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Today’s Question
- In the last issue, we learned about the simple and intuitive weighted average method, but it has a core flaw: it cannot quantify ‘uncertainty’. Today, we introduce the Multi-Bayesian Estimation Method, which no longer simply believes in ‘who has more’, but tells us ‘how likely things are’ through probability. How does it quantify fusion decisions under ‘uncertainty’?
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One-Sentence Answer
The Multi-Bayesian Estimation Method is based on Bayes’ theorem, transforming multi-sensor data into probability distributions through the recursive process of ‘prior probability + likelihood function → posterior probability’. It utilizes posterior probability normalization (ensuring the total probability sums to 1) to update and fuse the independent probabilities of all sensors into a unified, quantified optimal estimate, thereby making robust decisions in the face of uncertainty.
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Detailed Interpretation
The core of the Multi-Bayesian Estimation Method is the multi-source extension of Bayes’ formula. To understand the Multi-Bayesian Estimation Method, one must first grasp its foundation—Bayes’ theorem.
- Bayes’ Theorem
The core idea of this theorem is: to use new ‘evidence’ (observational data) to update our original ‘belief’ (probability) about a certain ‘hypothesis’ (environmental state). Its classic formula is: Posterior Probability = (Likelihood Probability × Prior Probability) / Evidence Probability.We use the example of flipping a coin to understand these four concepts. Suppose your friend gives you a coin, and you want to use the coin flip to determine whether it is a fair coin (with a 50% chance of heads and 50% chance of tails).
- Prior Probability: Your initial belief about the ‘hypothesis’ before seeing evidence
When you first receive the coin, without any other evidence, you can only judge based on your common sense and past experience, believing that there is a 90% chance that this coin is fair and a 10% chance that it is not. This is the prior probability, which is your initial judgment based on common sense, experience, or historical information when there is no observational data (no coin flips).
- Likelihood Probability: The probability of the ‘evidence’ occurring if the hypothesis is true
You flip the coin 10 times, and it comes up heads 8 times. If the coin is indeed fair, the probability of this extreme result occurring is only about 4.4% (calculated using the binomial distribution, P(observation = 8 heads | fair coin) = C(10,8) × (0.5)^8 × (0.5)^2 = 45 × (1/256) × (1/4) ≈ 0.0439 (i.e., 4.39%)). If the coin is unfair, the probability of observing this result is about 30.2% (using the same calculation). This result indicates that if the coin is an unfair coin biased towards heads, the probability of getting 8 heads in 10 flips is significantly higher, providing stronger support for the ‘unfair’ hypothesis. This is the core of likelihood probability, which answers ‘given a certain hypothesis, how likely is the current result to occur?’—it does not concern itself with whether the hypothesis is true, only with the probability of the observed data occurring if the hypothesis is true.
- Evidence Probability: The total probability of the evidence occurring (the key to normalization)
Evidence probability is the total likelihood of the observational data occurring without considering any hypothesis. Essentially, it is the weighted sum of ‘likelihood probability × prior probability’ across all hypotheses (the law of total probability), and its core function is to serve as a ‘normalization factor’ to ensure that the sum of posterior probabilities equals 1, adhering to the basic rules of probability.Substituting the values for the two hypotheses: P(8 heads) = P(8 heads | fair) × P(fair) + P(8 heads | unfair) × P(unfair) = (0.0439 × 0.9) + (0.302 × 0.1) = 0.03951 + 0.0302 = 0.06971. This means that regardless of whether the coin is fair, the total probability of getting 8 heads in 10 flips is about 6.97%. This value will be used to adjust the result of ‘likelihood × prior’ to ensure that the final posterior probability is comparable.
- Posterior Probability: The updated belief
By substituting the prior, likelihood, and evidence into Bayes’ formula, we can obtain the ‘final belief corrected by the data of 8 heads’—the posterior probability, which is also the quantified result of ‘uncertainty’:Posterior probability of a fair coin:P(fair coin | 8 heads) = (likelihood × prior) / evidence ≈ (0.0439 × 0.9) / 0.0697 ≈ 0.0395 / 0.0697 ≈ 0.567 (i.e., 56.7%);Posterior probability of an unfair coin:P(unfair coin | 8 heads) = (0.3020 × 0.1) / 0.0697 ≈ 0.0302 / 0.0697 ≈ 0.433 (i.e., 43.3%).After updating, the probability of the coin being fair drops from 90% to 56.7%, while the probability of it being unfair rises from 10% to 43.3%. This demonstrates how Bayes’ theorem quantifies uncertainty and updates beliefs using evidence. Although we still have uncertainty about ‘whether the coin is fair’, it is now clearer than in the initial state (the initial gap was 80%, now only 13.4%).
- Multi-Bayesian Estimation Method: Extending to Multi-Sensor Fusion
The Multi-Bayesian Estimation Method treats each piece of sensor data as a piece of ‘coin flip evidence’, continuously refreshing the posterior distribution of the ‘environmental state’ through chained Bayesian updates; normalization ensures that all uncertainties are strictly quantified, and the final decision is made under the framework of ‘probability + risk’ to achieve optimal expectations, allowing for robust operation even amidst conflicting or noisy multi-source information.Advantages: (1) Evidence-based: It outputs a probability rather than just a value, clearly expressing the system’s ‘confidence level’ in the result; (2) Handling Uncertainty: It can naturally deal with sensor noise and environmental ambiguity; (3) Iterative Fusion: New sensor data can be seamlessly integrated, continuously optimizing the estimation results, making the framework very flexible; (4) Theoretical Rigor: It is supported by strict probability theory.Limitations: (1) High Computational Complexity: Especially when the dimensionality of state variables is high, calculating the complete probability distribution requires substantial computational resources; (2) Dependence on Accurate Models: It requires prior knowledge of the reliability model of the sensors (i.e., likelihood probability P(observation | state)); if the model is inaccurate, the results will also be biased; (3) Independence Assumption: Classical methods often assume that observations from different sensors are conditionally independent, which may not hold true in reality (e.g., heavy rain affecting both cameras and LiDAR simultaneously).The Multi-Bayesian Estimation Method provides a powerful probabilistic framework, but its conclusions are always a probability distribution— even in cases of severe information deficiency or conflict, it will ‘force’ an answer. However, in safety-critical autonomous driving, sometimes it is more important to clearly express ‘I don’t know’ than to provide an uncertain ‘optimal guess’. This is precisely why D-S evidence theory comes into play.
Next Issue Preview
In the next issue, we will explore: ‘Fundamentals of Multi-Sensor Fusion Algorithms (III): D-S Evidence Theory, How to Solve the Fusion Dilemma of “Uncertainty” and “Conflict”?’, stay tuned!
Feel free to leave comments:
- In what daily decisions do you unconsciously use similar ‘Bayesian updating’ thinking?
- What do you think is the key role of ‘normalization’ in probability fusion?
- If the classification probabilities of a camera and LiDAR for a target are completely opposite (one says 90% is a car, the other says 10% is a car), who do you think the fusion system should trust?
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Editor | Li Meifang
Reviewer | Liu Guochen