M3-T2 — Bayes' Rule: Prior, Likelihood and Posterior

Learning objectives

After this lesson, you should be able to:

• Name the four parts of Bayes’ rule (prior, likelihood, evidence, posterior) and say what each means physically.
• Derive Bayes’ rule from the product rule and compute a posterior for a detector with missed detections and false alarms.
• Explain the base-rate effect: why a reliable detector can give a low posterior when its target is rare.
• Explain why a likelihood is not a probability of the hypothesis, and why a zero prior can never be overcome.

The intuition

Rover's camera looks for the dock marker. On a frame where the marker is in view, the detector fires with probability 0.8. On a frame where it is not, it still fires with probability 0.1 (a poster, a reflection). So the detector is right most of the time. Now Rover is driving down a corridor, where the marker is in view on only 5% of frames, and the detector fires. How likely is it that the marker is really there?

Most people say about 80%. The correct answer is about 30%. The reason is easy to see by counting. Imagine 10 000 frames from the corridor. In 500 of them (5%) the marker is in view, and the detector fires on 400 of those. In the other 9 500 frames the marker is not in view, and the detector fires on 10% of them: 950 false alarms. Of the 1 350 times the detector fires, only 400 are real.

This is Bayes' rule in words: the belief after a detection depends on two things, how well the hypothesis explains the detection (a real marker makes it likely, a missing one makes it unlikely) and how plausible the hypothesis was before. A rare event needs strong evidence. The curves show the other side of the same fact: the same detector gives 0.89 where the prior is 0.5 and only 0.07 where it is 0.01.

The four parts, with their physical meaning

Compare the two directions again. The likelihood answers “what does this sensor do in this world?”, which can be measured by experiment: put a marker in view a thousand times and count. The posterior answers “what is this world, given what the sensor did?”, which is what the robot has to act on, and which cannot be measured directly because the robot does not know the truth. Bayes' rule is the bridge, and the prior is the toll.

Posterior is proportional to prior times likelihood. Multiply how plausible the hypothesis was by how well it explains the reading, then rescale so the beliefs sum to 1.

Definitions

Hypothesis $H$

A statement about the world that the robot is trying to decide about: “the marker is present”, “the robot is in cell $i$”, “this cell is occupied”. With several candidate hypotheses $H_1, H_2, \dots$ they must be exclusive and exhaustive.

Data $z$

What the sensor reported. It is known, and fixed, when Bayes’ rule is applied.

Prior $P(H)$

The probability of the hypothesis before the data.

Likelihood $P(z \mid H)$

The probability (or density) of the data if the hypothesis were true. As a function of $H$ it is called the likelihood function; it does not have to sum to 1 over the hypotheses.

Evidence $P(z)$

The overall probability of the data, $\sum_{H} P(z \mid H)P(H)$. Also called the normalizer or marginal likelihood.

Posterior $P(H \mid z)$

The probability of the hypothesis given the data.

Derivation: Bayes' rule and the base-rate effect

The rule itself

1. $$P(H, z) = P(z \mid H)\, P(H)$$

   The product rule, writing the joint with the hypothesis first.

2. $$P(H, z) = P(H \mid z)\, P(z)$$

   The product rule again, with the data first. It is the same joint, because a joint does not care in which order its two events are named.

3. $$P(H \mid z)\, P(z) = P(z \mid H)\, P(H)$$

   Two expressions for the same number, so they are equal.

4. $$P(H \mid z) = \dfrac{P(z \mid H)\, P(H)}{P(z)}$$

   Divide by $P(z)$ (the data actually happened, so it is not zero). This is Bayes’ rule.

5. $$P(z) = \sum_{i} P(z, H_i) = \sum_{i} P(z \mid H_i)\, P(H_i)$$

   Marginalize the hypothesis (the law of total probability). It shows that the evidence is not a new quantity: it is the numerator summed over every hypothesis, which is why normalizing always works.

Why a rare target gives a low posterior

For a yes-or-no hypothesis with prior $p$, detection rate $d = P(z \mid H)$ and false-alarm rate $f = P(z \mid \neg H)$:

1. $$P(H \mid z) = \dfrac{d\,p}{d\,p + f\,(1-p)}$$

   Bayes’ rule, with the evidence written out by total probability over $H$ and $\neg H$.

