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From the time I heard I was accepted by the UCLA Motion Picture History and Criticism graduate program, I scoured the Cleveland Public Library for books about Southern California. One of them wrote at great length about mudslides along the Pacific Coast Highway (California Route 1) in Pacific Palisades and Malibu, and posited a fictional type of super-salesman called a "Pro-Cal," who regarded L.A. as the best of all places—diametrically opposed to the boo-birds from the East who were intent on digging up dirt about the Golden State.
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If I had to put my finger on what unites Houston and Los Angeles, it is a certain elusiveness as urban object. Both cities are opaque and hard to read. What is Houston? Where does it begin and end? Does it have a center? Does it need one? It's tough to say, even when you're there — even when you're looking directly at it.
The same has been said of Los Angeles since its earliest days. Something Carey McWilliams noted about L.A. in 1946 — that it is a place fundamentally ad hoc in spirit, "a gigantic improvisation" — is perhaps even more true of Houston. Before you can pin either city down, you notice that it's wriggled out of your grasp.
People who are accustomed to making quick sense of the world, to ordering it into neat and sharply defined categories, tend to be flummoxed by both places. And reporters at the New York Times are certainly used to making quick sense of the world. If there's one reason the paper keeps getting Los Angeles so spectacularly wrong, I think that's it. Smart, accomplished people don't like being made to feel out of their depth. Los Angeles makes out-of-town reporters feel out of their depth from their first day here.
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L.A. no longer wriggles out of my grasp. I recognize it as a conglomeration of landscapes, cultures, and even architectures. So don't think you can spend an afternoon on the Sunset Strip or the Bel Air Hotel and think you've scoped out the city: You haven't even begun.


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DoveFM is blessed to welcome Pastor Mike Nimer to our teaching lineup. You can hear Pastor Mike's program, Grace Infusion, each weekday at 3:00pm. Pastor Mike and his wife Lara are from Miami, Florida, and have been serving in ministry for over 20 years. He has planted a church in Fla., South Carolina and recently pastored a pre-established church in New Jersey for almost 12 years. He currently is a teaching pastor at Calvary Chapel in Lexington, SC. He has 3 grown daughters and one grandson. He loves to fish, read and is an avid Miami Hurricanes football fan.
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\section{Introduction}
Unbiased estimators are recipes for producing an estimate of a
quantity which, averaged over many realizations of the data from the
same underlying model, will average towards the true value of the
quantity we seek to measure (assuming the averaging is unweighted,
or symmetrically weighted).

A typical example of where unbiased estimators might be useful is the
estimation of cosmic shear. One can write the complete likelihood for
the observed galaxy image given the parameters of the galaxy
model. Such a model might include parameters describing the intrinsic
ellipticity of the galaxy, its size, etc. and also the quantities that
one wants to measure, such as shear. In general, the resulting
likelihood will be very non-Gaussian, i.e. it cannot be usefully
described by the position of maximum likelihood and the second
derivative matrix around that point in parameter space.  In order to
carry out an analysis in an unbiased manner, one would need to
propagate the full likelihood shape in the subsequent analysis of the
data. This is prohibitive in the limit of millions of galaxies whose
shear one hopes to measure in forthcoming surveys. One could attempt
to maximize the likelihood for each individual galaxy, but this
typically leads to wrong answers -- since galaxies are round on
average, a given galaxy might be best explained as a result of massive
shearing of an intrinsically round galaxy. But we know that a model
with a shear of say $0.3$ does not make much sense for a typical field
galaxy. In \cite{2014MNRAS.438.1880B} (BA14 hereafter), the authors
have argued for the expansion of the marginalized likelihood around
zero shear, i.e. compressing the likelihood to the value of the first
and second derivatives of the log-likelihood expanded around zero
shear. The fact that the likelihood for each individual galaxy is
highly non-Gaussian does not matter.  Since the shear is small, when
many log-likelihoods are added (i.e. likelihoods combined), the
resulting likelihood has to collapse to a Gaussian by the central
limit theorem. For such a collapsed likelihood, one can use a
Newton-Raphson step (using the first and second derivatives of the
combined likelihood) to calculate an estimate of the underlying
shear. In BA14, the authors show that this method works on a toy
example (also employed later in this paper), and
\cite{2014arXiv1403.7669S} demonstrates that it also performs as
expected in more realistic settings (e.g. working with real pixelated
galaxy images, but still using simulations).

