Finding the MAP estimate for an object's position

by tender   Last Updated October 20, 2019 03:19 AM

At an initial level I understand where MAP estimates come from. However, I'm having a problem determining how to find an optimal MAP estimate for a known prior, given data, and noisy range estimates... and I've been banging my head on my desk and I'm on my last leg here.

The problem is that I can come up with a MAP estimate function, but I can't seem to optimize it.

I found this post explaining a similar solution, albeit significantly more vaguely.

There's an object at a true position, $[x_T, y_T]^\top$ in a unit circle centered at the origin with radius 1, with known $K$ landmark coordinates evenly placed on the circle. The known ranges from the true value to the $K$ coordinates is $r_i = d_{Ti} + n_i$, where $d_{Ti} = ||[x_T, y_T]^\top - [x_i, y_i]^\top||$ and $n_i\sim \mathcal{N}(0, \sigma_i^2)$.

I am given the prior for any coordinate: $$P(\theta)=(2\pi|\Sigma|^{1/2})^{-1}e^{-\frac{1}{2}\theta^\top \Sigma^{-1}\theta}$$

where $\theta = [x, y]^\top$ and $\Sigma = \begin{bmatrix} \sigma_x^2 & 0\\ 0 & \sigma_y^2 \end{bmatrix}$.

In this situation, trying to find $\theta_{MAP}$ has led me to the equation $\text{argmax}_\theta P(\theta)P(R|\theta)$, where $R$ is list of range measurements and $P(R|\theta)\sim\mathcal{N}(d_{Ti}, \sigma_i^2)$.

This following equation is what I get, but I'm not able to find a closed form solution that gives a solution for $\theta$.

\begin{equation}\text{argmax}_\theta \left(-\log(2\pi\sigma_x\sigma_y) - \frac{\theta^\top\Sigma^{-1}\theta}{2}+\sum_{i=1}^K\left[ -\frac{1}{2}\log(2\pi\sigma_i^2)-\frac{(r_i - ||\theta - \theta_i||)^2}{2\sigma_i^2} \right]\right)\end{equation}

I assume it is possible, but how can $\theta_{MAP}$ be determined from this equation (assuming it's correct in the first place)?

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