2p2.tex
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\chapter{Bayesian Spectral Analysis of ``MAD'' Stars.}

%\chapterauthors{authors\footnote{affiliation}}
\chapterauthors{Nondas Sourlas\footnotemark[1], David A. van Dyk
\footnote{Department of Statistics, Harvard University, One Oxford
St., Cambridge, MA 02138},
Vinay Kashyap\footnotemark[2], Jeremy Drake\footnotemark[2], and Deron Pease
\footnote{Harvard-Smithsonian
Center for Astrophysics, 60 Garden St., Cambridge, MA 02138}}.


\section{Overview}
Getting reliable estimates of coronal metallicity ($Z$) from X-ray
spectra obtained with instruments such as ASCA/SIS and Chandra/ACIS is
very difficult, because the sole determinant of $Z$ is the ratio of
line to continuum fluxes, which is not well-determined for
low-resolution spectra.  Here we propose new Bayesian methods which
directly model the Poisson nature of the data.  Our model also accounts
for the Poisson nature of background contamination, blurring due to 
instrument response, and the absorption of photons in space.  The 
resulting highly structured hierarchical model is fit using the Gibbs 
sampler, data augmentation, and Metropolis-Hasting.  We demonstrate our 
methods with the X-ray spectral analysis of several apparently coronal 
metal abundance deficient ("MAD") stars.

\section{A Poisson Spectral Model}
  
The model is designed to summarize the relative frequency of the
energy of photons arriving at a detector.  We model the photon counts 
in each bin as independent Poisson random variables.  Specifically, we 
model the high energy tail of the ASCA spectrum (2.5-7.5 keV) as a 
combination of a Bremsstrahlung continuum and ten narrow emission lines, 
included at positions of known strong lines.  This source model is 
combined with instrument response, the effective area of the instrument, 
and background contamination to model actual observed counts 
\citep{vand:conn:kash:siem:01}.

Statistical analysis is based on two observations of the same source
and two of the same background.  We use sequential Bayesian analysis
for the two source observations; the posterior distribution from the
first analysis is used to construct an informative prior for the
second.  Non-informative priors were used in the first stage.  Model 
fitting proceeds by using the EM algorithm to check for multimodality in 
the posterior.  A Markov chain Monte Carlo algorithm is constructed to 
sample from that posterior.  Three chains are used at each step of the 
analysis in order to assess convergence.  Posterior inference is based 
on the second half of the draws of all the chains.  The sensitivity of 
our results to the choice of prior was investigated by altering the prior.  
We used residual plots for both source and background to diagnose the fit 
of our models.


\section{Results}

We have measured the coronal metallicity for 4 stars, $\alpha$ Aur
(Capella), $\sigma$ Gem, YY Gem, and II Peg.  Our result for Capella 
($Z=0.73_{>0.24}^{<2.4}$) are consistent with the independently 
determined value of $Z=0.57-0.78$ \citep{brick:etal:00}.  Based on our 
analysis of $\sigma$ Gem ($Z=0.81_{>0.59}^{<1.1}$), we find that contrary 
to classical analyses ($Z=0.25$) it is {\sl not} a MAD star (photospheric 
$Z=0.6$, \citet{rand:etal:94}).  YY Gem is found to have sub-Solar 
metallicity ($Z=0.46_{>0.23}^{<0.71}$) but higher than EUVE/SW measurements 
($Z \sim 0.1$; \citet{kash:etal:98}).  For II Peg, we find $Z=1.1_{>0.88}^
{<1.3}$, higher by a factor $\sim 10$ than preliminary results from 
Chandra/HETG; this discrepancy may be due to anomalies in the abundances 
of other elements, or to a high-temperature component to the emission measure.

\vspace{0.5cm}

{\it Acknowledgments:}
The author gratefully acknowledges funding for this project partially
provided by NSF grant DMS-01-04129 and by NASA contract NAS8-39073
(CXC).  This chapter is a result of a joint effort of the members of the
Astro-Statistics working group at Harvard University.

\bibliography{../../../../TeXfiles/BibTeX/my}{


\begin{thebibliography}{}

%\bibitem[van Dyk(2000)]{vand:00astro}
%van Dyk, D.~A. (2000).
%\newblock Fast new {EM}-type algorithms with applications in astrophysics.
%\newblock \emph{Technical Report} .

