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For $I_n={\int }_{{\mathbb{R}}^p}{e}^{nh(\theta )}a(\theta )d\theta$, suppose $h$ has a unique interior maximum $\hat{\theta }$, $H=-h^{\prime \prime }(\hat{\theta })$ is positive definite, the amplitude is smooth there, and contributions away from that maximum are negligible. Set $z=\sqrt{n}(\theta -\hat{\theta })$. The quadratic expansion $nh(\theta )=nh(\hat{\theta })-z^{T}Hz/2+o(1)$ yields the Laplace approximationThe determinant comes from diagonalizing the Gaussian integral; it is an inverse square root, not an inverse determinant. Under standard higher smoothness and domination, the relative error is $O(n^{-1})$.

For a regular model and a fixed smooth positive proper prior $\pi$, apply this to the marginal likelihood $m(y)=\int e^{\ell (\theta ;y)}\pi (\theta )d\theta$. With $j(\hat{\theta })=-{\ell }^{\prime \prime }(\hat{\theta })$,As $j$ is of order $n$, this produces the $p\log n$ complexity term in the Bayesian information criterion after multiplying log marginal likelihood by $-2$. A posterior mode can also be used with the corresponding combined log-density curvature. Boundary modes, singular information and multiple dominant modes invalidate the single interior Gaussian formula; distinct nondegenerate dominant modes must have their contributions added.

Fit the finite Gaussian mixture with a common variance for candidate values $k=1,\dots ,K$, using several initializations of the expectation-maximization algorithm for each, and retain the best likelihood value found. Maximized likelihood alone cannot select the number of components: a smaller mixture embeds in a larger one by giving an added component zero weight, so its likelihood cannot decrease.

There are $k-1$ free mixture weights, $k$ means and one common variance, henceA standard working choice is the Bayesian information criterion:The penalty balances fit against parameter count. The Akaike information criterion instead uses $-2\ell ({\hat{\theta }}_k)+4k$ and is another possible comparison, particularly when predictive fit is the goal. The parameter count would differ if component variances were separate, but that is not this model.

Likelihood fitting must first be made well posed. Even a shared variance does not prevent all degeneracy: if $k=N$, choose ${\mu }_j=x_j$, ${\alpha }_j=1/N$ and let the common variance tend to zero. Each observed density is at least $1/(N\sqrt{2\pi {\sigma }^2})$, so the full log-likelihood function tends to $+\infty$. The same construction works when $k$ reaches the number of distinct observed values. Thus one must not feed infinite fitted likelihoods into an information criterion. Restrict candidate sizes to a sensible range below that degeneracy, or impose a fixed positive variance floor and use that same constrained model in every comparison. Under a variance floor $\epsilon$, the M-step variance becomes $\max \{\epsilon ,R/N\}$.

There is also nonregular mixture model selection: at a missing component its weight is on a boundary and its mean is unidentified, while permuting labels leaves the distribution unchanged. Consequently an ordinary chi-squared calibration for a likelihood-ratio test between successive component counts is not generally justified. Information criteria are practical comparisons, not an automatic universal proof that the true number has been recovered. Predictive cross-validation or a suitably constructed parametric bootstrap comparison can supplement them. Select the component count by a penalized, well-defined mixture fit, rather than by unpenalized likelihood increase.

Allow each subject to have a separate baseline and slope through a correlated random-intercept and random-slope model:The subject random effects are independent of all measurement errors, and subjects are independent. The within-person covariance between times $s,t$ is ${\tau }_0^2+(s+t)\rho {\tau }_0{\tau }_1+st{\tau }_1^2$, with an additional ${\sigma }^2$ when the same reading is used twice.

Prefer the model with a random slope. The maximum likelihood estimation refits improve twice the log-likelihood by $227.9604$ while adding two covariance statistical parameters. Both the Akaike information criterion ($5165.779$ to $4941.819$) and Bayesian information criterion ($5183.367$ to $4968.200$) strongly favour it. The printed nominal likelihood-ratio test is overwhelming. The usual reference chi-squared distribution with two statistical degrees of freedom is not an exact regular calibration, since zero slope variance is a boundary and the intercept–slope correlation is then unidentified; a design-specific parametric bootstrap could calibrate it. This qualification does not undermine the substantial descriptive improvement shown by both information criteria.

The preferred fit estimates population mean baseline strength ${\hat{\beta }}_0=61.09286$ kg and weekly gain ${\hat{\beta }}_1=2.83143$ kg/week. The between-person baseline standard deviation is ${\hat{\tau }}_0=15.785582$ kg, and the between-person slope standard deviation is ${\hat{\tau }}_1=2.738666$ kg/week. Their estimated correlation $\hat{\rho }=0.117$ is weakly positive, giving random-effect covariance about $5.058$ kg${}^2$/week; its uncertainty is not supplied. The measurement-error standard deviation is $\hat{\sigma }=9.923280$ kg. The printed $-0.038$ instead describes the correlation between the estimated fixed effects, not between the subject random effects. These statistical parameter estimates come from restricted maximum likelihood; the model comparison uses the separate maximum likelihood estimation fits.