2. $$P(H \mid z) = \dfrac{\dfrac{d}{f}\cdot\dfrac{p}{1-p}}{\dfrac{d}{f}\cdot\dfrac{p}{1-p} + 1}$$

   Divide the top and bottom by $f(1-p)$. The result depends only on two ratios: the likelihood ratio $d/f$ (how much more likely the reading is if the hypothesis is true) and the prior odds $p/(1-p)$.

3. $$p \ll 1 \;\Rightarrow\; P(H \mid z) \approx \dfrac{d}{f}\, p$$

   When the prior is small the posterior is about the likelihood ratio times the prior. With $d/f = 8$ a prior of 0.01 becomes about 0.08, never anywhere near 0.8.

4. $$p = 0 \;\Rightarrow\; P(H \mid z) = 0 \quad\text{for every } z$$

   A zero prior stays zero whatever the data say, because the posterior is proportional to the prior. A hypothesis assigned zero probability can never be recovered by any amount of evidence.

The second line is the odds form of Bayes' rule, and M3-T4: Recursive Bayesian Updating builds the recursive filter on it. For now note what it says: a detector's usefulness is its likelihood ratio, and a good ratio can only multiply the odds the robot started with.

Posterior odds = likelihood ratio × prior odds. A reading changes belief by a fixed factor, so a rare hypothesis stays improbable after one reading and becomes certain only after several.

A worked example

Rover's marker detector fires on a frame with probability $d = 0.8$ when the marker is in view and $f = 0.1$ when it is not. In the corridor the marker is in view on 5% of frames.

1. By counting. Of 10 000 frames, 500 have the marker; the detector fires on 400 of them. Of the other 9 500, it fires on 950. A detection is real in $400/(400+950) = 29.6\%$ of cases.
2. By Bayes' rule. Numerator: $P(\text{fires} \mid \text{marker})\,P(\text{marker}) = 0.8 \times 0.05 = 0.04$. Evidence: $0.04 + 0.1 \times 0.95 = 0.135$. Posterior: $0.04/0.135 = 0.2963$. Identical to the count.
3. Change the prior. Near the dock, where the marker is in view half the time ($p = 0.5$), the same detection gives 0.8889. In a long corridor ($p = 0.01$), 0.0748. The detector has not changed, the robot's situation has.
4. Use a no-detection too. If the detector does not fire at $p = 0.05$, the likelihoods are $1-d = 0.2$ and $1-f = 0.9$, and the posterior falls to $0.2 \times 0.05 / (0.2 \times 0.05 + 0.9 \times 0.95) = 0.0116$. Silence is also evidence.
5. A better detector. Replace the single frame by the 3-of-5 vote of the previous module (detection 0.9421, false alarm 0.0086). At $p = 0.05$ a vote that fires gives $0.9421 \times 0.05 / (0.9421 \times 0.05 + 0.0086 \times 0.95) = 0.8522$. Cutting the false-alarm rate by a factor of ten moved the posterior from 0.30 to 0.85.
6. Several hypotheses. Recall the three rooms of the last lesson, with prior (0.3, 0.3, 0.4) and the light sensor's probabilities of reading “bright” in each room, (0.667, 0.167, 0.125). The products are (0.20, 0.05, 0.05), their sum 0.30 is the evidence, and the posterior is (0.667, 0.167, 0.167): the same answer the joint table gave. Note that the three likelihoods sum to 0.958, not 1: they are not a probability distribution over the rooms.

from robotprob import posterior_binary, bayes_update

d, f = 0.8, 0.1                      # P(fires | marker), P(fires | no marker)
prior = 0.05
evidence = d * prior + f * (1 - prior)
print('evidence P(fires)        =', round(evidence, 4))
print('posterior, prior 0.05    =', round(posterior_binary(prior, d, f), 4))
print('posterior, prior 0.5     =', round(posterior_binary(0.5, d, f), 4))
print('posterior, prior 0.01    =', round(posterior_binary(0.01, d, f), 4))
print('no detection, prior 0.05 =', round(posterior_binary(prior, 1 - d, 1 - f), 4))
print('natural frequencies      =', 10000 * prior * d, 'real,', 10000 * (1 - prior) * f, 'false')

d3, f3 = 0.9421, 0.0086              # the 3-of-5 vote of the previous module
print('3-of-5 vote, prior 0.05  =', round(posterior_binary(prior, d3, f3), 4))

post = bayes_update([0.3, 0.3, 0.4], [2 / 3, 1 / 6, 0.125])     # three rooms, reading "bright"
print('rooms after bright       =', [round(v, 4) for v in post])
print('sum of the likelihoods   =', round(2 / 3 + 1 / 6 + 0.125, 4))

evidence P(fires)        = 0.135
posterior, prior 0.05    = 0.2963
posterior, prior 0.5     = 0.8889
posterior, prior 0.01    = 0.0748
no detection, prior 0.05 = 0.0116
natural frequencies      = 400.0 real, 950.0 false
3-of-5 vote, prior 0.05  = 0.8522
rooms after bright       = [0.6667, 0.1667, 0.1667]
sum of the likelihoods   = 0.9583