However, one caveat to the method discussed above is that, in its
simplest incarnation presented in BA14, it only works when the shears
of all galaxies are assumed to be the same - something that is clearly
not true in reality. The method requires the likelihood to be combined
for a sufficiently large number of galaxies so that central limit
theorem ensures we can get a sufficiently Gaussian shear estimate for
the ensemble.  Therefore, in order to calculate a correlation function
or a power spectrum, one can either perform shear averaging in
cells where the shear can be roughly assumed constant,
or, alternatively, attempt to appropriately weight the estimates using cells in Fourier space to
recover individual Fourier modes of the shear field (see Section 2.2
in \cite{2014MNRAS.438.1880B}).

In this paper, we develop a related scheme.   In contrast to the BA14 method,
where one does not recover an estimate of the shear of a single galaxy,
the method in this paper does return an
unbiased estimate of the shear for each galaxy. For each individual
galaxy, we make no guarantee as to the probabilistic distribution for
the error $\boldsymbol{\epsilon}=\tilde{{\ibg}}-{\ibg}$ (where $\tilde{\ibg}$
is the shear estimate and ${\ibg}$ is the true shear), except that
$\left<\boldsymbol{\epsilon}\right>=0$, where the average is over all possible
realizations of the data.  Again, while the error properties for
a single galaxy are unknown, they must converge to a normal
distribution when many galaxies are considered by the central limit
theorem.  An important advantage in returning the shear of each galaxy, is that we are now not limited to the
case of constant shear and can calculate any correlation function using these
estimates, since it is trivial to show, for example, that $\left<
  \tilde{\ibg}_1 \tilde{\ibg}_2 \right>= {\ibg}_1 {\ibg}_2$,
where indices 1 and 2 correspond to two galaxies, $\tilde{\ibg}$ corresponds to the estimated shear, and
${\ibg}$ corresponds to the true shear.

In section \ref{sec:formalism}, we develop the formalism used in this
work, which is completely general and independent of any particular
inference problem. It will turn out that in general, an estimator can
be constructed that is unbiased to a certain order in the difference
between the true and assumed fiducial values for the theory
parameters. In Section \ref{sec:optim-quadr-estim}, we re-derive the
optimal quadratic estimator in our formalism, and in Section
\ref{sec:shear-estimation}, we apply our formalism to the toy problem
of BA14.

\section{Formalism}
\label{sec:formalism}

\newcommand{\mathcal{L}}{\mathcal{L}}
Consider a general likelihood function $L(\boldsymbol{D};\boldsymbol{\theta})$, which
is a function of a vector of $N$ theory parameters $\boldsymbol{\theta}$ and a
vector of $M$ observable data values $\boldsymbol{D}$.\footnote{We follow
  standard notation where vectors and matrices which are not
  explicitly indexed are denoted with bold-face italic
  font and bold-face roman fonts respectivelly.} We will denote the log
likelihood as $\mathcal{L}=\log L$. The likelihood is normalized as
\begin{equation}
  \int L d^M\boldsymbol{D} = \int e^\mathcal{L} d^M \boldsymbol{D}=1.
\label{eq:3}
\end{equation}
The above is true for \emph{any} set of theory parameters $\boldsymbol{\theta}$.  We
will write the average of any quantity over the likelihood at theory
parameter $\boldsymbol{\theta}$ as
\begin{equation}
  \left<X(\boldsymbol{D};\boldsymbol{\theta}') \right>_\theta = \int X(\boldsymbol{D};\boldsymbol{\theta}') e^{\mathcal{L}(\boldsymbol{D};\boldsymbol{\theta})} d^M\boldsymbol{D}
\label{eq:3x}
\end{equation}
Note that the function $X$ can in general be a function of both data and
the theory parameters, but the resultant average  $\left<X(\boldsymbol{D};\boldsymbol{\theta}')
\right>_{\boldsymbol{\theta}}$ is a function of $\boldsymbol{\theta}$ and $\boldsymbol{\theta}'$, but not $\boldsymbol{D}$.
Let us denote the derivative with respect to the theory parameters
with a comma, i.e.  $\mathcal{L}_{,i}=\frac{\partial \mathcal{L}}{\partial \theta_i}$.
The first derivative $\mathcal{L}_{,i}$ is a vector of size $N$, the second
derivative $\mathcal{L}_{,ij}$ is a symmetric matrix of size $N\times N$, etc.

