Take the following formula: $$ \mu_{ij}=\left(\alpha_i\Sigma_i^{-1}+\alpha_j\Sigma_j^{-1}\right)^{-1} \left(\alpha_i\Sigma_i^{-1}\mu_i+\alpha_j\Sigma_j^{-1}\mu_j\right). $$
Where the alphas are scalars, the sigmas are invertible matrices and the mus are vectors.
By defining:
$$ t = \frac{\alpha_i}{\alpha_i + \alpha_j} $$
And $s = \alpha_i + \alpha_j$
We can establish that $\alpha_i = st$ and $\alpha_j = (1-t)s$.
Using the above fact we can turn the first formula into:
$$ \mu_{ij}=\left(t\Sigma_i^{-1}+(1-t)\Sigma_j^{-1}\right)^{-1} \left(t\Sigma_i^{-1}\mu_i+(1-t)\Sigma_j^{-1}\mu_j\right). $$
What's important here is that both formulas are equal to each other. However, assuming that the only restrictions imposed on the variables are that $\alpha_i, \alpha_j >= 0$ and that $\alpha_i + \alpha_j$ > 0.
I find this result "impossible". The first equation has 2 independent variables, the value of $\alpha_i$ has nothing to do with the value of $\alpha_j$. This, in my head is a "2D manifold", it should be locally homeomorphic to the 2D plane.
The second equation however looks like a 1D manifold.
In other words, what makes me very confused here is, somehow the above formula seems to make 2 independent variables vary together such that they behave indistinguishably from a single variable. And I find this result very counter intuitive.
I hope the above conveys what I find confusing about this problem. And hopefully someone can enlighten me about it. I believe the result because the algebra doesn't lie, I am looking for a way to understand what's going on intuitively or visually.
A thing I just realized that is helping a bit is:
Consider $f(t) = t^2$ this is a 1D manifold, i.e it's one dimensional.
I can, if I want, define $t = x + y$ so $f(t) = (x + y)^2$. The result is a 2D manifold. However it's not that both formulas represent the same surface, the functions are different. The result is the same but the functions are different. One function is a set of ordered pairs where both values are scalars and the other is a set of order pairs where the input is a vector and the output is a scalar, i.e the 2 formulas above are NOT the same function, that's why one is a 2D manifold and the other is a 1D manifold.
Is that a correct way to think about it?