# Bayesian data combination?

I have $$N$$ sensors, where $$N$$ is typically of order 10. An object comes in, and each sensor activates.

Each sensor measures up to $$M$$ properties of a given object (the same set of possible measurements for all sensors) where $$M$$ is around 3. Some give back no measurements on a particular object, while others give all $$M$$, and some in between. (For example, most but not all measure the mass of the object.)

Each measurement from each sensor consists of a value, and an estimate of the measurement uncertainty. These can be assumed to be corresponding to a normal distribution. The different measurements from each sensor are correlated in a known way.

Some of the sensors randomly malfunction and give back noise. Some of the sensors are biased and give back poor estimates (wrong by more than their estimated uncertainty). Some sensors are considered highly accurate, but may not always activate. Some of the sensors use information from other sensors (and it is always known when and how they do this), so some of the sensor measurements are correlated.

I want to combine the data across all the sensors to get the best estimates of the $$M$$ measurements and their uncertainties. What is the best way to accomplish this?

Let's consider one sensor first; by using the Bayes theorem, one gets $$p(\vec{\theta} | d_i) \propto p(d_i | \vec{\theta}) p(\vec{\theta}),$$ where $$d_i$$ is the data obtained from the $$i$$-th sensor, with $$i=0, 1, 2, \dots, N-1$$. And $$\vec{\theta}$$ is the parameter of interest with a dimension of $$M$$.
To combine the data from multiple detections, again, one can use the Bayes theorem and get $$p(\vec{\theta} | \{d_i\}) \propto p(\{d_i\} | \vec{\theta}) p(\vec{\theta}),$$ where $$\{d_i\} = \{d_0, d_1, \dots, d_{N-1}\}$$. Because the data obtained from different sensors are correlated (in a known way), the likelihood is then expanded to be $$p(\{d_i\} | \vec{\theta}) = p(d_0 | \vec{\theta}, \{d_1, d_2 \dots\})p(d_1 | \vec{\theta}, \{d_2, d_3, \dots\})\dots p(d_{N-1} | \vec{\theta}).$$ In which the information describing the correaltion between sensors is encoded with in the likelihood function $$p(d_k | \vec{\theta}, \{d_j, d_{j+1} \dots\})$$.
For the randomly biased sensors, I would take a more ad-hoc approach to compare the outcomes between sensors. And quantify the coherence between sensors as the goodness of data. And select the data that go beyond a certain threshold for further analysis. Of course, this would induce selection bias to the analysis which requires correction as in the prior $$p(\vec{\theta})$$.