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Nonlinear regression fits a conditional mean whose dependence on its parameters is nonlinear. In a normal distribution observation model, the likelihood is based on residuals $y_i-m(x_i,\theta )$. Nonlinearity can create parameter trade-offs, asymmetric uncertainty and several posterior regions, so prior predictive checks and Markov chain Monte Carlo convergence diagnostics can be especially useful.

The hierarchical Bayesian model separates variability between individuals from variability between repeated measurements on an individual. In WinBUGS, dnorm takes a precision parameter, so inverse variance is the appropriate second argument.

Line (1) gives $\log {\sigma }_j^2\sim N(\varphi ,{\psi }^2)$, hence a positive log-normal distribution for each within-individual variance. Estimating the shared parameters supplies partial pooling of these variances without forcing them equal. Line (2) assigns a proper bounded flat prior to the between-individual standard deviation $\tau$, and line (3) does the same for the standard deviation $\psi$ on the log-variance scale. The broad bounded priors on $\mu$ and $\varphi$ are also proper.

This is a reasonable structure for heterogeneous repeated measurements, but flat does not mean uninformative on all transformed scales. The bounds must be wide enough for plausible values, while such enormous ranges can put substantial mass on implausible predictions. Use prior predictive checks and sensitivity analysis, and check the Markov chain Monte Carlo convergence diagnostics. Serial dependence or systematic trends would require an extension of the independent-measurement likelihood.

The priors are proper but difficult to justify as practically weak information. Uniform bounds $[-100,100]$ on log asymptotes cover fantastically small and large circumferences. The independent prior on the transformed location makes the centered fraction spend much of its mass near zero or one, while the slope prior permits decreasing growth. Independence also ignores likely relationships among curve parameters.

The $\mathrm{Gamma}(0.001,0.001)$ prior on error precision is proper, but its near-zero shape creates a highly skewed prior with consequential mass near very small precision. It is not automatically harmless. Use biologically scaled priors for asymptotes and timing, a positive growth-rate prior, and a defensible prior on the error standard deviation such as a suitably scaled half-normal distribution. Check simulated trajectories with prior predictive checks and assess prior sensitivity.

Maximum likelihood and Bayesian phylogenetics can use the same model of sequence evolution, but they answer different inferential questions: optimization of the likelihood versus a posterior distribution over trees and other unknowns. Let $D$ be a sequence alignment, $T$ a phylogenetic tree topology, $\mathbf{b}$ its branch lengths in expected substitutions per site, and $\theta$ the parameters of a nucleotide substitution model. Both methods can useThe likelihood function sums over unobserved ancestral states, rather than choosing one ancestral sequence and treating it as observed. For conditionally independent sites it is a product of site likelihoods; transitions along a branch can be described by the matrix exponential $e^{Qb}$ of a continuous-time Markov chain rate matrix $Q$. The phylogenetic pruning algorithm computes the ancestral sums efficiently.^[ref]

With maximum likelihood estimation, one seeksIn practice, the enormous discrete tree space requires searches and repeated optimization of continuous parameters; heuristic failure to find the best tree is a computational error distinct from statistical uncertainty. A tree's optimized likelihood is not the probability that it is correct. bootstrap analyses resample alignment columns and repeat inference to assess support under the resampling procedure; their frequencies are not automatically posterior probabilities. Independence assumptions matter, because linked sites and heterogeneous loci can make simple column resampling an imperfect representation of replication.

In Bayesian phylogenetic inference,The prior distribution supplies weights on topologies, branch lengths and model parameters. Integrating over nuisance parameters givesThus a very narrow likelihood peak can lose posterior support to a broader region with a slightly lower maximum. maximum a posteriori estimation is another optimization and does not replace this integration. Markov chain Monte Carlo provides dependent samples approximating the posterior distribution; a clade's posterior probability is the posterior mass of trees containing that clade. The early likelihood-based Bayesian formulation explicitly distinguished these probabilities from bootstrap support.^[ref]

Neither approach is protected against systematic error. Poor sequence alignment, incorrect orthology assignments, compositionally heterogeneous sequences, and an inadequate substitution process can support a wrong tree with high apparent confidence. A gene tree can differ from a species tree because of incomplete lineage sorting or introgression; extra sequence from that gene does not automatically fix the mismatch. Model comparison, biological examination of the data and sensitivity to locus sampling are needed in both approaches. In Bayesian analyses, agreement among short chains is not proof of convergence: independent runs, adequate mixing between topologies and an appropriate effective sample size of a Markov chain should be checked. A credible interval and a confidence interval also have different interpretations even if their numerical endpoints happen to resemble one another.

