typos rouen's slides

talks/202406-rouen/pln-background.qmd

@@ -161,7 +161,7 @@ Do as if $\hat{\theta}^{\text{ve}}$ was a MLE and $\bar{J}_n$ the log-likelihood
 
 ### Variational Fisher Information
 
-The Fisher information matrix is given by (from the Hessian of $J$) by
+For standard PLN, the Fisher information matrix is given by (from the Hessian of $J$) by
 
 $$I_n(\hat{\theta}^{\text{ve}}) = \begin{pmatrix}
   \frac{1}{n}(\mathbf{I}_p \otimes 
@@ -170,7 +170,7 @@ $$I_n(\hat{\theta}^{\text{ve}}) = \begin{pmatrix}
 \mathbf{\Omega}^{-1} 
   \end{pmatrix}$$
 
-and can be inverted blockwise to estimate $\mathbb{V}(\hat{\theta})$.
+where $A_{ij} = \exp{M_{ij} + 1/2 S_{ij}}$. 
 
 ### Confidence intervals and coverage
 
@@ -186,7 +186,7 @@ $B_{kj} =  \hat{B}_{kj} \pm \frac{q_{1 - \alpha/2}}{\sqrt{n}} \sqrt{\hat{\mathbb
 
 ### Theorem [@Westling2015]
 
-Under additional regularity conditions (still satisfied for example when $\theta$ and $\psi_i$ are restricted to compact sets), we have 
+Under additional regularity conditions (satisfied when $\theta$ and $\psi_i$ are restricted to compact sets), we have 
 $$
 \sqrt{n}(\hat{\theta}^{\text{ve}} - \bar{\theta})  \xrightarrow[]{d} \mathcal{N}(0, V(\bar{\theta})), \quad \text{where } V(\theta) = C(\theta)^{-1} D(\theta) C(\theta)^{-1}
 $$

talks/202406-rouen/slides.html

@@ -1688,14 +1688,14 @@ <h3 id="theorem-westling2015">Theorem <span class="citation" data-cites="Westlin
 <h2>Variance: naïve approach</h2>
 <p>Do as if <span class="math inline">\(\hat{\theta}^{\text{ve}}\)</span> was a MLE and <span class="math inline">\(\bar{J}_n\)</span> the log-likelihood.</p>
 <h3 id="variational-fisher-information">Variational Fisher Information</h3>
-<p>The Fisher information matrix is given by (from the Hessian of <span class="math inline">\(J\)</span>) by</p>
+<p>For standard PLN, the Fisher information matrix is given by (from the Hessian of <span class="math inline">\(J\)</span>) by</p>
 <p><span class="math display">\[I_n(\hat{\theta}^{\text{ve}}) = \begin{pmatrix}
   \frac{1}{n}(\mathbf{I}_p \otimes
 \mathbf{X}^\top)\mathrm{diag}(\mathrm{vec}(\mathbf{A}))(\mathbf{I}_p \otimes \mathbf{X}) &amp; \mathbf{0} \\
   \mathbf{0} &amp; \frac12\mathbf{\Omega}^{-1} \otimes
 \mathbf{\Omega}^{-1}
   \end{pmatrix}\]</span></p>
-<p>and can be inverted blockwise to estimate <span class="math inline">\(\mathbb{V}(\hat{\theta})\)</span>.</p>
+<p>where <span class="math inline">\(A_{ij} = \exp{M_{ij} + 1/2 S_{ij}}\)</span>.</p>
 <h3 id="confidence-intervals-and-coverage">Confidence intervals and coverage</h3>
 <p><span class="math inline">\(\hat{\mathbb{V}}(B_{kj}) = [n (\mathbf{X}^\top \mathrm{diag}(\mathrm{vec}(\hat{A}_{.j})) \mathbf{X})^{-1}]_{kk}, \qquad \hat{\mathbb{V}}(\Omega_{kl}) = 2\hat{\Omega}_{kk}\hat{\Omega}_{ll}\)</span></p>
 <p>The confidence intervals at level <span class="math inline">\(\alpha\)</span> are given by</p>
@@ -1704,7 +1704,7 @@ <h3 id="confidence-intervals-and-coverage">Confidence intervals and coverage</h3
 <section id="variance-sandwich-estimator" class="slide level2">
 <h2>Variance : sandwich estimator</h2>
 <h3 id="theorem-westling2015-1">Theorem <span class="citation" data-cites="Westling2015">(<a href="#/references" role="doc-biblioref" onclick>Westling and McCormick 2015</a>)</span></h3>
-<p>Under additional regularity conditions (still satisfied for example when <span class="math inline">\(\theta\)</span> and <span class="math inline">\(\psi_i\)</span> are restricted to compact sets), we have <span class="math display">\[
+<p>Under additional regularity conditions (satisfied when <span class="math inline">\(\theta\)</span> and <span class="math inline">\(\psi_i\)</span> are restricted to compact sets), we have <span class="math display">\[
 \sqrt{n}(\hat{\theta}^{\text{ve}} - \bar{\theta})  \xrightarrow[]{d} \mathcal{N}(0, V(\bar{\theta})), \quad \text{where } V(\theta) = C(\theta)^{-1} D(\theta) C(\theta)^{-1}
 \]</span> for <span class="math inline">\(C(\theta) = \mathbb{E}[\nabla_{\theta\theta} \bar{J}(\theta) ]\)</span> and <span class="math inline">\(D(\theta) = \mathbb{E}\left[(\nabla_{\theta} \bar{J}(\theta)) (\nabla_{\theta} \bar{J}(\theta)^\intercal \right]\)</span></p>
 <h4 id="practical-computations-chain-rule">Practical computations (chain rule)</h4>
@@ -1722,7 +1722,7 @@ <h4 id="caveat">Caveat</h4>
 </section></section>
 <section>
 <section id="handling-zeros-in-multivariate-count-tables" class="title-slide slide level1 center">
-<h1><br> <br> Handling <br> zeros in <br> multivariate <br> count tables</h1>
+<h1><br> Handling zeros in <br> multivariate count tables</h1>
 
