# Under which probabilistic assumptions is this estimation approach correct?

trying to tackle a problem, I ended up having built a model $$p(\boldsymbol{x}_{i}|\boldsymbol{y}_{i,j}, \boldsymbol{x}_{j})$$ which gives me for different $$\boldsymbol{y}_{i,j}$$ observations made by $$\boldsymbol{x}_{j}$$, a new PDF of $$\boldsymbol{x}_{i}$$. My ultimate goal is to obtain the best estimate for it, $$\hat{\boldsymbol{x}_{i}}$$, given my $$\boldsymbol{y}_{i,j}$$ observations. The most typical approach I have seen being used (from others who faced the same problem and made other models) is the:

$$$$\widehat{\boldsymbol{x}_{i}}=\arg \max_{\boldsymbol{x}_{i}} f(\boldsymbol{x}_{i}) \prod_{j \in \mathcal{O}} f(\boldsymbol{y}_{i,j} \mid \boldsymbol{x}_{i}, \boldsymbol{x}_{j})$$$$

However, I tried using instead simply the following and it worked just fine:

$$$$\widehat{\boldsymbol{x}_{i}}=\arg \max_{\boldsymbol{x}_{i}} \prod_{j \in \mathcal{O}} p(\boldsymbol{x}_{i} \mid \boldsymbol{y}_{i,j}, \boldsymbol{x}_{j})$$$$

I would like to ask why is that happening? What does the fact that both appear to work, suggest about their relationship? I did not mention the underlying assumptions (because I am not quite sure), therefore, I would like some help in identifying mathematically those.

*Some further information regarding $$\boldsymbol{x}_{i}$$, $$\boldsymbol{y}_{i,j}$$, $$\boldsymbol{x}_j$$ and their relation:

$$\boldsymbol{x}_{i}$$ is the position of some node in 3D space. Assume that this node is emitting one single impulse signal. $$\boldsymbol{y}_{i,j}$$ is the measurement of that signal from another node (namely node $$j$$), whose position is "known" to us. A measurement depends only on the distance between the transmitting and the receiving node (the model of this dependency is the same for all receiving nodes). Therefore, given some measurement $$\boldsymbol{y}_{i,j}$$ and the position of the receiver $$\boldsymbol{x}_j$$, there is a distribution about where $$\boldsymbol{x}_{i}$$ is.

The following graph depicts my system where my actual goal is to find the best position estimation for all nodes $$\boldsymbol{X}$$ (their positions are independent to each other) because, in fact, I do not have any prior knowledge about them (the knowledge gets built iteratively via my optimization process).

So in fact, after optimizing $$\boldsymbol{x}_{i}$$, I continue optimizing iteratively one by one all the rest of $$\boldsymbol{x}_{j}$$'s (which become the new $$\boldsymbol{x}_{i}$$ on each step) using the previous estimations of the $$\boldsymbol{x}_{j}$$'s when available (else I am using a random position). I have practically seen that this method converges to a correct solution where nodes are in a relative reference system (since my initial positions are random) placed.

• How are $x$ and $Y$ related? Are both random? What are the $f$s in your notation and how they relate with the $p$s? At any rate, $f(x) f(y_i \mid x)$ is arguably the joint distribution $f(x, y_i)$ and so, the likelihood principles makes it preferable over $p(x \mid y_i)$ (which is the density of $x$ given $Y = y_i$?). Jan 6, 2022 at 18:55
• Note that $p(x \mid y_i) \propto f(x, y_i)$ where the proportional constant is positive and "universal" (relative to $x$). So, the two maximisations should result in the same $\hat x.$ Did you get the same $\hat x$? Jan 6, 2022 at 19:00
• what is the function f? Jan 7, 2022 at 20:07
• f(x) I think is the prior belief about x position
– Gouz
Jan 7, 2022 at 20:53
• I made a change to also reflect the position of the observing node, because I think I had provided the formulations in a wrong way. Does this make some more sense?
– Gouz
Jan 7, 2022 at 21:01