Hello, I am fitting a spatio-temporal-and-informed-with-covariates bayesian model in Julia. I have a regression vector with prior distribution \vec{\beta}_t \sim \mathcal{N}_p(\vec{b_0},s^2I) (where p is the number of covariates, I the identity matrix). At every fitting iteration it has (as the other parameters of the model have) to be updated. The updating rule derive from the full conditional derivation, which corresponds to the following computations:

f(\vec{\beta}_t|-) \sim \exp \left\{ \vec{\beta}_t^T \underbrace{\left( \frac{1}{s^2}I + \sum_i \frac{\vec{x}_{it}\vec{x}_{it}^T}{\sigma^2_{it}}\right)}_{A_\star}\vec{\beta}_t -2 \underbrace{\left( \frac{\vec{b_0}}{s^2} + \sum_i \frac{\text{stuff}_{it} \vec{x}_{it}}{\sigma^2_{it}}\right)}_{\vec{b_\star}}\cdot \vec{\beta}_t \right\}

which is proportional to

\propto \exp \left\{ (\vec{\beta}_t -\vec{b}_\star)^T A_\star (\vec{\beta}_t -\vec{b}_\star)\right\}

which is the kernel of another multivariate gaussian with covariance matrix *the inverse* (sadly) of A_\star (you can see why the inverse from the picture).

That math translates into the following code:

```
sum_Y = zeros(p)
A_star = I(p)/s2_beta
for j in 1:n
X_jt = @view Xlk_covariates[j,:,t]
sum_Y += (Y[j,t] - muh_iter[Si_iter[j,t],t] - eta1_iter[j]*Y[j,t-1]) * X_jt / sig2h_iter[Si_iter[j,t],t]
A_star += (X_jt * X_jt') / sig2h_iter[Si_iter[j,t],t]
end
b_star = beta0/s2_beta + sum_Y
Am1_star = inv(Symmetric(A_star))
# Symmetric needed to solve an inversion and hermitian problem
# https://discourse.julialang.org/t/isposdef-and-eigvals-do-not-agree/118191/10
beta_iter[t] = rand(MvNormal(Am1_star*b_star, Am1_star))
```

which would work fine but for the inversion of the matrix required, which appears to be a major problem. Here is for example of an A_star matrix obtained during the fit and her inverse:

```
A_star = [89.53344398111983 18.907579544068415 -22.32917044622947 -26.313826364434874; 18.907579544068415 34.821884016362716 -18.103513531335015 -30.76131476315723; -22.32917044622947 -18.103513531335015 199.9111114786475 21.85563450347075; -26.313826364434874 -30.76131476315723 21.85563450347075 75.35950798333245]
# 89.5334 18.9076 -22.3292 -26.3138
# 18.9076 34.8219 -18.1035 -30.7613
# -22.3292 -18.1035 199.911 21.8556
# -26.3138 -30.7613 21.8556 75.3595
cond(A_star)
# 12.015947669881697, which doesn't seem too bad to me
A*inv(A_star)
# 0.130423 -0.0446254 0.00778601 0.0250668
# -0.0446254 0.47336 0.0190656 0.172111
# 0.00778601 0.0190656 0.0531558 -0.00491498
# 0.0250668 0.172111 -0.00491498 0.21313
```

So I was looking for some paths to avoid that inversion.