\subsection{Definitions}
\begin{itemize}
\item $\lambda$ : intrinsic coordinate, including masses and spins.

\item $\theta$ : extrinsic coordinate, including $d,RA,DEC,\iota,\psi_L,t,\phi_{\rm orb}$

\item $p_s(\theta)$: (joint) sampling prior in extrinsic dimensions

\item $p(\theta)$ : prior on extrinsic parameters

\item $\Like(\lambda,\theta)$ : likelihood.  In terms of individual detector strains $H_k$ and power spectra, provided by
\begin{eqnarray}
\ln L &\equiv \sum_k \ln L_k  = \ln L_{\rm model} + \ln L_{\rm data} \\
\ln L_{\rm model} &\equiv -\frac{1}{2} \sum_k \qmstateproduct{H_k}{H_k}_k  \\
\ln L_{\rm data} &\equiv  \sum_k \text{Re} \qmstateproduct{H_k}{\hat{H}_k}_k 
\end{eqnarray}

\item $Z(\lambda) \equiv L_{\rm red}(\lambda,\theta)$ : reduced or integrated likelihood, derived from $L$ via
\begin{eqnarray}
Z(\lambda) = L_{\rm red}(\lambda) = \int d\theta \; p(\theta) L(\lambda,\theta)
\end{eqnarray}

\item 
$w=Lp/p_s$ : weight

\item 
$n_{\rm eff}$ : ``effective number of samples''
\begin{eqnarray}
n_{\rm eff} \equiv  \frac{\sum_k w_k}{\text{max}_k w_k}
\end{eqnarray}

\item 
$h(t|\lambda,x)=h_+-i h_\times$ : complex gravitational wave strain

\item 
$h_{lm}(t)$: coefficients of a spin-weighted spherical harmonic decomposition
\begin{eqnarray}
\label{eq:def:hSpinWeightEmissionDirection}
h(t|\lambda,\theta) = \sum_{lm} h_{lm}(t|\lambda) e^{-2i\psi}\Y{-2}_{lm}(\theta_{JN}\phi_{JN})
\end{eqnarray}

\item 
$\tilde{h}(f)$ : two-sided Fourier transform of the complex function $h(t)$
\begin{eqnarray}
h(t) = \int_{-\infty}^{\infty} \frac{d \omega}{2\pi} \; e^{-i\omega t} \tilde{h}(\omega) 
\end{eqnarray}

%% \item
%% ${\cal I}$ : complex conjugation in time.  Provided to avoid confusion with $\tilde{h}^*$.  \textbf{Hopefully we won't
%%   need it.}


\item 
$\vec{x}_k$ : Position of the $k$th detector

\item 
$F_{+}$, $F_{\times},F$ : detector response function  to the $+,\times$ polarizations for sources visible in the
  $\hat{n}$ direction relative to detector
\begin{eqnarray}
F(\hat{n}) = F_+(\hat{n}) +i F_\times(\hat{n})
\end{eqnarray}

\item 
$\hat{H}_k$ : measured strain in  the $k$th detector

\item 
$H_k$ : strain response of the $k$th detector to an incident strain $h$
\begin{align}
H_k(t) &=F_{+,k}(t) h_+(t-\vec{x}_k(t)\cdot \hat{k}) + F_\times(t) h_\times(t-\vec{x}_k(t)\cdot \hat{k}) \\
 &=  \frac{F h(t-\vec{x}_k\cdot \hat{k}) }{2} + \frac{F^*h^*(t-\vec{x}_k\cdot \hat{k})}{2}
\end{align}

\item 
$S_k$ : noise power spectrum for the $k$th detector

\item 
$\qmstateproduct{a}{b}_k$ : complex-valued inner product defined by the $k$th detector's noise power spectrum:
\begin{eqnarray}
\qmstateproduct{a}{b}_k \equiv 2 \int_{-\infty}^{\infty} df \frac{[\tilde{a}(f)]^*\tilde{b}(f)}{S_h(|f|) }
\end{eqnarray}

\end{itemize}
