Metropolis-Hastings Acceptance Probabilities

Why the Proposal Correction Matters

In posterior computation, a proposal distribution suggests a candidate parameter value; the algorithm then decides whether to move there or remain at the current value. The acceptance decision is what makes the resulting chain have the desired posterior distribution as its invariant distribution. If proposals are asymmetric, the direction of a proposed move can be more or less likely than its reverse. The acceptance probability must account for that imbalance, or the chain can spend the wrong proportion of time in different regions.

The key correction compares the probability of proposing the reverse move with the probability of proposing the forward move. This tutorial derives that correction from detailed balance and works through both discrete and continuous examples. The posterior may be known only up to a normalizing constant; that constant cancels from the acceptance probability.

Setup and Acceptance Rule

Let $\theta$ denote the current parameter value and let $\pi(\theta\mid y)$ be the target posterior density or probability mass function for observed trial data $y$. Assume this target is proper. Given current value $\theta$, a proposal kernel $q(\theta'\mid\theta)$ generates a candidate $\theta'$. In a discrete state space, $q(\theta'\mid\theta)$ is a probability; in a continuous state space, it is a density with respect to a common dominating measure. For each current $\theta$, the proposal probabilities or density integrate or sum to one over candidate values.

Definition: The Metropolis-Hastings acceptance probability for a proposed move from $\theta$ to $\theta'$ is $$ \alpha(\theta,\theta')= \min\left\{1,\, \frac{\pi(\theta'\mid y)\,q(\theta\mid\theta')} {\pi(\theta\mid y)\,q(\theta'\mid\theta)} \right\}, $$ when the denominator is positive. If the forward proposal has positive probability or density but the reverse proposal has zero probability or density, the acceptance probability is zero. A candidate with zero target density is also rejected when the current value has positive target density.

The term $q(\theta'\mid\theta)$ is the forward proposal density, while $q(\theta\mid\theta')$ is the reverse proposal density. Their order matters. The formula assumes that the current value lies in the target’s support and that the proposal terms are evaluated under the same conventions and dominating measure in both directions.

In practice, the posterior often has an unnormalized kernel $\tilde{\pi}(\theta\mid y)=c\,\pi(\theta\mid y)$, where $c$ does not depend on $\theta$. Replacing both posterior terms by this kernel gives the same ratio, since $c$ cancels. This is useful when the posterior normalizing constant cannot be calculated.

Why This Formula Gives Detailed Balance

For distinct states, the probability flow from $\theta$ to $\theta'$ under the target distribution is the target probability or density at $\theta$, multiplied by the probability or density of proposing $\theta'$, multiplied by the probability of accepting it. Substituting the acceptance rule gives a symmetric expression for the flow in both directions.

\begin{aligned} \pi(\theta\mid y)q(\theta'\mid\theta)\alpha(\theta,\theta') &=\pi(\theta\mid y)q(\theta'\mid\theta) \min\left\{1,\frac{\pi(\theta'\mid y)q(\theta\mid\theta')} {\pi(\theta\mid y)q(\theta'\mid\theta)}\right\}\\ &=\min\left\{\pi(\theta\mid y)q(\theta'\mid\theta), \pi(\theta'\mid y)q(\theta\mid\theta')\right\}\\ &=\pi(\theta'\mid y)q(\theta\mid\theta')\alpha(\theta',\theta). \end{aligned}

This is detailed balance: the target-weighted flow from $\theta$ to $\theta'$ equals the reverse flow. Rejections add probability to staying at the current value, so the diagonal transition probabilities also satisfy balance. Summing the balance equations over starting states shows that the target distribution is invariant: if the chain currently has the target distribution, one transition leaves it with that same distribution.

Detailed balance establishes invariance, not by itself convergence from every starting value. For that, the chain must also be able to reach the relevant parts of the target support and satisfy suitable recurrence and aperiodicity conditions. A proposal that cannot reach a region of positive posterior probability can invalidate the intended computation even when the acceptance formula is implemented correctly.

