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I am working on a project that involves multiple sensors for measuring the one-dimensional position of a small car. Each sensor has its own error distribution, which is not necessarily Gaussian.

To be more specific, for example, I have two sensors, A and B:

Sensor A measures the car at position 3 with a certain known non-Gaussian error distribution.
Sensor B measures the car at position 4 with another known non-Gaussian error distribution.

If I know the specific error distributions for these two sensors, can I fuse them to get a more accurate estimate of the car's real position distribution by multiplying these two distributions together?

If this approach of cumulative multiplication is considered valid, could you please point me to any relevant literature or references?

I initially looked into Kalman filtering as it's a widely used technique for sensor data fusion. However, I found two key limitations with applying it to my problem:

Kalman filtering assumes Gaussian error distributions, but my sensors have known, non-Gaussian error characteristics. Kalman filtering is generally focused on estimating the expected value of the state (in my case, the position of the vehicle). However, I'm interested in the entire distribution of the vehicle's position to better understand the uncertainty involved.

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  • $\begingroup$ If I understand correctly, in your problem, sensors A and B provide error-prone measurements of a single latent variable ("car's position"). In this case, you might be dealing with the "repeated observations" case of error-in-variables models. See also this related Cross Validated question. $\endgroup$
    – Durden
    Sep 4, 2023 at 17:13
  • $\begingroup$ See stats.stackexchange.com/questions/344697/… for kalman with non_Gaussian errors, there is a reef in a comment there. Also stats.stackexchange.com/questions/369478/…, stats.stackexchange.com/questions/435825/… $\endgroup$ Sep 5, 2023 at 23:20
  • $\begingroup$ Thank you for your reply and for providing those links. While I've looked into them and they discuss issues around Kalman filtering with non-Gaussian errors, their focus seems to primarily be on obtaining a best point estimate. My question differs slightly in that I am actually interested in obtaining a more accurate or true distribution of the car's position, not just a best point estimate. $\endgroup$
    – wdq
    Sep 6, 2023 at 11:22
  • $\begingroup$ Correctly. In my problem, sensors A and B provide error-prone measurements for a single latent variable, which is the "car's position." Essentially, given the observations A and B, I know the distributions P(D∣A) and P(D∣B), and what I aim to compute is P(D∣A,B). The links you've provided are relevant, particularly the wiki page, but they don't completely correspond with my specific question. I'm currently reviewing the literature cited in the wiki for more insights. $\endgroup$
    – wdq
    Sep 6, 2023 at 11:29

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