\bibitem[van Dyk \emph{et~al.}(2001)van Dyk, Connors, Kashyap, and
  Siemiginowska]{vand:conn:kash:siem:01}
van Dyk, D.~A., Connors, A., Kashyap, V., and Siemiginowska, A. (2001).
\newblock Analysis of energy spectra with low photon counts via {B}ayesian
  posterior simulation.
\newblock \emph{The Astrophysical Journal} \textbf{548}, 224--243.

\bibitem[Brickhouse \emph{et~al.}(2000)Brickhouse, et al.]
{brick:etal:00}
Brickhouse, N.~S., Dupree, A.~K., Edgar, R.~J., Liedahl, D.~A.,
Drake, S.~A., White, N.~E., and Singh, K.~P. (2000)
\newblock \emph{The Astrophysical Journal} \textbf{530}, \textbf{387}.

\bibitem[Randich \emph{et~al.}(1994)Randich, Giampapa, and Pallavicini]
{rand:etal:94}
Randich, S., Giampapa, M.~S., and Pallavicini, R. (1994)
\newblock \emph{Astronomy and Astrophysics} \textbf{283}, \textbf{893}.

\bibitem[Kashyap \emph{et~al.}(1998)Kashyap, Drake, Pease and Schmitt]
{kash:etal:98}
Kashyap, V., Drake, J., Pease, D., and Schmitt (1998).
\newblock \emph{American Astronomical Society} \textbf{192}, 82.01.

\end{thebibliography}


\end{document}

PPP2.tex
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%\slideframe{none}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

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\spacingset{1.2}
\begin{center}
{\LARGE\bf Bayesian Spectral Analysis of Metal Abundance Deficient Stars}
\end{center}

\begin{center}
Epaminondas Sourlas and David A. van Dyk,
  Department of Statistics, Harvard University\\

%One Oxford St. Cambridge, MA 02138, USA, e-mail:{\tt sourlas@stat.harvard.edu}\\
\end{center}

\begin{center}
Vinay Kashyap, Jeremy Drake and Deron Pease, Harvard-Smithsonian Center for Astrophysics\\
%60 Garden Street, Cambridge, MA 02138
\end{center}

\kern -0,15in
\smallskip

\noindent{\large\bf Abstract}

\kern -0,10in

%The metalicity (the metal abundance) of stellar coranae is an important key 
%in the understanding of their astrophysics.  
{\footnotesize Metallicity can be measured by
analyzing the spectra in the X-ray region and comparing the flux in spectral 
lines to the flux in the underlying Bremsstrahlung continuum. 
In this paper we both explore the sampling properties of the standard
astrophysical estimates in the small-sample regime appropriate for 
the data sets of interest and propose new Bayesian methods which directly 
model the Poisson nature of the data and thus are expected to exhibit
improved sampling properties. Our model also accounts for the Poisson
nature of background contamination of the observations, image blurring
due to instrument response, and the absorption of photons in space.
The resulting highly structured hierarchical model is fit using the Gibbs 
sampler, data augmentation and Metropolis-Hasting. We demonstrate
our methods with the X-ray spectral analysis of several "Metal Abundance 
Deficient" stars.
The model is designed to summarize the relative frequency of the energy of
photons (X-ray or $\gamma$-ray) arriving at a detector.  Independent Poisson 
distributions are more appropriate to model the counts than the commonly used 
normal approximation. 
We model the high energy tail of the ASCA spectrum of each of the stars
as a combination of a Bremsstrahlung continuum and ten narrow emission lines.
The ten emission lines are included at positions of known strong lines.
Statistical analysis is based on two source observations and one background
observation.  We use sequential Bayesian analysis for the two source 
observations; the posterior distribution from the first analysis is used to 
construct a prior for the second.  Sensitivity of the final results to the 
choise of prior is investigated by altering the prior.}

\end{slide*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide*}

\heading{A Bayesian Poisson Spectral Model}

We use van Dyk et al.'s (ApJ, 2001) hierarchical model for spectral
analysis of high resolution X-ray and $\gamma$-ray spectra.

Bayesian methodology enables direct modelling of the Poisson nature of
the source counts and background contamination, instrument response,
absorbtion and varying effective area.