Worked Calculations

Worked Example: An Asymmetric Proposal on Two States

Suppose a hypothetical posterior puts probability $0.6$ on state A and $0.4$ on state B. From A, the proposal selects B with probability $0.8$; from B, it selects A with probability $0.2$. Find the probability of accepting a move from A to B.

The forward proposal probability is $q(B\mid A)=0.8$, and the reverse probability is $q(A\mid B)=0.2$. Therefore,

\alpha(A,B)= \min\left\{1,\frac{0.4(0.2)}{0.6(0.8)}\right\} =\min\{1,0.1667\}=0.1667.

A proposed move from A to B is accepted about 16.67% of the time. In the reverse direction, $\alpha(B,A)=\min\{1,0.6(0.8)/(0.4(0.2))\}=1$. The target-weighted flow in either direction is $0.6(0.8)(0.1667)$, approximately $0.0800$, or $0.4(0.2)(1)=0.0800$. The correction offsets the proposal’s greater tendency to move from A to B.

Worked Example: A Drifted Continuous Proposal

Consider a hypothetical posterior for a real-valued trial parameter $\theta$ with density proportional to the standard normal density, so its unnormalized kernel is $\tilde{\pi}(\theta)=\exp(-\theta^2/2)$. From current value $\theta=0$, propose $\theta'=1$ using a normal proposal with variance 1 and mean $\theta+0.5$. This proposal has a positive drift and is not symmetric.

The target-kernel ratio is $\tilde{\pi}(1)/\tilde{\pi}(0)=\exp(-0.5)$. The forward proposal density is the normal density at 1 with mean 0.5; the reverse is the density at 0 with mean 1.5. The normalizing constants cancel, giving

\frac{q(0\mid1)}{q(1\mid0)} =\exp\left[-\frac{(0-1.5)^2-(1-0.5)^2}{2}\right] =\exp(-1).

Thus the full ratio and acceptance probability are

r=\exp(-0.5)\exp(-1)=\exp(-1.5)=0.2231, \qquad \alpha(0,1)=\min\{1,r\}=0.2231.

The candidate is accepted with probability about 22.31%. Using only the posterior ratio would give $\exp(-0.5)=0.6065$, which is incorrect for this drifted proposal: it ignores that this particular move is more readily proposed in the forward than the reverse direction.

Worked Example: A Move With No Reverse Proposal

Suppose a hypothetical discrete proposal can move from A to B with probability $0.5$, but cannot move from B to A: $q(B\mid A)=0.5$ and $q(A\mid B)=0$. Let the target probabilities be $\pi(A\mid y)=0.7$ and $\pi(B\mid y)=0.3$. What happens to a proposed move from A to B?

The reverse proposal probability is zero, so the acceptance probability for the forward move is zero. This conclusion also follows from detailed balance: positive accepted flow from A to B would have no possible reverse flow from B to A. The chain rejects every such proposal and remains at A. If a proposal mechanism has this one-way structure, it may fail to explore the target support; changing the acceptance formula cannot repair a proposal that lacks needed reverse moves.

Using the Rule Reliably

A useful implementation check is to write down the ratio in words before substituting numbers: target at the candidate times reverse proposal, divided by target at the current value times forward proposal. For continuous parameters, proposal densities can have units, but their ratio is dimensionless when both are expressed with respect to the same measure. The final acceptance probability is always between zero and one.

When target kernels or proposal densities are very small, calculate the log ratio to avoid numerical underflow. If all terms are positive, define

\ell= \log\tilde{\pi}(\theta'\mid y)-\log\tilde{\pi}(\theta\mid y) +\log q(\theta\mid\theta')-\log q(\theta'\mid\theta).

