Bayes' Rule

Bayes’ rule converts probabilities of data given hypotheses into probabilities of hypotheses given data.

For events $H$ and $D$ with nonzero probabilities,

$$\mathbb P(H\mid D) = \frac{\mathbb P(D\mid H)\mathbb P(H)}{\mathbb P(D)}.$$

The left side is the posterior probability of $H$ after learning data $D$. The factor $\mathbb P(D\mid H)$ is the likelihood of the data under $H$. The factor $\mathbb P(H)$ is the prior probability before learning $D$. The denominator $\mathbb P(D)$ normalizes the result.

In quantum mechanics, Bayes’ rule is used for classical inference about unknown preparations, parameters, devices, and models. It should not be confused with the quantum state-update rule after a measurement outcome.

From Conditional Probability

The product rule says

$$\mathbb P(H\cap D) = \mathbb P(D\mid H)\mathbb P(H)$$

and also

$$\mathbb P(H\cap D) = \mathbb P(H\mid D)\mathbb P(D).$$

Equating the two expressions gives

$$\mathbb P(H\mid D) = \frac{\mathbb P(D\mid H)\mathbb P(H)}{\mathbb P(D)}.$$

The algebra is simple. The conceptual step is remembering which conditional probability is being asked for.

Hypotheses and Evidence

For mutually exclusive and exhaustive hypotheses $H_1,\ldots,H_n$,

$$\mathbb P(H_i\mid D) = \frac{ \mathbb P(D\mid H_i)\mathbb P(H_i) }{ \sum_j \mathbb P(D\mid H_j)\mathbb P(H_j) }.$$

The denominator is the total probability of the data:

$$\mathbb P(D) = \sum_j \mathbb P(D\mid H_j)\mathbb P(H_j).$$

This quantity is often called the evidence or marginal likelihood. It is not optional; it is what makes the posterior probabilities add to one.

Vocabulary

In a Bayesian calculation, the common terms are:

• Prior: $\mathbb P(H)$, the probability assigned before the data $D$ are included.
• Likelihood: $\mathbb P(D\mid H)$, the probability of the data under the hypothesis.
• Evidence: $\mathbb P(D)$, the total probability assigned to the data by the model family.
• Posterior: $\mathbb P(H\mid D)$, the updated probability after including the data.

The likelihood is a function of the hypothesis once the observed data are fixed. It is not, by itself, a normalized probability distribution over hypotheses.

Odds Form

For two hypotheses $H_1$ and $H_2$, Bayes’ rule can be written as an odds update:

$$\frac{\mathbb P(H_1\mid D)}{\mathbb P(H_2\mid D)} = \frac{\mathbb P(D\mid H_1)}{\mathbb P(D\mid H_2)} \frac{\mathbb P(H_1)}{\mathbb P(H_2)}.$$

The first factor on the right is the likelihood ratio. It says how strongly the data favor $H_1$ over $H_2$. The second factor is the prior odds.

Thus

$$\text{posterior odds} = \text{likelihood ratio} \times \text{prior odds}.$$

This form is often the cleanest way to see how evidence accumulates.

Base-Rate Example

Suppose a rare condition has prior probability

$$\mathbb P(H)=0.01.$$

A test has sensitivity

$$\mathbb P(+\mid H)=0.99$$

and false-positive probability

$$\mathbb P(+\mid H^c)=0.05.$$

After a positive test, the posterior is

$$\mathbb P(H\mid +) = \frac{0.99\cdot0.01}{ 0.99\cdot0.01+0.05\cdot0.99 }.$$

The numerator is $0.0099$, while the denominator is $0.0594$, so

$$\mathbb P(H\mid +) = \frac16 \approx 0.167.$$

The test was accurate, but the condition was rare. Ignoring the base rate would give a wildly overconfident conclusion.

Continuous Parameters

For a continuous parameter $\theta$, Bayes’ rule becomes a density formula:

$$\pi(\theta\mid D) = \frac{ \mathcal L(D\mid\theta)\pi(\theta) }{ Z }.$$

Here:

$$Z = \int \mathcal L(D\mid\theta)\pi(\theta)\,d\theta$$

is the evidence, $\pi(\theta)$ is the prior density, and $\pi(\theta\mid D)$ is the posterior density.

The posterior density must integrate to one:

$$\int \pi(\theta\mid D)\,d\theta = 1.$$

As with all probability densities, the value of the density at one exact parameter is not itself a probability.

Independent Data

If data points $d_1,\ldots,d_N$ are conditionally independent given $\theta$, then the likelihood factors:

$$\mathcal L(D\mid\theta) = \prod_{k=1}^N p(d_k\mid\theta).$$

Equivalently, the log-likelihood is a sum:

$$\log \mathcal L(D\mid\theta) = \sum_{k=1}^N \log p(d_k\mid\theta).$$

This is why repeated measurements can rapidly concentrate a posterior distribution, provided the model is appropriate and the data are genuinely informative.

Quantum Likelihoods

In quantum inference, the Born rule supplies likelihoods for classical data.

If a state model $\rho(\theta)$ is measured by a POVM with effects $E_d$, then

$$p(d\mid\theta) = \operatorname{Tr}[\rho(\theta)E_d].$$

For repeated conditionally independent outcomes $D=(d_1,\ldots,d_N)$,

$$\mathcal L(D\mid\theta) = \prod_{k=1}^N \operatorname{Tr}[\rho(\theta)E_{d_k}].$$

Bayes’ rule then updates a posterior over the classical parameter $\theta$:

$$\pi(\theta\mid D) \propto \mathcal L(D\mid\theta)\pi(\theta).$$

The symbol $\propto$ means “proportional to”; the evidence normalizes the posterior.

