0}{d}}\]

(from equation 4 in the Vignette (1)). The parameter p_0 is fixed at 2, and not treated as a hyperparameter.

The length-scale comes from a hyperparameter that is the sum (mixture) of two Gamma distributions, with prior parameter c(a1,g1,a2,g2) where g1 and g2 are scale (1/rate) parameters. As before, these parameters c(a1, g1, a2, g2) can be drawn from an exponential distribution hyperprior whose \(\lambda\) values are given by d.lam, or turned off.

  • \(\varepsilon\) is nug. It seems to have same shape prior as \(d\), specified by c(a1,g1,a2,g2), and optional hyperprior as well. Additionally it can be fixed to a constant by setting nug.p = 0 and specifying the staring value as gd[1].

  • The mean is not fixed at zero but is instead a linear model, we need hyperparameters for it as well. The key parameter there is the slope \(\beta\). The functional form for it’s prior is specified as bprior, which takes values bflat, b0, bmzt, or bmznot. A sensible choice is as a Gaussian b0, as shown in Equation 1 of the vignette

\[ \begin{align} Z | \beta, \sigma^2, K &\sim N_n(\mathbf{F} \beta, \sigma^2 \mathbf{K}) \\ \beta | \sigma^2, \tau^2, \mathbf{W}, \beta_0 &\sim N_m(\beta_0, \sigma^2 \tau^2 \mathbf{W} ) \\ \beta_0 &\sim N_m(\mathbf{\mu}, \mathbf{B}) \end{align} \]

Example comparison

Say, given some observed data pairs \(x,y\) and predictions desired on a grid \(X\),

X = seq(-5, 5, length=50)
x = c(-4, -3, -1,  0,  2)
y = c(-2,  0,  1,  2, -1)

Manually I could just do:

d = .5; epsilon = .1; sigma = 1; #fixed hyperparamaters
SE <- function(Xi,Xj, d) sigma * exp(-0.5 * (Xi - Xj) ^ 2 / d ^ 2)
cov <- function(X, Y) outer(X, Y, SE, d) 
cov_xx_inv <- solve(cov(x, x) + epsilon * diag(1, length(x)))
Ef <- cov(X, x) %*% cov_xx_inv %*% y
Cf <- cov(X, X) - cov(X, x)  %*% cov_xx_inv %*% cov(x, X)

And Ef and Cf are the estimated (krieging) mean and covariance.


Note: it appears that the mixed gammas for d and nug actually treat the second argument, g1 and g2, as the rate instead of the scale, so I invert the desired scale:

s2.p <- c(50,50)
tau2.p <- c(20,1)
d.p = c(10, 1/0.01, 10, 1/0.01)
nug.p = c(10, 1/0.01, 10, 1/0.01)

Define as curves and plot the priors:

s2_prior <- function(x) dinvgamma(x, s2.p[1], s2.p[2])
tau2_prior <- function(x) dinvgamma(x, tau2.p[1], tau2.p[2])
d_prior <- function(x) dgamma(x, d.p[1], scale = d.p[2]) + dgamma(x, d.p[3], scale = d.p[4])
nug_prior <- function(x) dgamma(x, nug.p[1], scale = nug.p[2]) + dgamma(x, nug.p[3], scale = nug.p[4])
beta0_prior <- function(x, tau) dnorm(x, 0, tau)

xx <- seq(.0001, 5, length.out=100)
priors <- data.frame(x = xx, s2 = s2_prior(xx), tau2 = tau2_prior(xx), beta0 = beta0_prior(xx, 1), nug = nug_prior(xx), d = d_prior(xx))
priors <- melt(priors, id="x")
ggplot(priors) + geom_line(aes(x, value)) + facet_wrap(~variable, scale="free")
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gp <- bgp(X=x, XX=X, Z=y, verb=0,
          meanfn="constant", bprior="b0", BTE=c(2000,6000,2), m0r1=FALSE, 
          corr="exp", trace=TRUE, beta = 0,
          s2.p = s2.p, d.p = d.p, nug.p = nug.p,
          s2.lam = "fixed", d.lam = "fixed", nug.lam = "fixed", 
          tau2.lam = "fixed", tau2.p = tau2.p)

Extract the posterior Gaussian process mean and the \(\pm 2\) standard deviations over the predicted grid from the fit:

V <- gp$ZZ.ks2
tgp_dat <- data.frame(x   = gp$XX[[1]], 
                  y   = gp$, 
                 ymin = gp$ - 1.96 * sqrt(gp$ZZ.ks2), 
                 ymax = gp$ + 1.96 * sqrt(gp$ZZ.ks2))

and likewise for our manual data:

manual_dat <- data.frame(x = X, 
                         y = Ef, 
                         ymin = (Ef - 1.96 * sqrt(diag(Cf))), 
                         ymax = (Ef + 1.96 * sqrt(diag(Cf))))

Compare the GP posteriors:

ggplot(tgp_dat) +
    geom_ribbon(aes(x, y, ymin = ymin, ymax = ymax), fill="red", alpha = .1) + # Var
    geom_line(aes(x, y), col="red") + # mean
    geom_point(data = data.frame(x = x, y = y), aes(x, y)) + # raw data
    geom_ribbon(dat = manual_dat, aes(x, y, ymin = ymin, ymax = ymax), fill = "blue", alpha = .1) + # Var
    geom_line(dat = manual_dat, aes(x, y), col = "blue")  + #MEAN    
    theme_bw() + theme(plot.background = element_rect(fill = "transparent",colour = NA))
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Posteriors of hyperparameters

hyperparameters <- c("index", "s2", "tau2", "beta0", "nug", "d", "ldetK")
posteriors <- melt(gp$trace$XX[[1]][,hyperparameters], id="index")
ggplot(posteriors) + geom_histogram(aes(value)) + facet_wrap(~variable, scales="free")
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