\bigskip

We use a generalized linear model for the source continuum (e.g., Bremsstrahlung 
continuum) and allow for several emission lines.

\end{slide*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide*}
\heading{Model Specification}

\begin{enumerate}
	\item Independent Poisson distributions are used to model the 
	unobserved uncontaminaited counts.  
	$$Y_j\sim {\rm Poisson}\l[\lambda_j(\theta)\r]$$
	We parameterize the intensity at energy $E_j$, as a mixture of a
	continuum term and $K$ 
	Gaussian distributions, 
	$$\lambda_j(\theta) = \delta_j f(\theta^C,E_j) + \sum_{k=1}^K \tl 
	p_j (\mu_k,\sigma_k^2), $$
        $f(\theta^C,E_j)$ represents the continuum and is typically a log linear model.
        
\smallskip

        \item Because of degradation due to the background, the detector,
        and absorption, we model the observed counts as independent
        Poisson variables
        $$Y_j^{obs}\sim {\rm Poisson}\l[\xi_l(\theta)\r]$$
        with parameters
	$$\xi_l(\theta) =\underbrace{\sum_{j=1}^J M_{lj} \lambda_j(\theta) 
	d_j g(\theta^A, E_j)}_{\rm{source}} + 
	\underbrace{\lambda_l^B(\theta^B)}_{\rm{bkg}}, $$

\eject

	where
\begin{itemize}
	\item The first term represents the source and the second term represents
	the background.
	\item $M_{lj}$ is instrument response, the probability that 
	a photon with energy $E_j$ is being detected in observed bin $l$.
	\item $d_j$ is the effective area of bin $j$, normalized to sum to 1.
	\item $g(\theta^A,E_j)$ is the probability a photon is not absorbed. 
	\item $\lambda_l^B(\theta^B)$ is the expected counts due to background.
\end{itemize}
\end{enumerate}

\end{slide*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide*}

\heading{Spectral Analysis of}
\heading{Metal Abundance Deficient Stars} 

\smallskip 

We model the high energy tail of the ASCA spectrum (2.5 - 7.5 kev)
of each of several stars as a combination of a Bremsstrahlung continuum and 
ten narrow emission lines.

\begin{enumerate}
	\item The continuum is parameterized as
	$$f(\theta^l,E_j) = {Norm\over\sqrt{T}}e^{-E_j/k_BT} $$
	\item The ten emission lines are included at positions of known strong 
	lines ($\mu^k$ are fixed, and their widths are fixed at zero). 
	\item The background counts are modeled as:
	$$Y^B_l\sim {\rm Poisson}(\theta^B_l)$$
	where the background intensities $\theta_l^B$, are modeled using
	a second log linear model, in particular a {\it Power law},
	$$\log\theta_l^B = \alpha - \beta\log{E_l}. $$
\end{enumerate}

\eject

\heading{The Stars}

We analyze the ASCA data of four stars: Capella, $\sigma$ Gem, YY Gem, and II Peg.

\smallskip

The motivation for choosing these specific stars is:
\begin{itemize}
	\item To validate our method comparing our results with known facts
	about certain stars (Capella, II-Peg).
	\item To compare our method to previous ones and point out the 
	modeling advantages ($\sigma$-Gem).  We are also planning to analyze
	a few other stars such as $\xi$ Uma and $\xi$ Boo.
	\item To get a better understanding of uncommon findings about certain 
	stars (YY-Gem).
\end{itemize}
\end{slide*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide*}
\heading{Prior Distributions}

\begin{enumerate}
	\item We used highly non-informative prior distributions in the first
	stage of the analysis:
	\begin{table}
	\begin{center}
	\begin{tabular}{lccc}
	\hline\hline
	& \footnotesize{$p({\log({Norm\over\sqrt{T}})})$} & 
	  \footnotesize{$p(\frac{1}{kT})$} & \footnotesize{$p(\tilde\lambda^k)$}\\
%	Parameter &  & YY-Gem & Capella & IIPeg\\[0.5ex]
	\hline
	\small{Capella} & \small{flat} & \small{flat} & 
		\small{$\gamma(\mu=1500,\sigma=1875e3)$}\\
	\small{$\sigma$ Gem} & \small{flat} & \small{$N(\mu=5.1,\sigma=2.5)$}
		& \small{$\gamma(\mu=1500,\sigma=1875e3)$}\\
	\small{YY Gem} & \small{flat} & \small{$N(\mu=5.1,\sigma=2.5)$} & 
		\small{$\gamma(\mu=1500,\sigma=1875e3)$}\\
	\small{II Peg} & \small{flat} & \small{flat} & 
		\small{$\gamma(\mu=150,\sigma=18750)$}\\
	\hline
	\hline
	\end{tabular}\\%[0.5ex]
	\end{center}
	\end{table}
	