The population mean fitted trajectory is $61.09286+2.83143t$ kg. HenceUsing the reported slope standard error, an approximate 95% confidence interval for the population mean gain is $28.3143\pm 1.96(2.984464)=(22.46,34.16)$ kg; a Student t confidence interval with 499 statistical degrees of freedom is almost identical.

Successful individuals can improve far more than the average. Their latent ten-week gains have fitted normal distributionA clear estimate for an upper-performing group uses a between-person slope quantile: the 95th percentile is $28.3143+1.645(27.38666)\simeq 73.4$ kg, and the 97.5th percentile is about $82.0$ kg. The upper 5% of fitted underlying gains begin around 73 kg. These are person-to-person performance quantiles, not confidence intervals for the population mean. Identifying the best observed trainee would require that person's data or fitted subject random effects; the aggregate output cannot identify a literal maximum. If performance means the observed difference of endpoint readings, add the measurement-error variance $2{\sigma }^2$ to the latent gain variance.

The three-component finite Gaussian mixture with a common variance has the largest displayed maximized log-likelihood, but it also has more fitted parameters. For $k$ components there are $k-1$ free weights, $k$ means and one common variance, making $d=2k$ residual-distribution parameters. Applying the nominal Akaike information criterion and Bayesian information criterion to the three residual fits givesBoth criteria favour the two-component model. It improves the likelihood substantially over one normal component; the third component gains only $1.5550$ in log-likelihood at a cost of two further parameters. The Akaike information criterion difference between two and three components is small, about $0.89$, so that criterion alone gives only a slight preference; the Bayesian information criterion preference for two is stronger. Their nearly indistinguishable overlaid curves also support retaining the simpler two-component fit.

These calculations are model-selection aids, not an exact test. Ordinary chi-squared calibration of a likelihood-ratio test for the number of mixture components is invalid in general: under a smaller-component null, some weights lie on the boundary and extra-component parameters are unidentified. This is nonregular mixture model selection. A parametric bootstrap or predictive cross-validation is preferable for a formal comparison, and should account for the mean-adjustment stage. The table uses the provided residual likelihoods and their nominal parameter counts; first-stage regression uncertainty and possible local EM maxima remain qualifications. On the supplied evidence, select two components without claiming that two distinct biological groups have been established.

For a causal autoregressive model of order $p$, the population partial autocorrelation function cuts off after lag $p$, whereas the autocorrelation function generally decays. For an invertible moving-average model of order $q$, the autocorrelation function cuts off after lag $q$, whereas the partial correlation coefficients generally decay. For a mixed autoregressive moving-average model, both generally decay. Exponential decay may alternate in sign or show damped oscillations. Sample correlation coefficients only approximate these patterns, so isolated crossings of the significance bands are not exact order tests.

In PDF Figure 1, the autocorrelation function alternates sign, with a large negative lag-one correlation coefficient and geometrically decreasing magnitude. The partial autocorrelation function has essentially one substantial spike, negative at lag one and about $-0.8$; later values mostly lie within the displayed bands. Thus a stationary AR(1) with a negative coefficient is the natural first model, with $\hat{\varphi }$ initially near $-0.8$. The trace fluctuates around an approximately constant mean and shows no evident deterministic trend.

Fit that model, compare nearby low-order alternatives using likelihood function and an Akaike information criterion or Bayesian information criterion, and inspect the residual autocorrelation function and residual variance. A few small later PACF spikes are expected in a plot with many lags. The roughly $\pm 1.96/\sqrt{T}$ white noise bands are a guide, rather than simultaneous guarantees for every lag.

The sign alternation also suggests concentration of spectral power near the high-frequency end: the AR(1) denominator $1+{\varphi }^2-2\varphi \cos \lambda$ is smallest near $\lambda =\pi$ when $\varphi <0$. From estimated $\gamma (0)$ and lag-one correlation coefficient one can estimate innovation variance as ${\hat{\sigma }}_{\epsilon }^2\approx \hat{\gamma }(0)(1-{\hat{\varphi }}^2)$. The persistence parameter controls the decay rate and long-run variance of a stationary process. These plots do not establish a normal distribution, independence, or the absence of nonlinear dependence; correlation coefficients of squared residuals can provide a separate variance diagnostic.

One suitable criterion among the three unpenalized normal linear models is the Akaike information criterion. With $d$ estimated mean coefficients and the variance also estimated, minimizing AIC is equivalent to minimizingThe values of $d$ are $2,2,3$ for model1, model2 and model3. This compares fit and complexity rather than choosing solely by individual coefficient significance. The Bayesian information criterion or a common-fold predictive comparison would also be reasonable, depending on the selection objective.

For the ridge regression tuning parameter, use K-fold cross-validation with squared prediction loss:Use the same folds for every candidate, estimate any centering or scaling within each training fold, choose a minimizing $\lambda$, and refit on all observations. This estimates mean squared prediction error while avoiding data leakage. The printed summaries do not supply all residual sums of squares or validation losses, so they do not determine which model or penalty wins.