I need to calculate the likelihood $L(\theta)$ of the Gaussian process:
$ y(t) \sim GP(m(t), K((t,t')|\theta)) $
where $m(t)$ is the mean function (this will be zero for my purposes) and $K((t,t')|\theta)$ the covariance function. If the interval $t$ is $[0,1]$ and for points $t_i,\ i = 1..n$ we have a time series of observations $y_i$, the likelihood can be obtained as the PDF of a multivariate normal distribution. However, the calculation becomes very expensive for large $n$ as it requires the calculation of $inv(K)$ and $det(K)$. This is done as part of a Monte Carlo simulation, making it necessary to speed up this calculation as much as possible. I know there are methods to make this less costly, but if possible I would like to avoid having to discretize the time series altogether.
Is there a way to calculate the likelihood for continuous data, i.e. for $y(t)$ being a continuous function over the interval [0,1] for a fully specified GPR? Drawing an analogy to a simpler case, to illustrate what I want to do:
If I have a realization $x$ of a random variable $X$ from a fully specified univariate distribution, the likelihood is the evaluation of the probability density function for that $x$. Is it possible to do something similar for the case of a GP with continuous measurements?