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For $k=m/2$ independent reads in each sum, read noise contributes $k{r}^2/g^2$ in squared ADU units. HenceUsing the shot-noise-only formula therefore givesThe condition for negligible contamination is $gN\gg k{r}^2$, equivalently a large number of collected Electrons per individual frame compared with $r^2$. Summing more identical-level frames does not change that ratio because both $N$ and the read variance scale with $k$. Measure read noise from paired bias images, or fit mean versus noise variance across several illumination levels: the shot-noise slope determines detector gain, while the read term gives an intercept. If $r$ is known, the corrected positive solution satisfies $E^2{N}^2{g}^2-2Ng-2k{r}^2=0$. Offset subtraction removes the mean bias but not the read variance. Correlated read errors need a different covariance calculation.

The stochastic objective is interpreted as its expectation; a literal pathwise minimization is generally impossible without knowing future noises. Let the value function from time $t$ be $V_t(x)=P_t{x}^2+d_t$, with $P_T=1,d_T=0$. Independence from past observations and zero contemporaneous covariance giveThe Bellman equation is $V_t(x)=x^2+{\min }_u\{c{u}^2+P_{t+1}{V}_{\xi }(Ax+u{)}^2+P_{t+1}{V}_{ϵ}+d_{t+1}\}$. Completing the square yieldsThis feedback is optimal among all controls measurable from the specified past, not merely linear controls. It depends on $V_{\xi }$ and does not depend on $V_{ϵ}$, though the minimal cost does. There is no cost-relevant $u_T$ in the displayed objective.

The backward Riccati recurrence iterates $F(P)=1+A^{2}c{V}_{\xi }P/(c+V_{\xi }P)$ from $1$. It is increasing, maps into $[1,1+A^{2}c]$, and has exactly one positive fixed point. Consequently it converges toAs $T\to \infty$, the finite-horizon feedback at any fixed time tends to this feedback. The closed-loop multiplier is $Ac/(c+V_{\xi }P)$, and its squared-mean factor $\alpha =V_{\xi }{A}^2{c}^2/(c+V_{\xi }P{)}^2$ is less than one, since $(P-1)/P=A^{2}c{V}_{\xi }/(c+V_{\xi }P)$ and $\alpha =(P-1)c/[P(c+V_{\xi }P)]$. Thus the second moment tends to $V_{ϵ}/(1-\alpha )$. The finite-horizon value has $d_1=V_{ϵ}{\sum }_{t=1}^{T-1}{P}_{t+1}$; its average converges by Cesaro summation. For a fixed initial state of finite second moment,The same average is attained by the stationary feedback, by the stationary Bellman equation and bounded second moments.

For deterministic step functions, define directly from Brownian motion incrementswith $s_0$ the left endpoint of the first interval and $f=0$ outside the listed intervals. The printed endpoint list omits $s_0$; the intended convention is $0\le s_0<s_1<\cdots <s_n$, usually $s_0=0$.

Given finitely many observation times, refine the deterministic partition of an interval to include all observation times and all $s_i$. Every $Z_{t_j}$ is a linear combination of the same finite family of independent centered normal Brownian motion increments. Thus every linear combination of $(Z_{t_1},\dots ,Z_{t_m})$ is normal, including possibly a degenerate constant: $Z$ is a Gaussian process. By independence, only increments in the common interval $[0,s\wedge t]$ contribute to the covariance, givingThe same calculation for two step functions $f,g$ proves, rather than assumes, the Itô isometryFor $f\in L^2({\mathbb{R}}_{+})$, choose deterministic step functions $f_n\to f$ in the L2 space. This identity makes $Z_t(f_n)$ a Cauchy sequence in the L2 space; the Riesz-Fischer theorem defines $Z_t(f)$ by mean-square convergence, independently of the approximation. The covariance formula passes to the limit by the Cauchy-Schwarz inequality. For finitely many times and coefficients $a_j$, the characteristic functions converge toso the extension remains a Gaussian process.

For completeness, the construction has a continuous version. Each approximating $Z(f_n)$ is a martingale, directly by the centered independent increments of Brownian motion; the Doob L2 maximal inequality givesTo prove the inequality used here without assuming Itô integral properties, a continuous martingale $N$ satisfies $\lambda \mathbb{P}(N_T^{\ast }\ge \lambda )\le \mathbb{E}(|N_T|1_{\{N_T^{\ast }\ge \lambda \}})$, by conditioning at the first crossing on finite deterministic grids and then using continuity; integrating in $\lambda$ and using the Cauchy-Schwarz inequality yields $\Vert N_T^{\ast }{\Vert }_2\le 2\Vert N_T{\Vert }_2$. A sufficiently fast subsequence therefore converges uniformly almost surely on every compact time interval, giving the continuous version of the stochastic integral.

