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For $c>0$ and a centered normal distribution prior, this conditional log-posterior has second derivative $-c{e}^{\alpha }-s^{-2}<0$. It is strictly concave but does not belong to a normal distribution or gamma distribution family. A Random-walk Metropolis algorithm, or a locally scaled proposal around its unique mode with the full proposal correction, can update it. Gaussian conjugacy is unavailable because the Poisson distribution likelihood contains an exponential rate.

Here the proposal is $Y=x+\eta$ with $\eta \sim N_2(0,I_2)$. Its probability density function depends only on $\Vert y-x{\Vert }^2$, so $q(y\mid x)=q(x\mid y)$. The determinant and normalizing constant of the target also cancel. With $Q$ as in the preceding entry,Equivalently, writing $u_j=(y_j-{\mu }_j)/{\sigma }_j$ and $z_j=(x_j-{\mu }_j)/{\sigma }_j$,One can also compute the exponent without subtracting two large quadratic forms:Generate two independent standard normal distribution proposal increments, compute this log ratio, and accept when $\log U\le \min (0,\log (\pi (Y)/\pi (x)))$. Retain the previous pair on rejection. Detailed balance gives the correct invariant distribution, and the everywhere-positive Gaussian proposal makes this Random-walk Metropolis algorithm chain irreducible; a rejection probability provides a holding step. The resulting target-distributed pairs are dependent, with proposal scale and target anisotropy controlling how quickly they mix.

Hold all parameters except ${\mu }_i$ fixed and propose ${\mu }_i^{\prime }={\mu }_i+\eta$ with $\eta \sim N(0,s_{\mu i}^2)$. This Random-walk Metropolis algorithm has a symmetric proposal distribution. DefineCancellation of unchanged likelihood and prior factors gives the Metropolis–Hastings acceptance probabilityDraw an independent uniform and accept if it is at most $a_i$; otherwise retain the old value. Update each component in turn, recomputing the needed mixture contributions. On a logarithmic scale, compare $\log U$ with $\min (0,{\sum }_j\log (D_j^{\prime }/D_j)-[({\mu }_i^{\prime }{)}^2-{\mu }_i^2]/(2{\tau }^2))$ to avoid numerical underflow.

The Metropolis–Hastings algorithm is useful when a target posterior can be evaluated up to a normalizing constant but direct independent sampling or simple conditional sampling is unavailable. It constructs a Markov chain with that posterior as invariant law. After adequate convergence, averages of functions of the chain estimate posterior expectations, and the retained draws approximate marginal distributions, quantiles and credible intervals.

For the Random-walk Metropolis algorithm, at current $\theta$ propose ${\theta }^{\prime }=\theta +Z$, where $Z$ has a symmetric distribution, often $N(0,\Sigma )$. With a symmetric proposal the acceptance probability isAccept the proposal when an independent uniform is below this probability; otherwise retain the current value. For a nonsymmetric proposal $q({\theta }^{\prime }\mid \theta )$, include $q(\theta \mid {\theta }^{\prime })/q({\theta }^{\prime }\mid \theta )$ in the ratio. Detailed balance follows because the accepted flow is the minimum of the two proposed directional flows; rejection supplies the remaining self-transition probability. Irreducibility, appropriate recurrence and aperiodicity are also needed for convergence.

In the displayed trace plots, the first parameter moves in small increments with pronounced long-term wandering and a shift from its early region to a later region. This is consistent with an undersized proposal or a strongly correlated slow direction. Measure its acceptance rate: if it is very high, increase that coordinate's proposal variance. The second parameter moves rapidly within regions but changes its level around iteration 4500; that is evidence against treating the full run as a settled stationary sample, and it suggests investigating dependence with the first parameter or separated posterior regions. Its local scale may already be reasonable, so a blanket reduction of all proposal variances would be unwarranted. The third parameter has conspicuous long plateaus, indicating many repeated states: if the recorded acceptance is low, reduce its proposal variance; also check constraints and dependence before deciding that scale alone is the cause.

Rerun with coordinate-specific scale changes, inspect joint dependence, consider a covariance-based block proposal, and run longer chains from dispersed starting points. The MCMC trace diagnosis and proposal tuning should include actual acceptance rates and effective sample sizes; the three marginal trace plots do not by themselves identify correlations or prove convergence. Further practical details are in the three subparts.

