Updated manuscript

document/manuscript.tex

@@ -279,25 +279,29 @@ \section{The Generative Model}
 informative, so conceptually a fit to the expected fluxes $E_{i}$ is generally 
 preferred wherever possible. For brevity I define $\bm{\kappa} \equiv (\bm{\psi},\{z,\sigma_s,\{c_k\}_{k=0}^{m},f\}_{0}^{N_{c}})$, and the likelihood for the mixture model is given by
 
- \begin{equation}
-\mathcal{L} = \prod_{i=1}^{N}\,\left[\left(1 - P_{o}\right)\times{}p_{model}\left(F_i|\lambda_i,\sigma_{i},\bm{\kappa}\right) + P_{o}\times{}p_{outlier}\left(F_i|\lambda_i,\sigma_i,\bm{\kappa},V_{o},P_o\right)\right]
-\end{equation}
+ \begin{multline}
+\mathcal{L} = \prod_{i=1}^{N}\,\left[\left(1 - P_{o}\right)\times{}p_{model}\left(F_i|\lambda_i,\sigma_{i},\bm{\kappa}\right)\right. \\
++ \left. P_{o}\times{}p_{outlier}\left(F_i|\lambda_i,\sigma_i,\bm{\kappa},V_{o},P_o\right)\right]
+\end{multline}
 
 \noindent{}where $p_{model}$ refers to $p$ in Equation \ref{eq:p_model} and 
 
-\begin{equation}
-p_{outlier}\left(F_i|\lambda_i,\sigma_i,\bm{\kappa},V_{o},P_o\right) = \frac{1}{\sqrt{2\pi\left(s_{i}^2 + V_{o}^2\right)}} \exp\left(-\frac{[F_i - C_i]^2}{2\left[s_{i}^2 + V_{o}^2\right]}\right)
-\end{equation}
+\begin{multline}
+p_{outlier}\left(F_i|\lambda_i,\sigma_i,\bm{\kappa},V_{o},P_o\right) = \dots \\
+  \frac{1}{\sqrt{2\pi\left(s_{i}^2 + V_{o}^2\right)}} \exp\left(-\frac{[F_i - C_i]^2}{2\left[s_{i}^2 + V_{o}^2\right]}\right)
+\end{multline}
 
 \noindent{}such that the likelihood $\mathcal{L}$ becomes:
 
-\begin{equation}
-\mathcal{L} = \prod_{i=1}^{N} \left[ \frac{1-P_o}{\sqrt{2\pi{}s_{i}^2}}\,\exp\,\left(-\frac{[F_i - E_i]^2}{2s_{i}^{2}}\right) + \frac{P_o}{\sqrt{2\pi\left[s_{i}^2 + V_o\right]}}\,\exp\,\left(-\frac{[F_i - C_i]^2}{2\left[s_{i}^{2} + V_o\right]}\right)\right]
+\begin{multline}
+\mathcal{L} = \prod_{i=1}^{N} \left[ \frac{1-P_o}{\sqrt{2\pi{}s_{i}^2}}\exp\left(-\frac{[F_i - E_i]^2}{2s_{i}^{2}}\right) \right.\\
+\left. + \frac{P_o}{\sqrt{2\pi\left[s_{i}^2 + V_o\right]}}\exp\left(-\frac{[F_i - C_i]^2}{2\left[s_{i}^{2} + V_o\right]}\right)\right]
 \label{eq:full_likelihood}
-\end{equation}
+\end{multline}
 
-I define the full parameter space with $\bm{\theta} \equiv \left(\bm{\psi},\{z,\sigma_s,\{c_{b_k}\}_{k=0}^{m},f\}_{b=0}^{N_{c}},V_o,P_o\right)$. From Bayes theorem the posterior probability 
+I define the full parameter space with ${\bm{\theta} \equiv \left(\bm{\psi},\{z,\sigma_s,\{c_{b_k}\}_{k=0}^{m},f\}_{b=0}^{N_{c}},V_o,P_o\right)}$. From Bayes theorem the posterior probability 
 distribution for $\bm{\theta}$ (up to a constant) is given by
+
 \begin{eqnarray}
 \mathcal{P} & \propto & likelihood \times prior \nonumber \\
 p(\bm{\theta}|\{F_i\}_{i=1}^{N}) & \propto & p(\{F_i\}_{i=1}^{N}|\bm{\theta})\,\times\,p(\bm{\theta})
@@ -385,7 +389,7 @@ \subsection{Monte-Carlo Markov Chain Sampling}
  by \citet{goodman;weare}, and implemented by \citet{emcee}. The Metropolis-Hastings 
  MCMC algorithm is employed by default. \sick{} allows for the model settings to 
  be specified in a human-readable \textsc{yaml}- or \textsc{json}-formatted 
- configuration file\footnote{The reader is referred to the online documentation 
+ configuration file\footnote{The reader is referred to the online documentation at http://astrowizici.st/sick/ 
  for an example.}, where the number of Goodman \& Weare walkers can be specified, 
  as well as the number of samples to perform. When the optimisation step is used, 
  the initial points are taken from a small multi-dimensional ball around the 
@@ -400,15 +404,15 @@ \subsection{Monte-Carlo Markov Chain Sampling}
 uninformative prior distributions are assumed (for all channels, where appropriate):
 
