Notation | Matrix calculus reminders | First-order matrix calculus | $$\begin{aligned}\frac{\partial\det(\boldsymbol{A})}{\partial p}&=\det(\boldsymbol{A}),\mathrm{tr}!\left(\boldsymbol{A}^{-1}\frac{\partial\boldsymbol{A}}{\partial p}\right),\\frac{\partial(\boldsymbol{U}\boldsymbol{A}\boldsymbol{V})}{\partial p}&=\boldsymbol{U}\frac{\partial\boldsymbol{A}}{\partial p}\boldsymbol{V},\\frac{\partial\boldsymbol{A}^{-1}}{\partial p}&= | $$\frac | \mathrm | $$\frac{\partial}{\partial p}\bigl((\boldsymbol{y}-\boldsymbol{x}p)^{\top}\boldsymbol{\Omega}(\boldsymbol{y}-\boldsymbol{x}p)\bigr) | Second-order matrix calculus | $$\frac | \boldsymbol | $$\frac | Inner products in the score | Unconstrained score and Hessian | $$\ell_{\boldsymbol{y}\mid\boldsymbol{\zeta}}(\boldsymbol{p}) | \tfrac{1}{2}\log\det\boldsymbol{\Omega}(\boldsymbol{p}) | The determinant term carries a factor $\tfrac{1}{2}$ | What “unconstrained” means here | Unconstrained gradient | Unconstrained Hessian | $$\begin | \underbrace | \boldsymbol | $$\begin | \underbrace | \boldsymbol | Constrained optimisation on the simplex | Log-sum-exp and softmax | $$s_ | \frac | e^{z_{i}-\operatorname{LSE}(\boldsymbol{z})},\qquad\operatorname{LSE}(\boldsymbol{z}) | $$H_ | s_{i}\bigl[(\delta_{ij}-s_{j})(\delta_{ik}-s_{k}) | Isometric log-ratio map | $$\boldsymbol | \mathbf | \mathbf | \operatorname{softmax}(\mathbf{V}\boldsymbol{\rho}) | ILR is a $C^{2}$ diffeomorphism | Why ILR is the default chart | Efficient derivatives of $\boldsymbol | $$\mathbf{J}_{\boldsymbol{\psi}}(\boldsymbol{\rho}) | \frac | $$\mathbf | Change of variables: ambient $\boldsymbol | Constrained score and Hessian | $$\nabla_{\boldsymbol{\rho}}\ell | \mathbf{J}{\boldsymbol{\psi}}^{\mathsf{T}}\nabla{\boldsymbol{p}}\ell | $$\mathbf | Implementation map | Expected Fisher information and Wald inference | Unconstrained expected Fisher information | $$I(\boldsymbol | Observed versus expected information | Constrained case: ILR delta method | $$I_ | \mathbf | Wald is the cheapest, not the safest, interval | Numerical speed-ups and solver safeguards | Cache a Cholesky factorisation of $\boldsymbol{\Sigma}(\boldsymbol{p})$ | Guard the box-constrained L-BFGS-B path | marqLevAlg Hessian sign under minimize = FALSE | Model objects and S3 accessors | Numerical consistency of the likelihood and its derivatives | Density against mvtnorm | Richardson extrapolation of the score and Hessian | Expected information versus Monte Carlo (\mathbb{E}[-\mathbf{H}]) | Structure-aware covariance backends | Decision tree | Computational complexity | Appendix: additive log-ratio chart | $$p_ | \frac | \frac | Why ALR is not the default | Structural zeros and folded simplex models | References

Generative model | Factorial design (972 scenarios) | Scenario ID | Log-likelihood along the simplex | Inference | Visualisations | Other one-sample log-likelihood slices (not drawn) | Likelihood geometry and solver behaviour | What is being optimised | One bulk, $N=1$ | Reading the twelve meta-scenarios | What the heatmaps still add | See also | References