MCMC Concepts - Statistics Graduate Level

1. Which statement about this sampler is correct?
   Answer: It cannot reach the second interval, so the chain is reducible even if it mixes well within the first interval. When evaluating MCMC samplers, the critical question is always irreducibility: can the chain reach every region of positive probability from any starting point? If not, the chain is reducible and will fail to explore the full target distribution, regardless of how well it behaves locally. Here, the target has support on $[-3,-2] \cup [2,3]$, separated by a gap of width 4 where the density is exactly zero. The random-walk proposal adds $U \sim \text{Uniform}[-1,1]$, so from any point $x \in [-3,-2]$, the farthest right a proposal can land is $-2 + 1 = -1$. That point lies in the zero-density gap, so the Metropolis acceptance probability is $\alpha = \min\left(1, \frac{\pi(y)}{\pi(x)}\right) = 0$. Every proposal that exits $[-3,-2]$ toward the gap is rejected. The chain is therefore trapped in its starting interval — it is reducible. This confirms B as correct. A is wrong because increments cannot "accumulate" across the gap in Metropolis. Each rejected proposal sends the chain back to its current state; there is no memory of partial progress toward the second interval. C is wrong because intermediate states with zero density are never accepted — the acceptance ratio is zero for any proposal landing in the gap, so no stepping-stone path through the gap exists. D is wrong because detailed balance within each interval is a local property. It guarantees correct stationary behavior conditional on staying in one component, but it says nothing about transitions between disjoint components. Irreducibility is a separate, global requirement. A useful study tip: whenever you see a disconnected support and a proposal with bounded step size, immediately check whether the step can bridge the gap. If it cannot, reducibility is virtually guaranteed.
2. Which asymptotic batching scheme is generally required for a consistent batch-means variance estimator?
   Answer: Let both the batch length and the number of batches increase as $N$ increases. Whenever you see a question about batch-means estimators for MCMC, focus on two competing requirements: the batches must be long enough to wash out serial correlation, and there must be enough batches to estimate variance reliably. These two requirements pull in opposite directions, and the correct batching scheme must satisfy both simultaneously as $N \to \infty$. For a batch-means estimator to be consistent, the batch length $b = b(N)$ must grow with $N$ — this ensures the autocorrelation within each batch decays, so individual batch means behave approximately like independent observations. At the same time, the number of batches $k = N/b$ must also grow with $N$ — this gives you an increasing number of "pseudo-observations" whose sample variance converges to the true asymptotic variance. A common sufficient condition is $b \to \infty$ and $k = N/b \to \infty$, for example $b \sim N^{1/2}$ or $b \sim N^{2/3}$. This confirms that C is correct: both quantities must diverge. Choice A fails because fixed batch length means the within-batch correlation never vanishes, so batch means remain dependent and the variance estimator is biased asymptotically. Choice B fails because fixing the number of batches at a constant $k$ means you never accumulate enough "observations" for the sample variance to concentrate — you're forever estimating variance from, say, 10 numbers regardless of $N$. Choice D is fundamentally confused: batch length one simply recovers the ordinary sample variance of the chain, which estimates the marginal posterior variance, not the Monte Carlo standard error that accounts for autocorrelation. A useful rule of thumb: on problems about MCMC variance estimation, ask yourself whether both the signal (batch independence) and the precision (number of batches) conditions are being satisfied simultaneously.
3. What is the probability that the sampler accepts the proposed move from $x$ to $y$?
   Answer: $3/4$, because both the target-density ratio and proposal asymmetry enter the ratio. Whenever you encounter a Metropolis–Hastings question, your first instinct should be to write down the full acceptance ratio — not just part of it. The MH acceptance probability is: $\alpha(x, y) = \min\!\left(1,\; \frac{\widetilde{\pi}(y)\, q(x \mid y)}{\widetilde{\pi}(x)\, q(y \mid x)}\right)$ This ratio has two components: the target-density ratio and the reverse-to-forward proposal ratio. Plugging in the given values: $\alpha = \min\!\left(1,\; \frac{3}{2} \cdot \frac{q(x \mid y)}{q(y \mid x)}\right) = \min\!\left(1,\; \frac{3}{2} \cdot \frac{0.2}{0.4}\right) = \min\!\left(1,\; \frac{3}{2} \cdot \frac{1}{2}\right) = \min\!\left(1,\; \frac{3}{4}\right) = \frac{3}{4}$ This confirms C is correct — both the target ratio and the proposal asymmetry combine to give an acceptance probability of $3/4$. A is wrong because it ignores the target-density ratio entirely. The proposal ratio alone ($0.2/0.4 = 1/2$) gives an incomplete picture and would only apply if the target were uniform. B inverts the target-density ratio (using $2/3$ instead of $3/2$), which would be appropriate only if you were computing the acceptance ratio for a move from $y$ back to $x$, not $x$ to $y$. D is a tempting but dangerous shortcut. A higher target density at $y$ does not guarantee automatic acceptance — proposal asymmetry can (and here does) reduce the acceptance probability below 1. As a study tip: always write the full MH ratio before substituting numbers. The two most common errors on exam questions are forgetting the proposal correction term or inverting one of the two ratios.