 \begin{eqnarray}
-p\left(\bm{\psi}_{dim}\right) &=& \mathcal{U}\left(\min\left[\bm{\psi}_{dim}\right], \max\left[\bm{\psi}_{dim}\right]\right) \\
-p\left(P_o\right) &=& \mathcal{U}\left(0, 1\right) \\
-p\left(\log{f}\right) &=& \mathcal{U}\left(-10, 1\right) \\
-p\left(V_o,\sigma_s\right) &=& \left\{
+p\left(\bm{\psi}_{dim}\right) &\,=\,& \mathcal{U}\left(\min\left[\bm{\psi}_{dim}\right], \max\left[\bm{\psi}_{dim}\right]\right) \\
+p\left(P_o\right) &\,=\,& \mathcal{U}\left(0, 1\right) \\
+p\left(\log{f}\right) &\,=\,& \mathcal{U}\left(-10, 1\right) \\
+p\left(V_o,\sigma_s\right) &\,=\,& \left\{
 \begin{array}{c l}      
     1\,, &\mbox{for values greater than zero}\\
     0\,, &\mbox{otherwise}
 \end{array}\right. \\
-p\left(z,\{c_k\}_{k=0}^{m}\right) &=& 1 \\
+p\left(z,\{c_k\}_{k=0}^{m}\right) &\,=\,& 1 \\
 \label{eq:default_priors}
 \end{eqnarray} 
 
@@ -449,15 +453,15 @@ \subsection{Self-consistent inference test}
 
 \begin{figure*}
 \label{fig:chains}
-\includegraphics[height=\textheight]{chains.pdf}
+\includegraphics[height=\textheight]{figures/chains.pdf}
 \caption{Points sampled by the 200 walkers at each step during the self-consistent 
 inference test. The true values are marked in blue. The first 1250 steps are 
 discarded as the burn-in period.}
 \end{figure*}
 
 \begin{figure*}
 \label{fig:corner-inference}
-\includegraphics[width=\textwidth,height=\textwidth]{corner.pdf}
+\includegraphics[width=\textwidth,height=\textwidth]{figures/corner.pdf}
 \caption{Marginalised posterior distributions for all parameters $\bm{\theta}$ 
 from a faux observation with spectral resolution $\mathcal{R} \sim 10,000$ and 
 S/N ratio $\sim{}7$\,pixel$^{-1}$. True values are marked in blue. This figure 
@@ -490,7 +494,7 @@ \subsection{Self-consistent inference test}
 
 \begin{figure*}
 \label{fig:spectrum-inference}
-\includegraphics[width=\textwidth]{spectrum.pdf}
+\includegraphics[width=\textwidth]{figures/spectrum.pdf}
 \caption{A faux observed spectrum (black) of $\mathcal{R} \sim 10000$ and S/N 
 ratio of $\sim7$ with an underestimated variance by $\sim$10\% which was used for 
 the self-consistent inference test. The recovered maximum likelihood model spectrum 
@@ -546,7 +550,7 @@ \subsection{Sol}
 
 \begin{figure*}
 \label{fig:solar}
-\includegraphics[width=\textwidth,height=\textwidth]{solar.pdf}
+\includegraphics[width=\textwidth,height=\textwidth]{figures/solar.pdf}
 \caption{Stellar parameter ($\bm{\psi}$: $T_{\rm eff}$, $\log{g}$, [Fe/H], 
 [$\alpha$/Fe])  posterior distributions for the GIRAFFE/FLAMES twilight spectrum. 
 Nuisance parameters are not shown. Maximum likelihood values for each parameter 
@@ -620,7 +624,7 @@ \subsection{Globular Clusters}
 
 \begin{figure}[h!]
 \label{fig:clusters}
-\includegraphics[width=0.5\textwidth]{clusters.pdf}
+\includegraphics[width=0.5\textwidth]{figures/clusters.pdf}
 \caption{Metallicities of cluster stars inferred from noisy, low-resolution 
 spectra obtained with the AAOmega spectrograph. The agreement with the 
 literature values (see text) is very reasonable.}
@@ -667,8 +671,8 @@ \section{Conclusion}
 which includes regular reproduction of the examples presented in this \article{}. 
 Complete documentation is available online\footnote{astrowizici.st/sick}, 
 which includes a number of additional examples and tutorials. In the spirit of 
-full scientific reproducibility, the online documentation is complemented with the 
-files necessary to reproduce all of the examples presented in this \article{}. 
+scientific reproducibility, the online documentation is complemented with the 
+files necessary to reproduce examples presented in this \article{}. 
 The source code is distributed using \textsc{git}, and hosted online at 
 GitHub\footnote{github.com/andycasey/sick}.