Use the random-walk proposal specified by its centre and covariance: for the current state $z=(x,y)$ draw $Z_1,Z_2$ independently standard normal and setIts properly normalized density of the proposal distribution isIt is symmetric: $q(z^{\prime }\mid z)=q(z\mid z^{\prime })$. This is a centred-at-the-current-state proposal, rather than an unshifted draw from the density notation in the source. The Random-walk Metropolis algorithm consequently needs only the target ratio. Put $H(x,y)=x^2+y^2-2\rho xy$. ThenAn independent uniform decides the move; on rejection both coordinates remain at their previous values. A proposal with lower $H$ is always accepted. This construction uses the joint density directly, without needing either conditional distribution. Its continuous positive proposal density supplies communication over the target support, and the holding probability avoids a deterministic periodic transition pattern.

The Metropolis–Hastings algorithm proposes $y\sim q(\cdot \mid x)$ and moves from the current state $x$ with acceptance probabilityOn rejection it retains $x$. The target need only be known up to its normalizing constant, but it must define a proper probability distribution. The accepted off-diagonal probability flow iswhich is symmetric in $x,y$ and proves detailed balance. This permits considerable flexibility in proposals, including joint moves when a posterior distribution has strongly correlated parameters, but requires proposal tuning and can waste computation on rejections.

A Gibbs sampler instead updates a coordinate or block from its exact full conditional distribution, holding the other coordinates fixed. This is a special Metropolis–Hastings algorithm update with acceptance probability one: the joint density factors into the unchanged marginal density and the proposed conditional density, which cancel in the acceptance ratio. The coordinate update preserves the target; composing such updates therefore preserves it too, although an entire deterministic sweep need not itself be reversible.

Use Gibbs sampling when the conditional laws are easy to simulate and suitably chosen blocks mix well. Scalar updates can mix poorly under strong posterior distribution correlation, whereas a blocked Gibbs sampler can alleviate that dependence. Prefer a flexible Metropolis proposal when conditional draws are difficult or a well-tuned joint proposal is more effective. Variations include random or systematic coordinate scans, Random-walk Metropolis algorithms, independence proposals, and using Metropolis updates for difficult blocks inside a Gibbs scheme. Both methods generate dependent draws; usable long-run estimates require an appropriate irreducible, aperiodic chain and assessment of mixing, not merely a formal update rule.

For the change-point calculation, put $v={\sigma }^2$ andMultiplication of the Gaussian likelihood function by the specified prior distribution gives the formal posterior distribution kernel, with respect to $d\theta d\varphi dv$ and counting measure on $m$,

There is an empty component under an improper prior at $m=n$. The sum $S_n$ is independent of $\varphi$, whose prior distribution is flat on all of $\mathbb{R}$. For every fixed finite $\theta$ and $v>0$, the kernel is a strictly positive constant in $\varphi$, henceThe configuration has positive prior probability $1/n$. Integrating over, for example, a bounded interval of $\theta$ and $1\le v\le 2$ already makes the joint normalization infinite. Thus the printed prior distribution does not produce a posterior probability distribution. This failure is present for every data set, not just for a special arrangement of observations.

The formal full conditionals expose exactly where the requested sampler breaks. For $m<n$, completing the two mean squares givesThe variance kernel is that of $\mathrm{IG}(n/2,S_m/2)$ when $S_m>0$, and the discrete conditional probabilities are proportional to $e^{-S_m/(2v)}$. At $m=n$ the conditional kernel for $\varphi$ is constant and cannot be normalized; it is not a normal distribution with an infinite variance. Therefore no valid Gibbs sampler, and no value of $P(m=n\mid x)$, exists for the model as printed. A chain that simply substitutes an arbitrary distribution for this missing conditional would be sampling a different model.

Here is a fully specified proper-prior repair retaining the no-change configuration. Choose fixed hyperparametersindependently, with $t_{\theta },t_{\varphi },a_0,b_0>0$. Its Gaussian change-point posterior with proper priors isIt is normalizable even for degenerate data: the likelihood function is at most $(2\pi {)}^{-n/2}{v}^{-n/2}$, and its integral against the variance prior distribution is bounded byThe normal mean priors integrate to one, and the normalizing constant is positive. This proves posterior propriety for the repaired model.

Completing the conditional squares gives the Gaussian change-point Gibbs updates. DefineThe four updates areAt $m=n$, $M_{\varphi }={\mu }_{\varphi },V_{\varphi }=t_{\varphi }$, giving its proper prior distribution unchanged. Starting from any finite means, positive variance and allowed $m$, cycle through these four updates using the latest values. The sum-of-squares terms can be evaluated from cumulative data sums; normalize the discrete log weights after subtracting their maximum to avoid numerical underflow. The proper positive conditionals allow movement among all split values and throughout the continuous parameter support.