 </section>
 <section id="a-zero-inflated-pln" class="slide level2">
@@ -1829,7 +1829,7 @@ <h4 id="more-accurate-variational-approximation">More accurate variational appro
 <p><span class="math display">\[\begin{aligned}
 q_{\psi_i}(\boldsymbol Z_i, \boldsymbol W_i) &amp; = q_{\psi_i}(\boldsymbol Z_i | \boldsymbol W_i) q_{\psi_i}(\boldsymbol W_i) \\
 &amp; = \otimes_{j = 1}^p \mathcal{N}(\boldsymbol x_i^\top \mathbf{B}_j, \Sigma_{jj})^{W_{ij}} \mathcal{N}(M_{ij},
-    S_{ij}^2)^{1-W_{ij}} W_{ij}, \quad W_{ij} \sim^\text{indep} \mathcal{B}\left(\rho_{ij}\right)\end{aligned}.\]</span></p>
+    S_{ij}^2)^{1-W_{ij}}, \quad W_{ij} \sim^\text{indep} \mathcal{B}\left(\rho_{ij}\right)\end{aligned}.\]</span></p>
 <h4 id="counterpart">Counterpart</h4>
 <p>We loose close-forms in M Step of VEM for <span class="math inline">\(\hat{\mathbf{B}}\)</span> and <span class="math inline">\(\hat{\mathbf{\Sigma}}\)</span> in the corresponding ELBO…</p>
 </section>

talks/202406-rouen/zi-pln.qmd

@@ -1,5 +1,5 @@
 
-#  <br/> <br/> Handling <br/> zeros in  <br/> multivariate  <br/> count tables
+# <br/> Handling zeros in  <br/> multivariate  count tables
 
 ## A zero-inflated PLN
 
@@ -135,7 +135,7 @@ $\rightsquigarrow$ Only an approximation of $Z_{ij} | Y_{ij}, W_{ij} = 0$ is nee
 $$\begin{aligned}
  q_{\psi_i}(\boldsymbol Z_i, \boldsymbol W_i) & = q_{\psi_i}(\boldsymbol Z_i | \boldsymbol W_i) q_{\psi_i}(\boldsymbol W_i) \\
  & = \otimes_{j = 1}^p \mathcal{N}(\boldsymbol x_i^\top \mathbf{B}_j, \Sigma_{jj})^{W_{ij}} \mathcal{N}(M_{ij},
-    S_{ij}^2)^{1-W_{ij}} W_{ij}, \quad W_{ij} \sim^\text{indep} \mathcal{B}\left(\rho_{ij}\right)\end{aligned}.$$
+    S_{ij}^2)^{1-W_{ij}}, \quad W_{ij} \sim^\text{indep} \mathcal{B}\left(\rho_{ij}\right)\end{aligned}.$$
 
 #### Counterpart