Finding most probable vector, given angles Assuming you have a series of known vectors $k_i$ in $\mathbb{R}^N$, each at an angle $\theta_i$ to a single unknown vector $v$.  All vectors have the same dimension $N$ which can be arbitrarily higher than the number of known vectors.
What is the best way to determine the expected / most probable value for $v$?
Additionally, let's say you have constraints on what $v$ is (std dev., mean, bounds, etc.), what would be the best way to determine a probabilistic value for $v$?
I'm tried pretty much all the scipy optimizers, but the results haven't been particularly impressive.   I've also tried to look at this from a purely math perspective, such as the intersection of subspaces, but the notes in this area rarely talk about probability or expected values and I get lost quickly and I am not sure how to approach it.
 A: We can convert this into a standard problem and you can go on from there.  To begin the conversion I will slightly change the notation into a conventional one in statistics.
Let $\beta = (\beta_1,\beta_2,\ldots, \beta_p)^\prime$ be the unknown vector.  You have $n$ observations of the angles $\theta_i$ made between $\beta$ and $n$ known $p$-vectors $x_1,x_2,\ldots,x_n.$  Because angles give no information about magnitudes, we can only hope to recover $\beta$ up to a positive multiple.  This allows us to assume that, say, $||\beta||=1.$  In that case the inverse cosine of the angle with $x_i = (x_{i1}, x_{i2}, \ldots, x_{ip})$ multiplied by the length $||x_i||$ gives the dot product,
$$y_i = \arccos(\theta_i) ||x_i|| = x_i\cdot \beta = \sum_{j=1}^p x_{ij}\beta_j .$$
Arranging the vectors as rows of a design matrix $X$ yields the relation
$$X\beta = y.$$
This is a standard underdetermined ("$n \ll p$") regression problem, subject to the constraint $||\beta||=1.$  However, even this constraint doesn't matter, because if you find any nonzero solution $\hat\beta$ it still works: remember, we cannot hope to determine $||\beta||$ in the first place.
Various methods exist to estimate $\beta,$ including Bayesian methods, Ridge Regression, and the Lasso.
