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Use the chain rule for conditional densities. With $x_0=x_1=0$, the likelihood function of the nondegenerate observations isThe first residual is $x_2$ and carries no parameter information. There is no ordinary joint Lebesgue density for $X_1,\dots ,X_n$ because $X_1=0$ deterministically; this expression is the conditional likelihood function given the fixed initial values, or the likelihood function on $x_2,\dots ,x_n$. The initial point mass is parameter independent and has no effect on inference. This is the conditional likelihood of an initialized Gaussian AR(2) process.

Define the sufficient data sums, all over $0\le t\le n-2$,Multiplying the likelihood function by the independent standard-normal priors and collecting the quadratic terms givesLet $\Lambda =\left(\begin{array}{cc}A& C\\ C& D\end{array}\right)$ and $\Delta =AD-C^2$. This precision matrix is $I+\sum v_t{v}_t^T$, where $v_t=(x_{t+1},x_t{)}^T$, so it is positive definite even for a short or singular design. Completing the square provesThe fully normalized posterior density isThis is Gaussian conjugacy for an initialized AR(2) regression.

Completing each one-dimensional square, or using the supplied conditional-normal identity, givesWhen $n=2$, all regressor sums vanish and the posterior remains the independent standard-normal prior. No stationary-parameter restriction is imposed: the specified prior is on all of ${\mathbb{R}}^2$, and the finite initialized chain is defined for all $(a,b)$.

The Bradley-Terry model assigns comparison probability $q_i={\theta }_i/({\theta }_i+{\theta }_{i+1})$ to each observed edge. Up to factors independent of the parameters, its likelihood function isThe comparison graph is a path graph, hence a tree. After fixing ${\theta }_n=1$, its edge ratios are unconstrained positive coordinates: every collection $r_i={\theta }_i/{\theta }_{i+1}>0$ determines uniquely ${\theta }_i={\prod }_{k=i}^{n-1}{r}_k$.

For a convenient strict-concavity calculation, use the edge log-ratios in a Bradley-Terry comparison tree ${\eta }_i=\log r_i$. The log-likelihood separates asEach derivative is $m(p_i-q_i)$, where $q_i$ is the logistic function of ${\eta }_i$, and each second derivative is $-m{q}_i(1-q_i)<0$. Because $0<p_i<1$, there is a unique finite global maximum atConverting back gives the Bradley-Terry maximum-likelihood estimate on a path:Equivalently, recurse backwards using ${\hat{\theta }}_i={\hat{\theta }}_{i+1}{p}_i/(1-p_i)$. All estimates are finite and positive. The sample size $m$ changes the curvature of the likelihood but not this maximizer; absence of comparison cycles is what permits every empirical edge proportion to be fitted simultaneously.

In the Bühlmann model, a latent risk parameter $\Theta$ is drawn from a population distribution. Conditional on $\Theta$, the yearly observations are independent and identically distributed random variables, with conditional expectation $m(\Theta )$ and conditional variance $v(\Theta )$. Define the structural parametersHere $v$ is the expected process variance, while $a$ is the variance of hypothetical means. The Bühlmann credibility premium is the best affine estimate of $m(\Theta )$ from the observed claims, under mean squared error. Predicting the next claim gives the same affine estimate: the extra conditional observation noise contributes the constant $v$ to the prediction error.

The law of total variance and conditional independence giveAn affine estimate can be written as $\hat{m}=m+{\sum }_{j=1}^n{b}_j(X_j-m)$: for any chosen $b_j$, optimizing the constant makes its expected value equal to $m$. The normal equations for the linear least-squares projection areFor $v>0$ they force all $b_i$ to agree, with $b_i=a/(v+na)$. Thus the credibility factor and premium areThe credibility factor increases with the observation count and between-risk variance, and decreases with within-risk variance. If $a=0$, the risk mean is known and $Z=0$; if $v=0$ and $a>0$, one observation reveals it and $Z=1$. If both vanish, the premium is the fixed value $m$ and the factor is immaterial.

