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The bounded uniform prior distributions on the slopes and population intercept are proper and broad on their stated scale. They allow either sign but are not invariant to changes of units. The residual precision parameter $q=1/{\sigma }^2$ has a proper shape-rate $\mathrm{Gamma}(0.001,0.001)$ prior. Equivalently, ${\sigma }^2$ has inverse-gamma distribution with the same shape and scale. This is a commonly used diffuse choice, but it is not literally noninformative and its tail behaviour merits sensitivity analysis.

The prior for $\tau$ is uniform on $(0,100)$, so the induced density of $v={\tau }^2$ isIt is integrable at zero, unlike the usual scale-invariant variance kernel $\pi (v)\propto 1/v$. The latter kernel, often called the standard variance Jeffreys prior, would cause an improper posterior from a log-uniform random-effect scale prior here. After integrating the child intercepts, each child's response vector has a normal likelihood with covariance matrix ${\sigma }^2{I}_3+v11^T$. For fixed positive ${\sigma }^2$, this covariance remains nonsingular at $v=0$, and the marginal likelihood has a positive finite limit there. On a compact set of the other parameters its positivity givesConsequently the scale-invariant kernel cannot be normalized to a posterior distribution. A proper prior on the standard deviation avoids this problem. This familiar scale prior is not necessarily the actual Jeffreys prior for an additive variance component derived from the observed-data likelihood; the latter can be finite at zero.

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 $\widehat{\mathrm{SD}}(w)/\sqrt{B_{\mathrm{eff}}}$.