4. Which statement correctly describes this chain?
   Answer: The distribution $\pi$ is invariant, but detailed balance fails and point-mass initial distributions do not converge to $\pi$. When analyzing a Markov chain, you should always separate three distinct properties: invariance, detailed balance, and convergence. These are related but not equivalent, and this question tests whether you can distinguish them precisely. Start with invariance. A distribution $\pi$ is invariant if $\pi P = \pi$, where $P$ is the transition matrix. For this deterministic cycle $1\to2\to3\to1$, the transition matrix sends each state's mass to the next state. Applying $\pi = (1/3, 1/3, 1/3)$: each state still receives exactly $1/3$ of the total probability mass, so $\pi P = \pi$. Invariance holds. Now check detailed balance: $\pi_i P_{ij} = \pi_j P_{ji}$ for all $i,j$. Consider states 1 and 2: $\pi_1 P_{12} = (1/3)(1) = 1/3$, but $\pi_2 P_{21} = (1/3)(0) = 0$. These are unequal, so detailed balance fails — the chain is not reversible. Finally, convergence: starting from a point mass on state 1, the chain cycles deterministically $1\to2\to3\to1\to\cdots$ and never settles. The distribution oscillates and does not converge to $\pi$. Answer A captures all three facts correctly. Answer B is wrong because invariance has nothing to do with whether transitions are deterministic — as shown above, $\pi$ is preserved. Answer C falsely claims detailed balance holds; it doesn't. Answer D confuses invariance with convergence — invariance is necessary but not sufficient for convergence from every initial distribution. Study tip: Memorize this hierarchy — invariance $\not\Rightarrow$ detailed balance, and invariance $\not\Rightarrow$ convergence. Convergence from all starting points typically requires aperiodicity and irreducibility, not just invariance.
5. Using large-sample variance approximations, which statement correctly compares the two estimators?
   Answer: The all-draw estimator has slightly smaller variance because the discarded draws still contribute information about the mean, even though they are correlated. When comparing MCMC estimators under a fixed computational budget, the key quantity is the variance of the sample mean, which depends on both the number of draws and their autocorrelation structure. The large-sample variance of a sample mean from a stationary chain is approximately $\text{Var}(\bar{X}) \approx \frac{\sigma^2}{N} \cdot \left(1 + 2\sum_{k=1}^{\infty} \rho_k\right)$, where the sum captures the penalty from autocorrelation. For the all-draw estimator using $N$ draws with $\rho_k = 0.9^k$, the effective sample size factor gives $\text{Var}(\bar{X}_{\text{all}}) \approx \frac{\sigma^2}{N} \cdot \frac{1+0.9}{1-0.9} = \frac{19\sigma^2}{N}$. For the thinned estimator using $N/10$ draws with lag-one autocorrelation $0.9^{10} \approx 0.349$, the variance becomes $\frac{10\sigma^2}{N} \cdot \frac{1+0.349}{1-0.349} \approx \frac{20.7\sigma^2}{N}$. The all-draw estimator wins — confirming C is correct — because discarding nine out of ten observations destroys more information than the reduction in autocorrelation recovers. Choice A is wrong because while thinning does reduce lag-one autocorrelation dramatically, it does not more than compensate — the tenfold reduction in sample size is too costly. Choice B is wrong because the two effects (fewer draws, lower autocorrelation) are not perfectly offsetting; that would require a very specific relationship between thinning interval and autocorrelation decay. Choice D is wrong on two counts: the thinned observations are not fully independent ($0.9^{10} \approx 0.35$), and even if they were, the variance would be $10\sigma^2/N$, not one-tenth of the all-draw variance. The study tip here: thinning almost never improves estimator efficiency under a fixed budget. It can reduce storage costs, but the default rule is to keep all draws and let the autocorrelation average out over a larger $N$.
