please bear with me, I'm really new to probability theory and inference, and I got stuck while pondering the following thought experiment. Reading through other similar questions confused me more than it helped. I hope that the notation that I used below is correct, please inform me if it's wrong.
Let's imagine I want to track my cat $C$ with a set of identical proximity sensors $i$ that I distributed uniformly along a 1D track. Each sensors activity depends on the distance of the cat from the sensor. I model this as a random variable $X_i \sim \mathcal{N}(0, \sigma^2)$: if the cat is on top of a sensor, the sensor's activity should indicate that the cat is there, but this activity drops of with distance from the sensor. To know where my cat is, I then simply look at the sensor $i$ which is maximally active, and look into my list where I wrote down the physical location of $i$, i.e. an inference or decoding problem.
Now, let's assume that I placed the sensors very densely due to their limited proximity range. This would give me a very high resolution of my cat's location. However, often I would like to get only a rough estimate. For instance, sensors $X_{1}$ and $X_{2}$ are in front of the fish tank (my cat likes to watch fish) and I'd like to know if she is there watching fish, but don't care about her "high-resolution location".
How would I model the fish tank variable $Z$ to inform me about if she's sitting there? My initial guess would be $Z = X_1 + X_2$, because somehow I need to integrate the sensor responses? I think it's not a mixture, because I don't randomly choose one of the two sensors over the other, but I'm not quite sure. I'm even more lost when thinking about how to infer the location. I'd appreciate any hints.
So you have a sensor which is a multiplication of Bernoulli Trial and something which says the probability of the cat being close to the sensor.
If the sensor detected the Cat - Bernoulli Trial with parameter $ p $.
The parameter $ p $ value is a function of the cat distance from the sensor (Which you can simulate by Gaussian Random Variable).
Now given measurements from few sensors you'll be able to build the PDF of the Cat location.
Then you can chose to chose the Mean (MMSE Estimation) of the Maximum of the PDF (MAP Estimation).