In the specified model, the conditional law is a gamma distribution with shape $\alpha$ and scale $\theta$. ThereforeThe prior is an inverse-gamma distribution with shape $k$ and scale $\lambda$. To obtain its moments directly, substitute $y=\lambda /\theta$ in the defining integral, obtainingThe Gamma function recurrence yieldsThe assumption $k>2$ makes both structural variances finite. HenceIt follows that the model-specific credibility estimate is

For the Bayes estimator under squared error loss, the quantity to estimate is $\alpha \Theta$, so the optimum is its posterior mean. The likelihood function, viewed as a function of $\theta$, is proportional toMultiplication by the prior shows gamma scale inverse-gamma conjugacy:Although the printed hint only mentions integer shapes, the same substitution and Gamma integral normalize this posterior for every positive real shape, so no integrality of $\alpha$ is needed. Its posterior mean gives the Bayesian estimate and comparisonThis exact Bühlmann credibility for gamma claims holds for every observed sample, not merely on average. Here the posterior mean is affine in the sample mean, so the best affine Bühlmann credibility premium is also the unrestricted Bayes estimator under squared error loss.

Use positive ability parameters in the Bradley-Terry model, so $\mathbb{P}(i\text{ beats }j)={\theta }_i/({\theta }_i+{\theta }_j)$. If the course instead denotes log abilities by ${\theta }_i$, apply the following calculation to their exponentials; the ordering is unchanged. Up to a factor independent of the abilities, the likelihood function isThe observed wins form a directed cycle, so a finite maximum exists. The Bradley-Terry likelihood Hessian is negative definite on contrasts of log abilities, giving uniqueness up to common scaling. We can therefore find the maximum-likelihood estimate through the Bradley-Terry score equations.

Player 1 has one observed win in two comparisons. Its score equation iswhich simplifies to ${\theta }_1^2={\theta }_2{\theta }_3$. By the model's scale invariance, set ${\theta }_2=1$ and write ${\theta }_1=r$, ${\theta }_3=r^2$, with $r>0$. Player 2's score equation becomesThe left side of the polynomial equation is strictly increasing on $r>0$, starts at zero, and tends to infinity. For $k=1$, its unique solution is $r=1$. For $k>1$, its value at one is $2k-1>1$, so its solution satisfies $0<r<1$. The Three-player Bradley-Terry comparison cycle consequently givesThus there is a complete tie when each directed edge is observed once, and otherwise the decreasing ranking is 2, 1, 3.

Write $S={\sum }_i{X}_i$. The likelihood function for the Poisson distribution is proportional to $e^{-n\theta }{\theta }^S$. For $S>0$, differentiation of the log-likelihood gives $-n+S/\theta =0$, and its second derivative is negative. For $S=0$, the likelihood decreases with $\theta$ and has its maximum at the boundary zero. HenceThe estimator is unbiased, since $\mathbb{E}\hat{\theta }=n^{-1}{\sum }_i\mathbb{E}{X}_i=\theta$. Its variance is $\theta /n$. If the parameter space excludes zero, an all-zero sample gives a supremum as $\theta \downarrow 0$ rather than an attained maximum.

Age changes during follow-up, so the generator matrix $Q(s)$ changes with time. The transition matrix must solve ${\partial }_{t_2}P(t_1,t_2)=P(t_1,t_2)Q(t_2)$, with $P(t_1,t_1)=I$. It is generally not the matrix exponential of one constant matrix; even $\exp (\int Q(s)ds)$ is not generally valid because the matrices at different times need not commute.

For example, write the age-dependent onset rate as $\lambda (s)$ and keep progression rate $\nu$ constant. ThenEven when the integrated onset rate is elementary, this remaining integral need not have an elementary closed form. Special cases can have closed forms; time dependence does not by itself prove universal impossibility.

For a tractable likelihood function, freeze age at its left-endpoint value, or a midpoint value, on each observation interval $[t_j,t_{j+1})$. The genetic covariate is already constant. This piecewise-constant covariate approximation gives a constant matrix $Q_j$ on each interval and a product of closed-form interval probabilities:Here $x_j$ is the observed state at $t_j$, and the likelihood function is conditional on the initial state. The approximation improves when intervals are short relative to the variation in the age effect.

Let $ϵ$ be the false positive rate for a genuinely cancer-free person, encompassing the harmless-tumour misclassification, and assume screening errors are conditionally independent given the latent states. A pre-clinical case tests positive with probability one; a clinical case is separately recognized. Thus a negative screen identifies latent state $1$, whereas a positive pre-clinical screen can arise from state $1$ or $2$. This is a Hidden Markov model with a screening misclassification model.