6. Which assessment of asymptotic validity is most appropriate?
   Answer: Diminishing adaptation is helpful but is not alone sufficient; a containment or related stability condition is also needed. When evaluating the asymptotic validity of adaptive MCMC algorithms, you should immediately think about the Adaptive MCMC validity conditions established by Roberts & Rosenthal (2007). The key insight is that valid adaptation requires two conditions working together, not just one. The correct answer is B because the theoretical literature establishes that diminishing adaptation alone — even at a well-behaved $1/t$ rate — is insufficient to guarantee ergodicity. You also need a containment condition (or similar stability requirement), which ensures the Markov chain doesn't "chase" a poorly adapted proposal into unvisited regions indefinitely. In the passage, the absence of any bound on covariance growth is a direct red flag: without containment, the chain could expand its proposal into unexplored territory in a way that disrupts convergence, even as adaptation magnitude shrinks. A is wrong because diminishing adaptation does not preserve detailed balance at every iteration — adaptation typically breaks detailed balance at each step. Detailed balance is a property of fixed-kernel chains; adaptive chains require different justification for ergodicity entirely. C is wrong because adaptation does not inherently change the target distribution. The posterior remains fixed; what changes is the proposal mechanism. Properly designed adaptive algorithms absolutely can be asymptotically valid. D is wrong because hitting the $0.234$ acceptance rate is a heuristic for mixing efficiency in fixed-kernel RWM, not a condition for ergodicity in adaptive settings. Acceptance rate convergence tells you nothing about whether the chain's stationary distribution is correct. As a study tip: whenever you see "adaptive MCMC," mentally check for both the diminishing adaptation condition and the containment condition — the exam frequently tests whether students conflate one with both.
7. Which condition most directly ensures that the resulting extended-state chain has the exact posterior for $\theta$ as its marginal stationary distribution?
   Answer: $\widehat{L}(\theta)$ is nonnegative and unbiased for $p(y\mid\theta)$ at every value of $\theta$. When you encounter a question about pseudo-marginal MCMC, anchor your thinking to the core theoretical result: the Beaumont (2003) / Andrieu & Roberts (2009) framework works by embedding the auxiliary random variables used to construct $\widehat{L}(\theta)$ into an extended state space. The joint chain over $(\theta, u)$ — where $u$ represents the auxiliary randomness — must have a well-defined stationary distribution whose $\theta$-marginal equals the true posterior $p(\theta \mid y)$. For this to hold exactly, you need $\widehat{L}(\theta)$ to be nonnegative and unbiased for $p(y \mid \theta)$, making D correct. Nonnegativity ensures the acceptance ratio stays well-defined (a negative "likelihood" is meaningless). Unbiasedness — in the likelihood scale, not the log scale — is what guarantees the extended-state detailed balance equations integrate correctly to yield the exact posterior as the marginal. This is not an approximation; it is an exact equality under these two conditions. Option A is wrong because variance of the estimator doesn't govern correctness of the stationary distribution — it affects mixing speed, not validity. Option B is a common and dangerous trap: unbiasedness on the log scale does not imply unbiasedness on the likelihood scale (by Jensen's inequality, $E[\log \widehat{L}] = \log p$ actually implies $E[\widehat{L}] > p$, the wrong direction), so the posterior target would be distorted. Option C describes a large-sample property, but pseudo-marginal theory requires exactness at fixed simulation effort — consistency alone gives no finite-sample guarantee. Remember: in pseudo-marginal methods, unbiasedness lives in the likelihood scale, not the log scale. This is the single most common misconception tested on this topic.
