# Likelihood of Gaussian Process for continuous data

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?