	\item
	The prior for the total count from all the lines, $\lambda^L$ 
	= $\sum_{k=1}^{10}$ $\tilde\lambda^k$ was truncated to be less
	 than 70\% of the total counts in the selected energy range 
	 of 2.5 - 7.5 kev; which translates into the following truncation points:
  	\begin{table}
	\begin{center}
	\begin{tabular}{lcccc}
	\hline\hline
	data set & Capella & $\sigma$ Gem & YY Gem & II Peg\\[0.5ex]
	\hline
	gis2 & N/A & 2071 & 516 & N/A\\
	sis0 & 349 & 2470 & 627 & 2331\\
	sis1 & 284 & N/A & N/A & 1826\\
	\hline
	\hline
	\end{tabular}\\[0.5ex]
	\end{center}
	\end{table}

	\item Flat priors were assumed for background parameters.

\end{enumerate}

\end{slide*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}
\heading{Model Fitting}

Statistical analysis is based on two observations of the same source and 
two of the same background.  The background observations are strongly
informative for $\theta^B$.  

\bigskip	

We use sequential Bayesian analysis for the two source observations;
the posterior distribution from the first analysis is used to construct
an informative prior for the second.

\end{slide*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\heading{A Sample Statistical Analysis}

The analysis was done by:

\begin{enumerate}
	\item Fitting the model using the Markov Chain Monte Carlo 
	algorithm (Gibbs Sampler) to sample from the posterior distributions.  
	\item Posterior inference is based on the second half of 4000
	iterations of three Markov chains for a total of 6000 draws for 
	each source observation.  Each chain uses different, over-dispersed,
	starting values.
	\item Convergence of the MCMC algorithm is verified using the method 
	of Gelman and Rubin (1992). 
	\item Sensitivity of the results to the choice of prior distribution
	is examined by altering the prior. 
	\item The standard deviation of the posterior distribution resulting 
	from the first source observation is inflated by a factor of 2 to form the 
	prior distribution for analysis in the second source observation.
\end{enumerate}
\end{slide*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide*}

\heading{$\sigma$ Gem}

Analysis I:

\begin{figure}[h]
  \begin{center}
    \includegraphics[width=3.4in,height=1.2in]
	{SigGem.g2ph_postfig1.eps}
  \end{center}
\end{figure}

Analysis II:

\begin{figure}[h]
  \begin{center}
    \includegraphics[width=3.4in,height=1.2in]
	{SigGem.s0b1s_postfig.eps}
    \caption{\footnotesize{Posterior distributions of some of the parameters
	from the two instruments.  Lambda is the expected model counts from
	the lines, and omega is the ratio of the total counts in the lines
	to the total counts in the spectrum.  Transformations of the 
	parameters are used to produce the distribution nearest to the Gaussian.
	The dashed lines in Analysis I represent the non-informative priors,
	and the solid lines the posterior distribution after having inflated
	its standard deviation by a factor of two to construct the priors for
	Analysis II (solid lines).}}
  \end{center}
\end{figure}