Conditional on the negative baseline screen, the state at year $0$ is $1$. The negative screen at year $2$ requires staying in state $1$ for two years and then testing negative. From that state, the positive screen at year $4$ can arise either with probability $ϵ$ while still disease-free or through pre-clinical disease. Consequently the conditional likelihood isIf the study instead specifies an initially disease-free patient and includes the baseline screening result in the likelihood function, its probability contributes a further factor $1-ϵ$, givingThe formulas use the positive emission for a pre-clinical screen, with no clinical diagnosis in this record. A persistent harmless-tumour subtype producing correlated screening errors would need an additional latent-state or emission specification; its likelihood is not determined by the single parameter $ϵ$.

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.

First, the absolute values in the printed limit are an error: the correct long-run variance of a stationary process is the signed sum of autocovariances. Directly,soEach coefficient tends to $1$ and has absolute value at most $1$. Absolute summability and the dominated convergence theorem therefore giveThe sum of absolute values is an upper bound, not the general limit. For a concrete counterexample to the printed assertion, take $X_t={\eta }_t-{\eta }_{t-1}$ with unit-variance iid noise. Then $\gamma (0)=2$, $\gamma (\pm 1)=-1$, and other covariances vanish. The absolute sum is $4$, but ${\overline{X}}_T=({\eta }_T-{\eta }_0)/T$, so $T\mathrm{Var}({\overline{X}}_T)=2/T\to 0$.

For correlated data the central limit theorem can still hold under suitable strong mixing of a stationary process and moment conditions, but the limiting variance is the long-run variance of a stationary process, rather than the one-observation variance. When it is positive,Positive serial dependence usually increases the standard error, while negative dependence can reduce it. The approximate effective sample size of a stationary sample is $T{\gamma }_X(0)/{\sum }_h{\gamma }_X(h)$ when that denominator is positive. A long-memory time series can require a different normalization or a different limit law. Absolute covariance summability alone does not prove a CLT: if $X_t=V{\eta }_t$ with a common independent random scale $V$ taking values $1$ and $2$ with equal probabilities and iid ${\eta }_t$ with the standard normal distribution, the off-diagonal covariances are zero, yet $\sqrt{T}{\overline{X}}_T$ has the nonnormal scale-mixture law $VN(0,1)$. The example is not an ergodic stationary process. If the long-run variance of a stationary process is zero, the usual nondegenerate square-root-$T$ CLT is unavailable.

For the causal AR(1), $|\varphi |<1$ andThe conditional distribution follows because the current innovation is independent of the past.

For a fixed known $\varphi$, condition on the observed $X_1$ and use the $n=T-1$ transitions $t=2,\dots ,T$. Their conditional maximum likelihood criterion is, up to constants,Differentiating in $\mu$ yieldsSince $X_t-\varphi X_{t-1}=(1-\varphi )\mu +{\epsilon }_t$,Thus it is an unbiased estimator, even conditionally on $X_1$, andIt has statistical consistency with mean-square convergence, and the iid-noise strong law of large numbers also gives almost-sure statistical consistency. Its conditional distribution is exactly normal with the displayed mean and variance. If $X_0$ is also observed and all $T$ transitions are used, replace $T-1$ by $T$.

The fixed-$\varphi$ qualification is necessary for the exact finite-sample claims. If $\varphi$ is jointly estimated, conditional likelihood function is linear regression with an intercept: writing ${\bar{X}}_{-}=n^{-1}{\sum }_{t=2}^T{X}_{t-1}$ and ${\bar{X}}_{+}=n^{-1}{\sum }_{t=2}^T{X}_t$, the unconstrained estimators areThis ratio is not generally an unbiased estimator and does not have the preceding finite-sample variance. Under the usual stationary regression conditions it has statistical consistency; its asymptotic variance is ${\sigma }_{\epsilon }^2/(1-\varphi {)}^2$. Indeed, it differs from ${\bar{X}}_{+}$ by $\hat{\varphi }(X_T-X_1)/(n(1-\hat{\varphi }))$, an asymptotically negligible endpoint term. Profiling an unknown innovation variance does not change the fixed-$\varphi$ estimate of $\mu$.