Choose an orthonormal basis $(e_n)$ of the Dirichlet energy space and independent variables $({\xi }_n)$ with the standard normal distribution. The zero-boundary Gaussian free field is the formal seriesIts precise energy-indexed interpretation is the isonormal Gaussian processFor every fixed $f$, this sum converges in $L^2$ of the probability space by Parseval identity, and it is a centered Gaussian random variable. For $f,g\in H_0^1(D)$,This covariance characterizes its law independently of the chosen basis. The series does not define an $H_0^1$-valued random element: ${\sum }_n{\xi }_n^2=\infty$ almost surely. It is realized as a random distribution, and its pairing with ordinary test functions is constructed in part (b).

The violated assumption is independence of random variables across observations: repeated measurements on the same rat form clustered data. A persistent rat-specific weight difference induces positive within-rat covariance, even after adjustment for time and diet. Treating all 176 measurements as independent is pseudoreplication.

For the claimed lack of identifiability in the second fit, Rat must be a categorical variable, as intended by the rat-effect interpretation. Each rat receives only one diet, so this is confounding of nested fixed factors. If $z_r$ is the rat-indicator column, the diet-$d$ indicator is ${\sum }_{r:D_r=d}{z}_r$. Thus a diet effect can be increased by $c_d$ and every corresponding rat effect decreased by $c_d$, leaving all fitted values unchanged. With an intercept, two diet columns, fifteen rat contrasts and time, there are 19 columns but only 17 independent columns: the intercept and rat contrasts already span the diet columns, while time varies within rats. This gives rank-deficient ordinary least squares.

The first model ignores within-rat dependence; the second cannot separate unrestricted fixed diet and rat effects. If Rat were instead encoded as a single numeric predictor, the asserted rank deficiency would not follow merely from nesting.

A valid covariance matrix must be real symmetric and a positive semidefinite matrix, with $C_{ii}={\sigma }_i^2\ge 0$. Conversely, every such matrix is the covariance of a possibly degenerate multivariate normal distribution. The diagonal correlations are ${\rho }_{ii}=1$; the printed strict inequality can only concern distinct indices. Pairwise bounds do not suffice: the $3$ by $3$ correlation matrix with diagonal $1$ and every off-diagonal entry $-0.9$ has eigenvalue $-0.8$ along $1$ and is invalid. Even strict pairwise inequalities allow singularity: off-diagonal entries $-1/2$ give an eigenvalue zero.

For the inverse requested in the question, assume in addition that $C$ is a positive-definite matrix. Write $x=({\hat{\mu }}_i)$, $\Lambda =C^{-1}$ and $S=1^T\Lambda 1>0$. Up to a constant, the log-likelihood is $-\frac{1}{2}(x-\mu 1{)}^T\Lambda (x-\mu 1)$. Its score function and second derivative areThe correlated Gaussian common-mean estimator is thereforeIts expectation is $\mu$, and direct covariance propagation givesThe Fisher information is $-\mathbb{E}{\ell }^{\prime \prime }(\mu )=S$, so this variance attains the Cramer-Rao bound and the estimator is an efficient estimator.

If $C$ is singular, an ordinary Lebesgue density and $C^{-1}$ are unavailable. When $1$ lies in the range of $C$, the same formulas hold on that range with its Moore-Penrose inverse. Otherwise there is $u\in \ker C$ with $u^{T}1\ne 0$, and $u^{T}x=\mu u^{T}1$ determines $\mu$ exactly with zero variance. The usual nonsingular Cramér–Rao calculation then does not apply.

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

$\boxed{(b)⟷P3}$. This is a Gaussian kernel covariance. Near zero, $2A^2(1-e^{-h^2/{\tau }^2})\sim 2A^2{h}^2/{\tau }^2$, consistent with a finite mean-square derivative variance. The kernel has derivatives of every order, and the resulting Gaussian process admits a smooth version. P3 is smooth but does not repeat, unlike P1.