\smallskip

\end{slide*}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide*}

\heading{The Fitted Spectrum of $\sigma$ Gem}
\begin{figure}[h]
  \begin{center}
   \includegraphics[width=3.5in,height=3.5in]
	{SigGem.s0b1_stabilityfig.eps}
   \caption{\footnotesize{Overview of the fitted model.  The first plot compares the fitted
		model with the observed PHA counts.  The second one gives the residual
		for each PHA channel after subtracting the background and standardizing
		by the model standard deviation.  The third illustrates the stability 
		of the continuum.}}
  \end{center}
\end{figure}
\end{slide*}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\heading{Evaluating the Model for $\sigma$ Gem}
\begin{figure}
\begin{center}
%   \includegraphics[width=3.5in,height=3.5in]
\begin{turn}{-90}
\epsfig{figure=SigmaGem.s0b1_modeldiag.ps,width=2.8in}
\end{turn}
   \caption{\footnotesize{Those are diagnostics plots for our fitted model.  The upper 
		plots are the fit of the source model and the corresponding residual plot,
		and the lower two plots are the fit of the background model and its
		residual plot.  The solid lines in the residual plots represent plus
		and minus two standard deviations.}}
  \end{center}
\end{figure}
\end{slide*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide*}

%\begin{table}
%\begin{center}
\begin{sidewaystable}
\centering
\heading{Results}
\setlength{\doublerulesep}{1pt}
\setlength{\extrarowheight}{-3pt}
\setlength{\textwidth}{7pt}
\setlength{\columnsep}{1pt}
\begin{tabular}{||@{}c@{}|@{}c@{}||@{}c@{}||@{}c@{}||@{}c@{}||@{}c@{}||@{}c@{}||}
\hline\hline


\multicolumn{1}{||l|}{Star} & \multicolumn{1}{l||}{Data}
& \multicolumn{4}{c||}{Parameters} & \multicolumn{1}{c||}{Fe/H}\\\hline

& & \multicolumn{1}{c||}{Norm} & \multicolumn{1}{c||}{kT} 
& \multicolumn{1}{c||}{Exp Line Cnt} 
& \multicolumn{1}{c||}{\%-From Lines} & \\\hline\hline


\multirow{4}{17mm}{Capella} 
& \multirow{2}{6mm}{sis1} & 0.00442 & 0.78 & 212 & 0.26 & \\
	& & \scriptsize (9.3e-4, 1.9e-2) & \scriptsize(0.6, 1.13) & 
		\scriptsize (127, 280) & \scriptsize (0.124, 0.437) &
		-0.137\\\cline{2-6}
& \multirow{2}{6mm}{sis0} & 0.00798 & 0.726 & 177  & 0.147 & \scriptsize (-0.6135, 0.375)\\
	& & \scriptsize (2.7e-3, 2.2e-2) & \scriptsize (0.6, 0.93) & 
		\scriptsize (87, 290) & \scriptsize (0.06, 0.27) & \\\hline

\multirow{4}{17mm}{$\sigma$ Gem} 
& \multirow{2}{6mm}{gis2} & 8.5e-5 & 1.52 & 929 & 0.26 & \\
	& & \scriptsize (5.4e-5, 1.3e-4) & \scriptsize (1.35, 1.78) & 
		\scriptsize (690, 1180) & \scriptsize (0.186, 0.34) &
		-0.08956\\\cline{2-6}
& \multirow{2}{6mm}{sis0} & 7.4e-5 & 1.28 & 785  & 0.14 & \scriptsize(-0.227, 0.0394)\\
	& & \scriptsize (5.63-5, 9.7e-5) & \scriptsize (1.18, 1.39) & 
		\scriptsize (579, 1010) & \scriptsize (0.1, 0.19) & \\\hline

\multirow{4}{17mm}{YY Gem (3-7.5keV)} 
& \multirow{2}{6mm}{gis2} & 0.000613 & 1.67 & 144 & 0.178 & \\
	& & \scriptsize (2.4e-4, 1.57e-3)  & \scriptsize (1.24, 2.5) & 
		\scriptsize (91, 204) & \scriptsize (0.098, 0.273) &
		-0.3363 \\\cline{2-6}
& \multirow{2}{6mm}{sis0} & 0.00102 & 1.44 & 101  & 0.099 & \scriptsize (-0.634, -0.151)\\
	& & \scriptsize (5.3e-4, 1.9e-3) & \scriptsize (1.18, 1.84) & 
		\scriptsize (49, 186) & \scriptsize (0.048, 0.174) & \\\hline