Under this specified repair, $P(m=n\mid x)$ is the posterior probability of no change within the observed sequence: all $n$ data points belong to the first mean regime. For $N$ retained draws after initialization has ceased to affect the estimates, its Markov chain Monte Carlo estimator isAlternatively average the displayed conditional probability of $m=n$ at the sampled continuous parameters, giving an estimate by Rao-Blackwellization. Autocorrelation must be accounted for in its Monte Carlo uncertainty. These are probabilities and estimators for the explicitly repaired model; the original unnormalizable kernel supplies neither.

A log-flat improper prior $p(v)\propto 1/v$ is not appropriate when all known $r_s={\sigma }_{m,s}^2$ are positive: the integrated likelihood in (iii) has a positive finite limit as $v\downarrow 0$, so its integral against $dv/v$ diverges. One defensible noninformative choice is the scalar Jeffreys prior for an additive variance component, calculated holding the mean parameters fixed:Indeed the normal variance score has information $I_{vv}=\frac{1}{2}{\sum }_s(v+r_s{)}^{-2}$. This is a scalar conditional-information prior, not a claim that its product with the flat mean prior is the joint Jeffreys prior. It is bounded near zero and is $O(v^{-1})$ at infinity. The integrated likelihood is uniformly bounded near zero and is bounded by a constant times $v^{-(N-1)/2}$ at infinity, independently of the mean-shape parameters, since $e^{-Q/2}\le 1$. Thus $N>1$ and a proper external prior on the physically admissible $(w,{\Omega }_M)$ ensure posterior propriety. Proper weak scale priors are another option. If known errors vanish, the boundary argument and appropriate prior need separate reconsideration.

For an efficient parameterization, use $\theta =(\beta ,u,w,{\Omega }_M)$ with $u=\log v$ and $\beta =M_0+h(H_0)$. A chain on this joint reduced parameter space targetsThe factor $e^u$ is the Jacobian determinant. At each iteration propose ${\theta }^{\prime }=\theta +\epsilon$, with $\epsilon \sim N(0,{\Sigma }_q)$ for a fixed nonsingular proposal covariance, and accept with probability $\min \{1,\tilde{\pi }({\theta }^{\prime })/\tilde{\pi }(\theta )\}$. Proposals outside the physical prior domain have target zero. Tune ${\Sigma }_q$ during warmup and freeze it for the retained Random-walk Metropolis algorithm. To obtain samples of the full original parameter vector, independently draw $H_0$ from its retained positive prior and reconstruct $M_0=\beta -h(H_0)$; the resulting vector is $(M_0,v,H_0,w,{\Omega }_M)$. Analytically marginalizing $\beta$ using (iii) would also be valid.

Use several dispersed chains, trace plots, rank-normalized split $\hat{R}$, and the effective sample size of a Markov chain for each parameter and for $w,w^2$. These Markov chain Monte Carlo convergence diagnostics reveal poor mixing and disagreement but do not prove convergence. Estimate the integrated autocorrelation time ${\tau }_{\mathrm{int}}$; with $B$ retained draws, $B_{\mathrm{eff}}\approx B/{\tau }_{\mathrm{int}}$. Default to no thinning of a Markov chain, so the thinning factor is $1$. If storage requires thinning, choose a spacing after inspecting the autocorrelation and verify the retained-chain effective sample size; no finite spacing guarantees independent draws.

Using the retained $w_b$ samples, computeThese are posterior summaries, provided the corresponding moments exist, not uncertainties of the numerical estimates. A proper prior alone does not ensure finite second moments; choose or check the external prior accordingly. The Monte Carlo standard error of the posterior mean is approximately $\widehat{\mathrm{SD}}(w)/\sqrt{B_{\mathrm{eff}}}$.

Use broad proper uniform priors for $\Delta t$, $\Delta m$, and $c$ over physically plausible ranges, and broad log-uniform priors for the positive scales $A_f$ and ${\tau }_f$. ThenA Random-walk Metropolis algorithm can update $(\Delta t,\Delta m,c,\log A_f,\log {\tau }_f)$ with a multivariate Gaussian proposal distribution. Initialize several dispersed chains near plausible cross-correlation delays and near the marginal-likelihood optimum; reject proposals outside the prior bounds; discard warm-up while adapting only the proposal scale and covariance; then freeze the kernel and retain a long run. Evaluate trace plots, autocorrelations, acceptance rates, between-chain agreement, and the effective sample size of a Markov chain. Posterior predictive quasar light curves provide a model check.

With $D=(y_1,y_2,t)$ and $\nu =(\Delta m,c,A,\tau )$,A Random-walk Metropolis algorithm proposes ${\theta }^{\prime }=\theta +Z$ from a symmetric multivariate normal increment, conveniently using logarithmic coordinates for $A,\tau >0$, and accepts with probabilityAfter burn-in, retain a suitably long chain and assess convergence and effective sample size of a Markov chain.