At the fitted hyperparameters, define $K=k_0(t,t)$, $K_{\ast \ast }=k_0(t^{\ast },t^{\ast })$, and $K_{\ast t}=k_0(t^{\ast },t)$, with the matrices formed by pairwise kernel evaluations. Conditional independence of observation noise and future latent values givesThe residual $f_{\ast }-K_{\ast t}{V}^{-1}y$ has zero covariance with $y$, so the joint normal law makes it independent of $y$. This derives the conditional multivariate normal distribution and the Gaussian process regression posterior:The covariance is a Schur complement and is positive semidefinite. When nonsingular, its joint density is $(2\pi {)}^{-M/2}|{\Sigma }_{\ast }{|}^{-1/2}\exp [-\frac{1}{2}(f_{\ast }-m_{\ast }{)}^T{\Sigma }_{\ast }^{-1}(f_{\ast }-m_{\ast })]$. Repeated phases can make it singular, in which case the displayed normal law is interpreted on its support.

This predicts the true latent light curve, so no future measurement variance is added. For noisy future observations, add their independent noise covariance to ${\Sigma }_{\ast }$. Fixing the hyperparameters omits their estimation uncertainty, as requested; integrating over their posterior would add it. The periodic model extrapolates by phase, even though all prediction times follow the observed times.

Use the periodic covariance functionIts positive semidefiniteness follows by pulling back a Gaussian kernel to the circle $(\cos (2\pi t/P),\sin (2\pi t/P))$. A zero Gaussian process mean is an ensemble statement, whereas an exactly zero long-term average for every periodic light curve is stronger. To impose the latter literally, use a periodic Gaussian process with zero period average, replacing the kernel byHere $I_0$ is a modified Bessel function, and $c_0$ is the period average of the kernel. Subtracting each process's random period average produces this covariance, so it remains positive semidefinite. We use $k_0$ to enforce the stated mean subtraction; using $k_P$ gives the usual ensemble-zero-mean formulation and the same likelihood derivation.

Let $K_{ij}=k_0(t_i,t_j)$ and $R=\mathrm{diag}({\sigma }_1^2,\dots ,{\sigma }_N^2)$. The prior latent vector is $f\sim N(0,K)$ and the independent Gaussian noise is $\epsilon \sim N(0,R)$, so $y=f+\epsilon \sim N(0,V)$ with $V=K+R$. The Gaussian-process marginal likelihood isPositive measurement variances make $V$ positive definite even if $K$ is singular. Maximize $\ell (\mathcal{H})=-\frac{1}{2}{y}^T{V}^{-1}y-\frac{1}{2}\log |V|-\frac{N}{2}\log (2\pi )$ over all three hyperparameters, using a broad search in $P$ to find competing aliases, followed by joint optimization. Cholesky decomposition evaluates the likelihood without explicitly forming an inverse.

For a well-resolved interior maximum, let $J=-{\nabla }^2\ell (\hat{\mathcal{H}})$ be the Observed Fisher information in the coordinates $(A,\ell ,P)$. Then $\boxed{\hat{P}=(\hat{\mathcal{H}}{)}_P,{\sigma }_P\approx \sqrt{(J^{-1}{)}_{PP}}}$. Equivalently, the profile-likelihood curvature in $P$ gives the same local nuisance-adjusted variance; keeping the other hyperparameters fixed would generally underestimate it. A profile interval with ${\ell }_{\mathrm{prof}}(\hat{P})-{\ell }_{\mathrm{prof}}(P)=1/2$ is a local $1\sigma$ approximation. If the period likelihood is multimodal or poorly constrained, fit the complete posterior with proper hyperparameter priors and report its modes and marginal credible intervals rather than a single Hessian error bar. The sampling pattern can create period aliases that a single local search misses.

Assume $m^2>0$ and take $\lambda \to 0^{+}$. The free Gaussian integral gives $Z_0=\sqrt{2\pi }/|m|$, and the normalized free Gaussian measure has covariance $C=m^{-2}$. Expanding only the interaction exponential gives the asymptotic expansionIn this zero-dimensional scalar field theory, the path integral has become an ordinary integral, but its combinatorics are precisely those of sextic scalar field theory.

The free generating functional is $⟨e^{J\varphi }{⟩}_0=e^{C{J}^2/2}$. Differentiating it proves Wick theorem: odd moments vanish, while each even moment is the sum over pairings of its factors. Each Wick contraction contributes $C$, soThus the Feynman rules are a quantum field theory propagator $C$ for each internal line and a factor $-\lambda$ for each six-valent interaction vertex. The factorials $V!$ and $(6!{)}^V$ remove vertex and half-edge labels; the remaining weight is $(-\lambda {)}^V{C}^I/S$, where $S$ is the Feynman-diagram symmetry factor, the order of the automorphism group of the resulting diagram, including its half-edge symmetries. A tadpole diagram has an additional interchange symmetry of the two ends of its loop.