8. Which conclusion is best supported by these diagnostics?
   Answer: The chains agree within one mode, but the diagnostic cannot establish that the other posterior mode was explored. When evaluating MCMC convergence diagnostics, you must distinguish between within-chain mixing and global exploration of the posterior. The split rank-normalized $\hat{R}$ statistic measures whether multiple chains have converged to the same distribution — but it can only detect disagreement between chains. If all chains are stuck in the same region, they will appear to agree perfectly, and $\hat{R} \approx 1.00$ will result regardless of whether the full posterior was explored. This is exactly the trap in this problem. All four chains entered the first mode and mixed well within it. From $\hat{R}$'s perspective, the chains look identical — low variance between chains relative to within-chain variance — so it reports near-unity values. But the second mode, carrying equal posterior probability, was never visited. The diagnostic has no information about what it never saw. B is correct because it accurately reflects what the diagnostics can and cannot establish: the chains agree within one mode, but the diagnostic is silent on the second. A is wrong because $\hat{R} \approx 1.00$ is a necessary but not sufficient condition for convergence. It confirms chain agreement, not global coverage. C is a dangerous overclaim — the second mode's absence from the draws reflects the sampler's failure, not the mode's posterior mass. Assigning negligible probability based on non-exploration is unjustified. D is wrong because overdispersed initialization is a strategy intended to encourage global exploration, but it doesn't guarantee it — all four chains here still ended up in the same mode. As a study tip: whenever $\hat{R}$ appears in a question, ask yourself "could all chains be stuck in the same wrong place?" If yes, $\hat{R}$ is uninformative about global convergence.
9. Using the usual integrated-autocorrelation approximation, what are the effective sample size and the Monte Carlo standard error of the sample mean, respectively?
   Answer: Effective sample size $100$ and Monte Carlo standard error $0.20$. When MCMC draws are autocorrelated, your effective sample size (ESS) shrinks because correlated draws carry less information than independent ones. The key formula is: $\text{ESS} = \frac{n}{1 + 2\sum_{k=1}^{\infty} \rho_k}$ With $\rho_k = 0.8^k$, the sum in the denominator is a geometric series: $\sum_{k=1}^{\infty} 0.8^k = \frac{0.8}{1-0.8} = 4$. So the denominator becomes $1 + 2(4) = 9$, giving: $\text{ESS} = \frac{900}{9} = 100$ The Monte Carlo standard error (MCSE) of the sample mean is then: $\text{MCSE} = \sqrt{\frac{\text{Var}(\theta)}{\text{ESS}}} = \sqrt{\frac{4}{100}} = \sqrt{0.04} = 0.20$ This confirms answer A is correct. Each wrong answer reflects a specific miscalculation. B (ESS = 180) likely comes from using only the lag-1 autocorrelation $\rho_1 = 0.8$ in the denominator — $900/(1+2\times0.8) \approx 346$, or some similar partial-sum error — and doesn't match the full geometric series result. C (ESS = 500) appears to use just $1/(1-\rho_1) = 1/0.2 = 5$ as the inflation factor rather than the correct factor of 9. D (ESS = 900) is the trap of ignoring autocorrelation entirely, treating all draws as independent — this gives the naive variance $\sqrt{4/900} \approx 0.067$, which dangerously underestimates uncertainty. Your study tip: always evaluate the full geometric sum $\frac{\rho}{1-\rho}$, not just the lag-1 term. Partial sums are the most common trap on ESS problems.
10. Which modification most directly improves mixing while preserving the intended target distribution?
    Answer: Update both coordinates jointly from their bivariate conditional block, rather than using two scalar conditional updates. When a Gibbs sampler targets a highly correlated distribution — like a bivariate normal with correlation $\rho = 0.99$ — the chain gets trapped along a narrow ridge. Each scalar conditional update can only move perpendicular to the ridge in tiny steps, because the conditional variance $\text{Var}(\theta_1 \mid \theta_2) = 1 - \rho^2 \approx 0.02$ is nearly zero. The marginals look correct (the chain is stationary), but autocorrelation between successive samples is enormous, meaning the chain explores the joint distribution extremely slowly. This is the classic slow mixing problem, and the fix must address how the chain moves through the joint space — not just how it looks marginally. Answer A is correct because drawing $(\theta_1, \theta_2)$ jointly from their bivariate distribution in a single block update allows the sampler to move freely along the correlation ridge, not just orthogonally to it. This is called blocking or a block Gibbs update, and it directly resolves the bottleneck by letting both coordinates move together in large, correlated steps — all while still targeting the exact same stationary distribution. Answer B is wrong because thinning (keeping every tenth draw) doesn't change how the underlying chain moves — it just discards samples. You're wasting computation without improving mixing. Answer C is wrong and actually breaks the sampler entirely: replacing random draws with conditional means eliminates the stochasticity required for valid MCMC, causing the chain to collapse to a deterministic sequence that cannot explore the distribution. Answer D is wrong because burn-in length affects initialization bias, not stationary autocorrelation; extending burn-in does nothing to improve mixing once the chain has converged. The key study tip: whenever you see high correlation in a target distribution paired with slow mixing, think blocking — group correlated parameters and update them jointly.

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