Use $v={\sigma }^2$ as the variance parameter, with prior density relative to $d\mu dv$. Let $\bar{x}=n^{-1}\sum x_i$ and $S=\sum (x_i-\bar{x}{)}^2$. Multiplying the prior by the likelihood function gives the unnormalized Bayesian posteriorThe printed request for an “improper posterior” should not be read as a claim that this posterior is always improper. Although the prior is improper, integrating over $\mu$ gives a marginal kernel $v^{-(n+1)/2}{e}^{-S/(2v)}$, which is integrable exactly when $n>1$ and $S>0$. Under these usual nondegenerate-data conditions it is a proper posterior. If those conditions fail, the displayed kernel is improper and there is no posterior probability distribution for a sampler to target.

Completing the square and using the inverse-gamma distribution give the full conditionalsThe inverse-gamma distribution convention is density proportional to $v^{-\alpha -1}{e}^{-\beta /v}$. Equivalently the precision satisfiesusing shape and rate, as in the question.

A Gibbs sampler starts from $v^{(0)}>0$ and alternatesThis is the normal mean-variance posterior with a log-uniform variance prior. For a marginal check, $v\mid x\sim \mathrm{IG}((n-1)/2,S/2)$: integrating out $\mu$ reduces the shape by $1/2$. This marginal shape is different from the full-conditional shape $n/2$.

The Gauss-Markov theorem states that with full-column-rank design, mean-zero errors and covariance ${\sigma }^{2}I$, the ordinary least squares estimator is the best linear unbiased estimator: its covariance is no larger than any other such estimator's in the Loewner order. Normality is unnecessary for that theorem.

In the given normal linear model, maximizing the likelihood function is equivalent to minimizing $\Vert Y-X\beta {\Vert }^2$. The normal equations and full rank giveFor a direct variance comparison, write ${\beta }^{\ast }=AY$. Unbiasedness for every $\beta$ requires $AX=I_p$. Set $B=(X^{T}X{)}^{-1}{X}^T$ and $D=A-B$, so $DX=0$. The cross covariance terms vanish because $B{D}^T=(X^{T}X{)}^{-1}(DX{)}^T=0$. ThereforeThis proves the requested comparison for every contrast $t$ without relying only on the theorem's name.

Let $I_0=\{i:y_i=0\}$ and $I_{+}=\{i:y_i>0\}$. Under the proposed zero-inflated negative binomial model, independence gives the likelihood functionThe positive-count factor uses the ordinary, untruncated negative binomial distribution. With $f_i=f_{\mathrm{NB}}(y_i;r,{\lambda }_i)$, the log-likelihood isFor latent indicators $a_i$, the complete-data log-likelihood iswhere $a_i=1$ is allowed only when $y_i=0$. This separation is what makes the expectation-maximization algorithm convenient.

The marginal Akaike information criterion is $\mathrm{AIC}=-2\ell ({\hat{\vartheta }}_{\mathrm{ML}})+2k$, where $k$ counts estimated population parameters. Here they are ${\beta }_0,{\beta }_1,{\sigma }^2,{\tau }^2$, so $k=4$. The realized incubator slopes have been integrated out of the likelihood function; they are not four additional free population coefficients. From the marginal Gaussian log-likelihood, the expression isAll hats here denote the ordinary maximum likelihood estimation fit.

The displayed value $14041.1$ is instead a restricted maximum likelihood criterion. Thus it cannot simply be substituted for the ordinary maximum-likelihood deviance. The logLik method defaults to the fitting convention. If one applies AIC(fly.model) directly to the displayed REML object, the default restricted log-likelihood and the same parameter count give the software's restricted-likelihood-based value $14041.1+8=14049.1$. This number should be labelled as such; it is not the ordinary marginal AIC derived above. To obtain the latter, refit and calculateThe output does not contain that numerical answer. In particular, restricted criteria must not be used to compare models with different fixed-effect designs. A conditional AIC targeting predictions for existing groups is yet another criterion and does not follow by counting each latent slope as an ordinary parameter.

The selected model is a moving-average process of order one, with the plus-sign convention used by R:The Gaussian likelihood function models innovations as independent $N(0,{\sigma }_{\epsilon }^2)$. Its autocovariance is $\gamma (0)={\sigma }_{\epsilon }^2(1+{\theta }^2)$, $\gamma (1)={\sigma }_{\epsilon }^2\theta$, and zero at larger lags, with autocorrelation $\rho (1)=\theta /(1+{\theta }^2)\simeq 0.4631$. The estimated coefficient has modulus less than one, giving invertibility of a moving-average model; a finite moving-average process is stationary regardless of this invertibility restriction.

