Update docstring

Author: simsurace

src/expectations.jl

@@ -29,29 +29,18 @@ default_expectation_method(_) = GaussHermiteExpectation(20)
         y::AbstractVector,
     )
 
-This function computes the expected log likelihood:
-
-```math
-    ∫ q(f) log p(y | f) df
-```
-where `p(y | f)` is the process likelihood. This is described by `lik`, which should be a
-callable that takes `f` as input and returns a Distribution over `y` that supports
-`loglikelihood(lik(f), y)`.
-
-`q(f)` is an approximation to the latent function values `f` given by:
+This function computes the sum of the expected log likelihoods:
 ```math
-    q(f) = ∫ p(f | u) q(u) du
+    ∑ᵢ ∫ qᵢ(f) log pᵢ(yᵢ | f) df
 ```
-where `q(u)` is the variational distribution over inducing points.
-The marginal distributions of `q(f)` are given by `q_f`.
+where `pᵢ(yᵢ | f)` is the process likelihood and `yᵢ` the observation at location/site `i`.
+The argument `q_f` is the vector of normal distributions `qᵢ(f)`, and the argument `y` is
+the vector of observations `yᵢ`. The argument `lik` is one of the following:
+- A callable that takes `f` as input and returns a Distribution over `y` that supports `loglikelihood(lik(f), y)`. This corresponds to the case where `pᵢ` is independent of `i`.
+- A vector of such callables. Here, each element of the vector is the corresponding `pᵢ`.
 
 `quadrature` determines which method is used to calculate the expected log
 likelihood.
-
-# Extended help
-
-`q(f)` is assumed to be an `MvNormal` distribution and `p(y | f)` is assumed to
-have independent marginals such that only the marginals of `q(f)` are required.
 """
 expected_loglikelihood(quadrature, lik, q_f, y)