This is common in quantum parameter estimation, calibration, and tomography. The quantum probabilities come from states and measurement operators, but the inference over $\theta$ is ordinary Bayesian inference. For the local sensitivity and Cramér–Rao viewpoint, see Fisher Information.

Tomography Interpretation

In state tomography, the unknown object may be a density operator $\rho$ itself. A Bayesian treatment assigns a prior density over allowed density operators and updates it using measurement data:

$$\pi(\rho\mid D) \propto \mathcal L(D\mid\rho)\pi(\rho).$$

For counts $n_d$ of outcomes $d$ from one fixed POVM, a multinomial likelihood has the form

$$\mathcal L(D\mid\rho) \propto \prod_d \left[ \operatorname{Tr}(\rho E_d) \right]^{n_d}.$$

This posterior is a state of knowledge about an unknown preparation or device. It is not the same object as the post-measurement quantum state of one system after a single outcome. The latter is governed by a measurement instrument or state-update rule.

Bayesian Update versus Quantum State Update

Bayesian updating changes a probability distribution over hypotheses:

$$\pi(\theta) \longrightarrow \pi(\theta\mid D).$$

Selective quantum measurement update changes the quantum state assigned after an outcome in a specified measurement model:

$$\rho \longrightarrow \frac{M_d\rho M_d^\dagger}{ \operatorname{Tr}(M_d\rho M_d^\dagger) }.$$

Both updates include conditioning and normalization. They answer different questions. Bayesian updating asks, “Which model or parameter is plausible after the data?” Quantum state update asks, “What state should be assigned to the system after this measurement outcome, given the measurement model?”

The distinction becomes essential when the same observed data are used to infer an unknown source, calibrate a detector, or predict a later measurement on the same system.

Common Mistakes

• Confusing $\mathbb P(D\mid H)$ with $\mathbb P(H\mid D)$.
• Ignoring prior probabilities or base rates.
• Forgetting the evidence denominator.
• Treating a likelihood as a normalized posterior.
• Assigning zero prior probability to a hypothesis and then expecting data to revive it.
• Comparing continuous posterior densities by point height alone without considering volume.
• Confusing Bayesian inference about an unknown quantum preparation with quantum state update after a measurement.
• Treating tomography estimates as exact states without reporting uncertainty, model assumptions, and measurement design.

Cross-Links

• Conditional Probability
• Probability Densities
• Expectation Values
• Fisher Information
• Born Rule
• POVMs: First Encounter
• Trace Rule for Expectation Values
• State Update Rule

References

• E. T. Jaynes, Probability Theory: The Logic of Science, Cambridge University Press, 2003.
• D. S. Sivia and J. Skilling, Data Analysis: A Bayesian Tutorial, 2nd ed., Oxford University Press, 2006.
• A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis, 3rd ed., CRC Press, 2013.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, 1976.

Exercises

1. A box is chosen at random. Box 1 is chosen with probability $0.7$ and contains a red ball with probability $0.2$. Box 2 is chosen with probability $0.3$ and contains a red ball with probability $0.8$. If a red ball is observed, what is the posterior probability that Box 2 was chosen?

Solution

Let $H_2$ be “Box 2” and $R$ be “red.” Then

$$\mathbb P(H_2\mid R) = \frac{ \mathbb P(R\mid H_2)\mathbb P(H_2) }{ \mathbb P(R\mid H_1)\mathbb P(H_1) + \mathbb P(R\mid H_2)\mathbb P(H_2) }.$$

Substitute:

$$\mathbb P(H_2\mid R) = \frac{0.8\cdot0.3}{ 0.2\cdot0.7+0.8\cdot0.3 } = \frac{0.24}{0.38} = \frac{12}{19}.$$

2. Two hypotheses have prior probabilities $\mathbb P(H_1)=0.6$ and $\mathbb P(H_2)=0.4$. The likelihoods for data $D$ are $\mathbb P(D\mid H_1)=0.1$ and $\mathbb P(D\mid H_2)=0.3$. Compute the posterior odds $H_2$ to $H_1$.

Solution

The prior odds $H_2$ to $H_1$ are

$$\frac{0.4}{0.6} = \frac23.$$

The likelihood ratio is

$$\frac{0.3}{0.1}=3.$$

Thus the posterior odds are

$$3\cdot\frac23=2.$$

So $H_2$ is twice as likely as $H_1$ after observing $D$.

3. A qubit source is modeled by a parameter $q$, the probability of outcome $0$ in a computational-basis measurement. If $N$ independent measurements produce $n_0$ zeros and $n_1$ ones, write the likelihood for $q$.

Solution

The probability of a zero is $q$ and the probability of a one is $1-q$. Ignoring the combinatorial factor independent of $q$, the likelihood is

$$\mathcal L(D\mid q) \propto q^{n_0}(1-q)^{n_1}.$$

4. A POVM has effects $E_1,E_2$. A model state is $\rho(\theta)$. Write the likelihood for observing outcome sequence $(1,2,1)$ under conditional independence.

Solution

The Born probabilities are

$$p(i\mid\theta) = \operatorname{Tr}[\rho(\theta)E_i].$$

Thus

$$\mathcal L(D\mid\theta) = \operatorname{Tr}[\rho(\theta)E_1]\, \operatorname{Tr}[\rho(\theta)E_2]\, \operatorname{Tr}[\rho(\theta)E_1].$$

5. Why is a Bayesian posterior over $\rho$ in tomography not the same thing as the post-measurement state of a single quantum system?

Solution

The posterior over $\rho$ is a probability distribution over possible source or model states after data are observed. It describes uncertainty about an unknown preparation or device. A post-measurement quantum state is the state assigned to a particular system after a specified measurement outcome and measurement instrument. Both involve conditioning, but they refer to different objects.