There is one empty Vacuum Feynman diagram, contributing $1$. At one interaction vertex, the six half-edges form three loops. Its Feynman-diagram symmetry factor is $2^{3}3!=48$, so its contribution is $-\lambda C^3/48$.

At two interaction vertices let $r$ count the lines joining them. Each interaction vertex has $(6-r)/2$ self-loops, so $r$ can only be $0,2,4,6$. These four possibilities exhaust all Vacuum Feynman diagrams at this order. Their Feynman-diagram symmetry factors areHere the leading $2$ exchanges the two interaction vertices, $r!$ permutes the joining lines, and the remaining factors permute and reverse the self-loops. The $r=0$ graph is disconnected and must be included in $Z/Z_0$; only a connected generating functional would discard it.

Consequently,Indeed ${\sum }_r{S}_r^{-1}=77/7680$, independently agreeing with $11!!/[2(6!{)}^2]$ from Wick theorem.

This is an asymptotic expansion, rather than a convergent perturbation series: the coefficient of ${\lambda }^V$ contains $(6V-1)!!/[V!(6!{)}^V]$ and grows too rapidly for a nonzero radius of convergence. For any fixed truncation order, the Taylor expansion remainder bound for $e^{-x}$ on $x\ge 0$, followed by the finite Gaussian moment integral, proves the stated remainder estimate. For negative real $\lambda$ the original integral diverges.

The covariance matrix isIf $\mathrm{Cov}(X_1,X_2)=0$, the moment-generating function in part (ii) factors into the product of the two marginal moment-generating functions. Thus $X_1$ and $X_2$ are independent random variables; this is the independence of uncorrelated jointly normal variables.

By bilinearity of covariance,ChooseThen $(X_1,Y)$ is bivariate normal by part (i) and has zero covariance, so $Y$ is a normal random variable independent of $X_1$. Hence $X_2=a{X}_1+Y$.

Use a stationary, non-streaming tracer population with spherical symmetry in its full velocity distribution. In particular, the mean velocities vanish, the mixed covariances vanish, and the two tangential components have equal velocity dispersions. WriteHere ${\sigma }_t^2$ is the variance of one tangential component. In the alternative convention using the sum of both tangential variances, the same velocity-anisotropy parameter is $1-({\sigma }_{\theta }^2+{\sigma }_{\varphi }^2)/(2{\sigma }_r^2)$. These definitions agree; they must not be mixed by inserting an extra factor of two.

The time derivative and mean-velocity terms in the supplied Jeans equation vanish, as do its angular mixed-moment terms. Its remaining geometric term is $2\nu ({\sigma }_r^2-{\sigma }_t^2)/r$. Thus the Spherical Jeans equation becomesThe no-streaming assumption excludes a steady radial flow; stationarity of the density alone would not remove its advective terms.

By the shell theorem, the gravitational potential of a spherical mass distribution satisfies $d\Phi /dr=GM(<r)/r^2$. Expanding the derivative above, dividing by $\nu$, and using logarithmic derivatives givesThe tracer density $\nu$ need not be the density of the gravitating matter. The circular speed is $V_c^2=rd\Phi /dr=GM(<r)/r$, whether or not the tracers themselves follow circular orbits.

Now let $\beta$ be constant, $\nu ={\nu }_0(r/r_0{)}^{-\gamma }$, and $V_c^2=V_0^2(r/r_0{)}^{2\alpha }$. With $y={\sigma }_r^2$, the Spherical Jeans equation readsFor a power-law ansatz $y=K{V}_c^2$, $y^{\prime }=2\alpha y/r$, so $(\gamma -2\beta -2\alpha )y=V_c^2$. Therefore the scale-free spherical Jeans solution isThe positivity condition is necessary for this particular solution to be a physical velocity dispersion.