Then $\alpha(\theta,\theta')=\min\{1,\exp(\ell)\}$. In particular, if $\ell\geq0$, accept with probability one; if $\ell<0$, accept with probability $\exp(\ell)$. Log calculations do not remove the need to handle zero densities explicitly, since the logarithm of zero is not finite.

Check the candidate.
Confirm that the candidate is a valid parameter value and evaluate its target density or kernel. A candidate outside the target support has zero acceptance probability.

Evaluate both proposal directions.
Calculate $q(\theta'\mid\theta)$ and $q(\theta\mid\theta')$ using the proposal rule at the appropriate current state. Do not assume they are equal.

Form and apply the ratio.
Multiply the candidate target by the reverse proposal term, divide by the current target times the forward proposal term, and cap the result at one.

Special Cases, Alternatives, and Failure Modes

For a symmetric proposal, $q(\theta'\mid\theta)=q(\theta\mid\theta')$, and the proposal terms cancel. The acceptance probability then depends only on the target ratio. Symmetry is a special case, not a general property of proposals. For an asymmetric proposal, dropping the correction changes the transition flow and generally changes the invariant distribution. For example, in the two-state calculation above, omitting the correction would accept A-to-B with probability $0.4/0.6$, rather than $0.1667$.

Direct posterior sampling is an alternative when it is available: independent draws from the target do not require a Metropolis-Hastings accept-or-reject step. In many trial models, however, direct sampling is not available, and the general Metropolis-Hastings rule allows proposals suited to the parameter and target. Its flexibility comes with a limitation: the proposal must permit adequate movement across the target support, and rejections can produce repeated values.

Reversing the proposal terms. Using forward divided by reverse rather than reverse divided by forward can reverse the correction. Label each term by its conditioning state before calculating.
Ignoring a proposal asymmetry. The posterior ratio alone is not sufficient unless the proposal is symmetric. Verify symmetry rather than assuming it.
Using a normalizing constant unnecessarily. A common target normalizer cancels, but parameter-dependent factors do not. Keep every term that changes with $\theta$.
Overlooking support. Check that proposed values are valid and that reverse moves needed for exploration are possible. A zero reverse proposal density means the corresponding forward move must be rejected.
Treating acceptance as proof of a good chain. The formula ensures detailed balance under its conditions; it does not guarantee that the proposal explores the posterior efficiently or that simulation has converged.

A practical diagnostic is to test a pair of states in both directions. Compute the accepted flow from each direction and verify they agree, as in the worked two-state example. If they do not, the most likely causes are an omitted proposal correction, swapped conditional arguments, or an inconsistent target calculation. Also inspect whether the proposal and target are defined on compatible supports before running a chain.

Key takeaway: For an arbitrary proposal, the Metropolis-Hastings ratio is the target ratio multiplied by the reverse-to-forward proposal ratio. The minimum with one gives the acceptance probability; the correction is what makes the probability flow satisfy detailed balance.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A target assigns probabilities $0.25$ and $0.75$ to states C and D. The proposal probabilities are $q(D\mid C)=0.5$ and $q(C\mid D)=0.25$. What is $\alpha(C,D)$?
In a continuous example, the target-kernel ratio for a proposed move is $0.4$, the forward proposal density is $0.2$, and the reverse proposal density is $0.1$. Calculate the acceptance probability.
Why is a move with positive forward proposal density and zero reverse proposal density rejected?
When can the proposal terms be omitted from the acceptance ratio?

Solutions. (1) $\alpha(C,D)=\min\{1,0.75(0.25)/[0.25(0.5)]\}=\min\{1,1.5\}=1$. (2) The ratio is $0.4(0.1)/0.2=0.2$, so the acceptance probability is $0.2$. (3) Accepting such a move would create positive target-weighted flow in one direction with no reverse flow, violating detailed balance; the acceptance probability is therefore zero. (4) They cancel when the proposal is symmetric, meaning the forward and reverse proposal probabilities or densities are equal for the pair of states.