In robotics

Detections must be read against how common the target is

A person detector that fires correctly on 99% of people and falsely on 1% of empty frames looks excellent. In a warehouse aisle where one frame in a thousand contains a person, a detection is real only about 9% of the time (0.99 × 0.001 against 0.01 × 0.999). That is not a reason to ignore it: it is a reason to slow down, look again, and combine several frames, and M3-T4: Recursive Bayesian Updating shows how the posterior then grows quickly. The perception course discusses the same effect for tracks in Uncertainty and Conflicting Observations.

Loop-closure and place recognition

A SLAM system compares each new scan or image with thousands of past ones. Almost every comparison is a non-match, so a match score that is right 99% of the time can still produce mostly wrong loop closures. Real systems follow a similarity hit with a geometric check (a second, independent piece of evidence) before adding a constraint, because one bad closure can bend the whole map (Loop Closure).

Localization: the correction step

For a localizer the hypotheses are the possible poses, the prior is the belief after moving, the likelihood is the sensor model evaluated at the actual reading for each candidate pose, and the posterior is the corrected belief. Every filter in modules 6 to 9 is this one formula, computed on a grid (module 6), in closed form for Gaussians (modules 7 and 8) or with samples (module 9). The SLAM course gives its own introduction in Bayesian Estimation.

Priors come from somewhere, and can be wrong

A prior from a map says “the cell behind the wall is impossible”, and if the map is wrong the filter can never learn the truth. Practical localizers therefore floor the prior at a small positive value, inject a few random particles, or keep a small probability for “none of the above”. Treat a prior of exactly zero or one as a decision never to learn.

Common misconceptions

Key takeaways

1. Bayes’ rule: posterior = likelihood × prior / evidence. Posterior ∝ prior × likelihood; the evidence only rescales so the beliefs sum to 1.
2. The likelihood describes the sensor (how probable this reading is under each hypothesis). The posterior describes the robot’s belief. They are different quantities and differ in direction.
3. The base rate matters: a rare target gives a low posterior even for a good detector. Posterior odds = likelihood ratio × prior odds.
4. Silence (no detection) is evidence too, and is computed with the complementary likelihoods.
5. A prior of zero can never be overturned. Keep a small positive floor wherever the model might be wrong.

References and further reading

Each source is marked free or paid. Free to read does not mean free to reuse: the course resources page lists every source with its licence.

• 3Blue1Brown, “Bayes theorem, the geometry of changing beliefs” . A visual explanation. Free to watch.
• J. Tsitsiklis and P. Jaillet, MIT RES.6-012 Introduction to Probability , lecture 2 (conditioning and Bayes’ rule) and lecture 14 (introduction to Bayesian inference). Free videos and notes, licence CC BY-NC-SA 4.0.
• J. Blitzstein and J. Hwang, Introduction to Probability, 2nd edition, CRC Press, 2019 , chapter 2. Free to read online; the book is copyrighted.
• C. Stachniss, “Bayes Filter” (video lecture) , which applies the rule to robot localization. Free to watch.
• S. Thrun, W. Burgard and D. Fox, Probabilistic Robotics, MIT Press, 2005, section 2.2 (Bayes rule). Paid book.

Knowledge check

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What’s next

Bayes’ rule combines a prior with one reading. To combine several readings, we need to know when it is legitimate to multiply their likelihoods, and that is the question of independence.

Related content

Nothing below is required to finish this lesson.

• Before
  M3-T1: Joint, Marginal and Conditional Probability

  Build a joint table for two variables, marginalize it, condition on an observation and use the product rule and the law of total probability.

• Lab
  M3-L1: Bayesian Landmark Detection

  A marker detector fires near the charging dock, but it also fires by mistake. Compute how much each detection should change the belief that the robot is at the dock.

Other courses on this site treat neighbouring topics from their own angle:

• SLAM M2-T5: Bayesian Estimation
• UAV M3-T3: Bayesian estimation
• Physical AI: Belief under uncertainty