The original plots show fluctuations around a roughly constant level, no persuasive deterministic trend, and a conspicuous positive lag-one correlation with the other plotted correlations within the approximate sampling bands. They make a stationary nonseasonal working model reasonable for this short series. stationary=TRUE excludes differenced alternatives, and seasonal=FALSE excludes seasonal ARIMA terms. Neither option is proved by a 60-day plot: annual temperature seasonality cannot be excluded from two months, and a longer record or residual diagnostics might require a different model. The options are reasonable local simplifications, not universal statements about temperature dynamics.

Let $\mathcal{D}=(x^{(1)},\dots ,x^{(n)})$ contain independent complete observations of the $p$-dimensional vector, and let $G$ be a Directed acyclic graph. A Gaussian Bayesian network uses, for node $j$,Here ${\alpha }_j$ is an intercept, ${\gamma }_j$ the vector of parent coefficients, and ${\sigma }_j^2>0$ the conditional variance; the vector ${\theta }_G$ collects these statistical parameters. Its likelihood function factors asChoose a proper statistical parameter prior distribution $\pi ({\theta }_G\mid G)$ and a graph prior distribution $\pi (G)$. Bayes theorem gives the Bayesian network structure score, up to the normalization common to graphs,The integral is Bayesian model evidence: it averages over nuisance parameters rather than substituting their best-fitting values. A graph prior can favor sparse graphs, while the evidence balances fit against the amount of prior statistical parameter space that predicts the observations well. Compare these scores over admissible Directed acyclic graphs, using enumeration when feasible or a search procedure otherwise; search need not find a global maximum, and observational data need not identify a unique causal orientation.

A conjugate prior makes the integral analytic. For example, for ${\eta }_j=({\alpha }_j,{\gamma }_j)$ take ${\eta }_j\mid {\sigma }_j^2\sim N(m_j,{\sigma }_j^2{V}_j)$, ${\sigma }_j^2\sim \mathrm{IG}(a_j,b_j)$, where $V_j$ is positive-definite matrix and $a_j,b_j>0$; these are independent local normal-inverse-gamma priors. Under prior independence between nodes, the evidence is a product of local regression evidences. Each posterior has the same family, so the local integral is the ratio of prior and posterior normalization constants, with the likelihood's constants included. This avoids costly numerical integration and makes local graph updates inexpensive. Hyperparameters must be specified coherently if score equivalence between observationally equivalent graphs is desired; arbitrary local priors do not automatically have that property.

Set $S=n_{11}+n_{22}$, $R=n_{12}+n_{21}$ and $N=S+R$. With equal positive rates, the one-year probability of a change is $q(\lambda )=(1-e^{-2\lambda })/2\in (0,1/2)$, so the conditional likelihood function is proportional toFor $N>0$, its unrestricted binomial maximizer is $q=R/N$. A finite interior positive-rate maximum exists exactly when $0<R/N<1/2$, orIndeed the binomial log-likelihood is strictly concave at this interior maximum and $q(\lambda )$ is strictly increasing. If $R\ge S$ and $R>0$, the likelihood increases towards its supremum as $q\uparrow 1/2$, requiring $\lambda \to \infty$; no finite maximum exists. Equality $R=S>0$ also has only this limiting maximum.

If $R=0<S$, allowing the boundary $\lambda =0$ gives a finite maximum there; insisting on $\lambda >0$ gives only a supremum as $\lambda \downarrow 0$. Thus if “valid” includes zero rates, the condition is $R<S$ with $N>0$; if rates must be strictly positive, it is $0<R<S$. If $N=0$, every rate has the same empty-data likelihood and the statistical parameter is not identifiable. These boundary qualifications prevent a negative or infinite plug-in rate from being called a valid interior estimate.

Condition on each patient's initial observed state and assume independent patient trajectories and observation times that do not carry additional information about the unobserved path. The Markov property factors each trajectory's likelihood function into its successive one-year transition probabilities. ThusMore explicitly, with $q=\lambda +\mu >0$ and $E=e^{-q}$,Successive transitions within one patient need not be independent unconditionally: this is a product of conditional factors justified by the Markov property. If initial-state probabilities are modeled rather than conditioned on, their likelihood contributions must also be included. Zero-count terms contribute zero; a positive count attached to a zero model probability makes the log-likelihood equal to $-\infty$.