Combining Multivariate Normal Variables

Question

Consider a case where I have $n$ datasets and from each data set I am estimating a multivariate normal distribution denoted as $$\textbf{y}_i=[y_{i1},y_{i2},y_{i3}]^t \sim N \Big( [a_{i1},a_{i2},a_{i3}]^t, \Sigma_i \Big) $$

Note that $\textbf{y}_i$ independent of $\textbf{y}_j$

Is it correct to average the predictions as follows $$\textbf{y}=\frac{1}{n}\sum\limits_{i=1}^n \textbf{y}_i \sim N \Big( \Big[\frac{1}{n}\sum\limits_{i=1}^na_{i1},\frac{1}{n}\sum\limits_{i=1}^na_{i2},\frac{1}{n}\sum\limits_{i=1}^na_{i3}\Big]^t, \frac{1}{n^2} \sum\limits_{i=1}^n \Sigma_i \Big)$$

where the summation for the covariance is element wise

Also is there a better way to average the predictions ?

Answer 1

"Is it correct to average the predictions as follows". Yes, it is correct.

One way to prove it is writing all of $y$ in a vector, $Y=(y_{11}, y_{12}, y_{13},...,y_{n1}, y_{n2}, y_{n3})^t$. Let $$A=\frac 1 n \begin{pmatrix}1&0&0&...&1&0&0\\ 0&1&0&...&0&1&0\\ 0&0&1&...&0&0&1 \end{pmatrix}$$

Then your $y$ can be expressed as$$y=AY$$ We have $$y \sim N(A\mathrm{E}(Y), A\mathrm{Var}(Y)A^t)$$ which is the same as yours after some simplifications.

I do not understand your last question. Average is just average and there is no better way.