How can probabilistic graphical models improve decision-making in uncertain environments for autonomous vehicles?
Instruction: Explain how a graphical model represents sensor dependencies. In a toy binary-hazard model with prior probability 0.5, one alert has sensitivity 0.9 and false-positive rate 0.1. Compare the posterior after one alert with two alerts that are exact copies, and with a model that incorrectly assumes the copies are conditionally independent.
The second alert is an exact duplicate of the first, not a second independent observation. Treat the supplied probabilities as exact for the exercise; no real driving threshold follows.
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Worked answer
The prior odds are 1. One positive alert has likelihood ratio 0.9/0.1=9, so the posterior odds are 9 and the probability is 9/(1+9)=0.9.
An exact copy adds no information: the correct posterior remains 0.9. If a model wrongly treats the two copies as conditionally independent given the hazard state, it multiplies by 9 twice and reports 81/(1+81)=81/82, approximately 98.78%. The inflated confidence comes from a false dependency assumption, not stronger evidence.
A graphical model can represent latent states, observations and conditional dependencies explicitly. In a vehicle, weather, a shared clock or incorrect association can correlate observations. Separate hardware does not establish statistical independence, and the dependency structure itself requires evidence.
Use appropriate inference approximations and assess their error and computational cost. Validate predictive calibration and relevant event outcomes on held-out conditions, including common-cause faults. The posterior is an estimate under assumptions; it is not a guarantee of physical truth or a complete control decision.