Time inference adds a particularly strong role for assumptions. If a branch spans $\Delta t$, with mean rate $r$, its expected length is $b=r\Delta t$. Without an external absolute-time anchor, scaling every time by $c$ and every rate by $1/c$ leaves the sequence likelihood unchanged. Even a very long alignment can identify branch lengths accurately without identifying an absolute calendar scale. For example, a lineage length $b=0.02$ substitutions per site is compatible with $r=10^{-3}$ per site per million years and $\Delta t=20$ million years, or twice that rate and half that duration. This is an identifiability issue, not merely a shortage of sampled bases.

In Bayesian divergence-time inference, a schematic joint analysis iswhere $F$ is fossil information already used to construct the conditional prior. An alternative generative analysis includes a fossil-sampling likelihood explicitly; the same fossil evidence must not be counted twice. A strict molecular clock equates rates across branches; a relaxed molecular clock allows variation, whose distribution and correlations also require specification. A narrow rate prior favors particular absolute ages, and a strong rate-smoothing assumption can redistribute inferred time among branches. A birth-death process prior on the tree encodes assumptions about lineage origination, extinction and sampling; it is not a neutral choice just because it is mathematically convenient.

fossil calibration must separate a fossil's geological age from the age of the lineage split it constrains. A confidently placed crown fossil gives a minimum age for the relevant crown, whereas a stem fossil does not automatically set that crown minimum. Uncertain placement and the incompleteness of the record complicate upper bounds. hard calibration bounds assign zero probability outside the permitted range; enough sequence cannot overcome a zero prior. soft calibration bounds allow small tail probabilities, exposing rather than eliminating the possibility of conflict. Multiple-calibration analyses demonstrate that changing the treatment of bounds changes inferred times and uncertainty.^[ref]

Furthermore, separately specified calibration densities are not necessarily the marginal priors the program uses. The rule that an ancestor must be older than its descendant truncates their joint possibilities. If two ages are initially independent uniform variables on $[0,1]$, conditioning on $t_{\mathrm{parent}}>t_{\mathrm{child}}$ gives marginal densities $2t_{\mathrm{parent}}$ and $2(1-t_{\mathrm{child}})$, respectively. Neither remains uniform. Combining tree-process priors, calibration densities and ordering constraints can therefore introduce substantial information unintentionally.^[ref]

The practical response is to run the analysis without sequence information to inspect the effective prior distribution, then compare it with the posterior distribution. Repeat under defensible calibration tails, rate models, tree priors and fossil placements; report which ages change and whether the data sharpen relative ages more than absolute ages. Examine prior predictive checks and posterior predictive checks, and keep numerical mixing diagnostics separate from biological model adequacy. Priors can legitimately supply the time information missing from sequences, but narrow posterior ages are persuasive only when that information and its sensitivity are explicit.

Prior probability describes uncertainty before the current observations are incorporated. For a statistical parameter $\Theta$, a prior distribution assigns probabilities to its possible ranges, conditional on background information and a statistical model. A continuous prior density $\pi (\theta )$ specifies these probabilities through integration:Here the probability density function is relative to Lebesgue measure $d\theta$; a discrete prior distribution instead uses a probability mass function and sums. Density at one point is not the probability of that point. A nonnegative kernel with infinite total mass is an improper prior, whose use requires a separate check of posterior propriety.

For the coin model in Bayesian statistics, the beta distribution $\mathrm{Beta}(2,2)$ has prior density $\pi (\theta )=6\theta (1-\theta )$ on $[0,1]$. Its mean is $1/2$ and its variance is $1/20$. It favors moderate heads probabilities without ruling out values near zero or one. For example,The prior curve in Bayesian posterior makes this preference visible; moment matching for a beta prior explains how to choose a beta distribution from a desired mean and variance.

A prior distribution lets earlier evidence or elicited knowledge contribute to statistical inference when the current sample size is limited. It also exposes assumptions that can be checked by prior elicitation and a prior predictive check. Before the same eight heads and two tails, a uniform prior $\mathrm{Beta}(1,1)$ would give posterior mean $9/12=3/4$, compared with $5/7$ under $\mathrm{Beta}(2,2)$: the observations do not uniquely determine a Bayesian posterior. Even a uniform prior makes a choice of parameter coordinates; its child article explains why changing coordinates can destroy uniformity.