2D MAP Estimation with a Uniform Prior

[Master1022]

Homework Statement

Two radar tracking stations provide independent measurements $x_1$ and $x_2$ of the landing site, $\mathbf{x} = (x, y)$, of a returning space probe. Both have Gaussian sensor models, $p(x_i|X_i ) = N(X_i, \Sigma_i)$. We also have a uniform prior, $U(x_p ; R)$, which is a constant within a distance $R$ of $x_p$ , and zero beyond. Determine a constraint on the location of the MAP estimate when $|x_{MLE} - x_p| > R$

Relevant Equations

MAP

Hi,

I was attempting the following question, but got confused on this part:

Question:
Two radar tracking stations provide independent measurements $x_1$ and $x_2$ of the landing site, $\mathbf{x} = (x, y)$, of a returning space probe. Both have Gaussian sensor models, $p(x_i|X_i ) = N(X_i, \Sigma_i)$. We also have a uniform prior, $U(x_p ; R)$, which is a constant within a distance $R$ of $x_p$ , and zero beyond. Determine a constraint on the location of the MAP estimate when $|x_{MLE} - x_p| > R$

Attempt:
I think the constraint is that the MAP estimation must lie on the radius of the circle (at a distance $R$ from $x_p$) because the prior suggests it would be impossible (at least the model thinks so) for the probe to have landed outside the circle.

However, I cannot think of a more rigorous to support this statement. I was told to consider iso-likelihood contours, but am not quite sure how that may be of help in this scenario.

Any help is greatly appreciated.

 

[KataKoniK]

Thank you for your question. I would approach this problem by considering the joint distribution of the measurements, $p(\mathbf{x}|x_1, x_2)$. This joint distribution can be written as the product of the individual sensor models and the prior: $p(\mathbf{x}|x_1, x_2) = p(x_1|\mathbf{x})p(x_2|\mathbf{x})U(\mathbf{x}; R)$.

Now, in order to find the MAP estimate, we need to maximize this joint distribution with respect to $\mathbf{x}$. This can be done by finding the point where the gradient of the log of the joint distribution is equal to zero. This point is known as the stationary point and is the MAP estimate.

When we do this calculation, we will find that the MAP estimate is indeed constrained to lie on the radius of the circle with a distance of $R$ from $x_p$. This can be seen by considering the derivatives of the log of the joint distribution with respect to $x$ and $y$. These derivatives will be zero at the point on the circle where the probe is most likely to have landed.

Furthermore, we can also consider the iso-likelihood contours of the joint distribution. These are curves that connect points with the same likelihood value. The shape of these contours will depend on the sensor models and the prior. However, we can see that for points outside the circle of radius $R$, the likelihood values will be very low, and the iso-likelihood contours will be far apart. This means that the probe is unlikely to have landed outside the circle, and the MAP estimate will be constrained to the circle.

I hope this helps to clarify the reasoning behind the constraint on the location of the MAP estimate. Please let me know if you have any further questions or need any additional clarification.