# Gaussian process with interval observations

The stochastic process $(X_t)_{t \in T}$ is a Gaussian process if the marginal distribution of $X_{t_1}, \ldots, X_{t_n}$ is a multivariate Gaussian distribution for all $t_1, \ldots, t_n \in T$. Let us write $\mathbb{E}[X_t] = \mu(t)$ and $\operatorname{Cov}(X_s, X_t) = \sigma(s, t)$ for the parameters and assume that $T \subset \mathbb{R}^d$.

In the usual setting we are given data $\mathcal{D} = (t_i, x_i)_{i = 1}^{n}$ and compute the (Gaussian) posterior distribution $\Pr(X_{t^*} = x^* | X_{t_1} = x_i, \ldots, X_{t_n} = x_n)$. I would be interested in a slightly more complicated form of observations, namely

$$\Pr(X_{t^*} = x^* | \forall t \in T_1\colon X_t \in [x_1, y_1], \ldots, \forall t \in T_n\colon X_t \in [x_n, y_n]) \text,$$

where each $T_i$ is a hyperrectangle, i.e. we are given some (possibly overlapping) hyperrectangles $T_i \subseteq T$ and observe that for each point in $T_i$, the value of the process lies in the interval $[x_n, y_n]$.

Is there any (exact or approximate, e.g. a variational bound) expression for the posterior distribution either in general or for specific choices of $\mu$ and $\sigma$? I would like some references or hints to try deriving it.

• Have you found any relevant references? I'm also interested in this topic. – Truong Apr 2 at 22:01
• Unfortunately not, but I am still very much interested myself. – Kristóf Marussy Apr 4 at 15:12