There is a boundary condition implicit in selecting this solution. The integrating factor $r^{2\beta -\gamma }$ gives, when $D=\gamma -2\beta -2\alpha \ne 0$,The homogeneous term represents an additional radial-pressure boundary contribution. It is removed, for example, by requiring $r^{2\beta }\nu {\sigma }_r^2\to 0$ at infinity. Under that condition,and convergence requires $D>0$, giving the boxed result. If $D=0$, the solution instead has the logarithmic formso the quoted constant proportionality does not apply. A finite outer boundary can retain a homogeneous term even when $D\ne 0$.

To derive the heliocentric projection of a spherical velocity dispersion, put the Sun at vector ${\mathbf{R}}_{\odot }$ of length $R_{\odot }$, and a tracer at $\mathbf{r}=r\hat{\mathbf{r}}$. Its heliocentric line of sight isIf $\chi$ is the angle between $\hat{\mathbf{\ell }}$ and the Galactocentric radial direction, thenAfter correction for the Sun's motion, the intrinsic covariance matrix has eigenvalues ${\sigma }_r^2,{\sigma }_t^2,{\sigma }_t^2$ in the local spherical basis. Projection onto the line of sight therefore givesAt fixed $r$, ${\sigma }_r$ and $\beta$ are constant across the spherical shell. Averaging its squared line of sight velocity, with zero mean, givesFor a uniformly sampled Galactocentric shell, the solid angle measure makes $u$ uniform on $[-1,1]$, giving the geometric correction for heliocentric velocity dispersionTo evaluate it, set $a=r^2+R_{\odot }^2$ and $b=2r{R}_{\odot }$, and divide the integrand:The odd term integrates to zero, while ${\int }_{-1}^{1}du/(a-bu)=b^{-1}\ln [(a+b)/(a-b)]$. For $r>0$, $R_{\odot }>0$, and $r\ne R_{\odot }$, this givesThe removable limit at $r=R_{\odot }$ is $A=1/2$. As $r/R_{\odot }\to \infty$, $A\simeq 2R_{\odot }^2/(3r^2)$, while as $r/R_{\odot }\to 0$, $A\to 2/3$. If $R_{\odot }=0$, $A=0$ and the observed velocity dispersion is exactly radial. For isotropic velocities, $\beta =0$, the correction vanishes at every radius.

The averaging measure matters. If directions are instead uniform in heliocentric solid angle at a fixed Galactocentric radius $r>R_{\odot }$, let $\vartheta$ be the angle between the line of sight and ${\mathbf{R}}_{\odot }$. Geometry gives ${\sin }^2\chi =(R_{\odot }/r{)}^2{\sin }^2\vartheta$, so that particular weighting gives $A=2R_{\odot }^2/(3r^2)$ exactly. The logarithmic expression above uses uniform tracer weighting on a Galactocentric shell. For an observational sample, use its actual angular selection weights in $A=⟨{\sin }^2\chi ⟩$; the boxed dispersion relation remains valid.

Equal LD and SD efficacy is equivalent tobecause both efficacy definitions use the same control probability. In this two-outcome table that is also equivalent to equality of the LD and SD log-odds interactions,The summary gives the two estimates and their individual standard error, but a test of their difference needsThe required covariance is absent from the displayed table, so the two separate coefficient p-values cannot test equality.

Instead, fit the equal-efficacy Poisson log-linear model. For example, create one indicator for a Worse outcome in either treated group:The reduced model has one common treated-versus-control outcome interaction, while retaining separate LD and SD main effects for their different group totals. The full model has two such interactions. Their analysis of deviance for nested generalized linear models is therefore a one-degree-of-freedom likelihood-ratio test of equal efficacy.

The Zero-boundary Gaussian free field on the unit disc is the centered Gaussian process indexed by finite Borel measures of finite Green energy, with covarianceThis covariance determines all its finite-dimensional distributions.

The Domain Markov property of the Gaussian free field says that for every suitable open $U\subset \mathbb{D}$ one can writewhere $h_U$ is a zero-boundary Gaussian free field on $U$, independent of $h^{\mathbb{D}\setminus U}$, while $h^{\mathbb{D}\setminus U}$ is harmonic on $U$ and carries the information from the field outside $U$.

The field $\psi$ is centered and jointly Gaussian. Since the Gaussian free field has covariance $\mathbb{E}[{\phi }_x{\phi }_y]=g(x,y)$ and $h_x=g(x,0)/g(0,0)$,Also ${\psi }_0=0$ andJoint Gaussianity turns this zero covariance into independence. Therefore $\psi$ is a Pinned Gaussian free field at $0$, independent of ${\phi }_0$, with covariance equal to the Green function of the walk killed on hitting $0$.