\multirow{4}{17mm}{II Peg} 
& \multirow{2}{6mm}{sis1} & 0.0037 & 1.32 & 540 & 0.112 & \\
	& & \scriptsize (0.00255, 0.0052) & \scriptsize(1.19, 1.48) & 
		\scriptsize (355, 752) & \scriptsize (0.071, 0.163) &
		0.0299\\\cline{2-6}
& \multirow{2}{6mm}{sis0} & 0.0028 & 1.42 & 1040  & 0.167 & \scriptsize (-0.056, 0.1135)\\
	& & \scriptsize (0.0022, 0.0036) & \scriptsize (1.31, 1.54) & 
		\scriptsize (872, 1230) & \scriptsize (0.135, 0.199) & \\\hline\hline

\end{tabular}
\end{sidewaystable}
%\end{center}
%\end{table}
\end{slide*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{slide*}

\heading{Conclusion}

\begin{itemize}
	\item Capella coronal Z(Fe) is 0.57-0.78 (Brickhouse et al., 2000, ApJ
	539, 387), which is consistent with our measurements.  Our analysis
	is also consistent with Chandra grating data of Capella.

	\item $\sigma$ Gem XSPEC measurement is [Fe/H] = -0.6, compared to the
	photospheric abundance of -0.3 (Randich, Giampapa, \& Pallavicini, 
	1994, A\&A 283, 893), but we measure the abundance to be approximately
	Solar, and hence conclude that this should not be considered a MAD 
	(Metal Abundance Deficient) star."

	\item Based on line-to-continuum measurements using EUVE/SW data,
	Kashyap et al. $1998$ (AAS 192, 82.01) determined the metallicity of 
	YY Gem to be very low, [Fe/H] $\sim -1$.  This low value may have 
	been due to an underestimation of line flux contribution and an 
	overestimation of the continuum level.  Our analysis instead shows
	that [Fe/H] is larger, though still sub-Solar.

	\item Metallicity derived for II Peg is approximately Solar, which
	is inconsistent with preliminary results from Chandra grating data
	(Drake, J.J., 2001, private communication), which suggest metallicities
	that are almost a factor of 10 lower.  The discrepancy may be due to 
	anomalies in the abundances of other elements such as Si and Mg, or due
	to a high-temperature component to the emission measure.

	\item Our findings are inconsistent with 2 or 3 temperature, global
	fits to the spectra (usually carried out over a wider energy range of
	0.5-7 keV).  This inconsistency may be attributed to atomic line 
	incompleteness at low energies, uncertainty in element abundances,
	and lack of weight given to the high-energy tail in standard analyses
	becausse of the low counts at these energies.
% and presumably the 
%	wrong model assumptions (like normality of the photon-counts) of 
%	standard fitting algorithms.
\end{itemize}

\end{slide*}

\end{document}



astro-ph0203165.extracted_bib
\begin{thebibliography}{}

%\bibitem[van Dyk(2000)]{vand:00astro}
%van Dyk, D.~A. (2000).
%\newblock Fast new {EM}-type algorithms with applications in astrophysics.
%\newblock \emph{Technical Report} .

\bibitem[van Dyk \emph{et~al.}(2001)van Dyk, Connors, Kashyap, and
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van Dyk, D.~A., Connors, A., Kashyap, V., and Siemiginowska, A. (2001).
\newblock Analysis of energy spectra with low photon counts via {B}ayesian
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\newblock \emph{The Astrophysical Journal} \textbf{548}, 224--243.

\bibitem[Brickhouse \emph{et~al.}(2000)Brickhouse, et al.]
{brick:etal:00}
Brickhouse, N.~S., Dupree, A.~K., Edgar, R.~J., Liedahl, D.~A.,
Drake, S.~A., White, N.~E., and Singh, K.~P. (2000)
\newblock \emph{The Astrophysical Journal} \textbf{530}, \textbf{387}.

\bibitem[Randich \emph{et~al.}(1994)Randich, Giampapa, and Pallavicini]
{rand:etal:94}
Randich, S., Giampapa, M.~S., and Pallavicini, R. (1994)
\newblock \emph{Astronomy and Astrophysics} \textbf{283}, \textbf{893}.

\bibitem[Kashyap \emph{et~al.}(1998)Kashyap, Drake, Pease and Schmitt]
{kash:etal:98}
Kashyap, V., Drake, J., Pease, D., and Schmitt (1998).
\newblock \emph{American Astronomical Society} \textbf{192}, 82.01.

\end{thebibliography}