 Taking $n$ derivatives of Equation \eqref{eq:3} with respect to
 theory parameters, we find that
\begin{equation}
  \left<^{n}\mathbf{U}(\boldsymbol{\theta}) \right>_{\boldsymbol{\theta}} = 0\\
\label{eq:4}
\end{equation}
where we have introduced the shorthand notation
\begin{eqnarray}
  ^{1}U_i &=& \frac{L_{,i}}{L} = \mathcal{L}_{,i} \label{eq:u1}\\
  ^{2}U_{ij} &=& \frac{L_{,ij}}{L} = \mathcal{L}_{,ij}+\mathcal{L}_{,i}\mathcal{L}_{,j}\\
  ^{3}U_{ijk} &=& \frac{L_{,ijk}}{L} = \mathcal{L}_{,ijk} + \mathcal{L}_{,ij}\mathcal{L}_{,k}  + {\rm cyc} + \mathcal{L}_{,i} \mathcal{L}_{,j} \mathcal{L}_{,k}\\
  ^{n}\mathbf{U} &=& \frac{1}{L} \frac{\partial^n L}{\partial \boldsymbol{\theta}^n} =
  \frac{\partial}{\partial \boldsymbol{\theta}} {^{n-1}\mathbf{U}} + {^{n-1}\mathbf{U}}{^1\mathbf{U}}
\end{eqnarray}
Note that Equation \ref{eq:4} only holds when both the $\boldsymbol{\theta}$ inside the brackets and
outside the brackets are the same.
In general, however, in Equation \ref{eq:3x}, the $\boldsymbol{\theta} '$ appearing in $X$ need not be at the same
position in theory space as the $\boldsymbol{\theta}$ appearing in $L(\boldsymbol{D};\boldsymbol{\theta})$.

The first of the above equations, namely $\left<\mathcal{L}_{,i}\right>=0$ has a
very clear physical interpretation. It is telling us, that if one chooses
a theoretical model specified by $\bithetat$, generates a set of observed
data points $\boldsymbol{D}$ given that model, calculates the first
derivative of the log-likelihood at the true model value
$\mathcal{L}_{,i}(\boldsymbol{D};\bithetat)$, and then averages this quantity over all possible
realizations of the data, then the result will be zero.  In fact, this must intuitively
be so: if one has access to many realizations of the data from the
same theory available, multiplying likelihoods (or equivalently adding
log-likelihoods) will result in a Gaussian likelihood that will become
increasingly tightly centered on the true value. In the limit of the
infinite number of data realizations, it becomes a delta function at
the true value.

Of course, this is not very helpful, since if we knew the true value,
we would not need to measure it. So, let us assume that the true value
is at some nearby position $\bithetat=\boldsymbol{\theta} + \Delta \boldsymbol{\theta}$. If we expand
the likelihood around $\boldsymbol{\theta}$ (note that we are not expanding around
the true model, but around a chosen fiducial model), we find
\begin{equation}
e^{\mathcal{L} (\bithetat)} =  e^{\mathcal{L}(\boldsymbol{\theta})}\left(1 +
  {\sum_{n=1}^\infty} \frac{1}{n!}  {^n \mathbf{U}}(\boldsymbol{\theta}) \Delta \boldsymbol{\theta}^n \right).
\label{eq:8}
\end{equation}
Note that the $n$-th term in the Taylor expansion is a product of $^nU$,
which has $n$ indices, with $\Delta \boldsymbol{\theta}^n=\Delta \theta_i \Delta
\theta_j \ldots \Delta \theta_l$, which also has $n$ indices.

Substituting the right side of Equation \ref{eq:8} into Equation
\ref{eq:3x} gives
\begin{equation}
\left<^m\mathbf{U}(\boldsymbol{\theta})\right>_{\bithetat} =  \sum_{n=1}^{\infty}
\frac{1}{n!}  {^{mn}\mathbf{W}} \Delta \boldsymbol{\theta}^n,
\label{eq:rax}
\end{equation}
where
\begin{equation}
  ^{mn}\mathbf{W} = \left<^m\mathbf{U}  ^n\mathbf{U} \right>_{\boldsymbol{\theta}}
\end{equation}
Note that the $^{mn}\mathbf{W}$ object has $m+n$ indices and is only a function
of $\boldsymbol{\theta}$, not $\boldsymbol{D}$.  We see that quantities $^n\mathbf{U}$ are special. They
average to zero, if we are sitting on a true model
($\left<^{n}\mathbf{U}(\bithetat) \right>_{\bithetat} = 0$ as in Equation \ref{eq:4}
since $\Delta \boldsymbol{\theta} = 0$ when $\boldsymbol{\theta} = \boldsymbol{\theta}^T$).  However, as the
true model slips away, those averages analytically respond to the
difference between the true and the fiducial model (as described by
Equation \ref{eq:rax}).

The motivation for all this may be opaque at this point. The important thing
to recognize is that both $^m \mathbf{U}(D;\boldsymbol{\theta})$ and $^{mn}\mathbf{W}(\boldsymbol{\theta})$ are things that
we can compute, given data and a {\rm choice} of fiducial parameters
$\boldsymbol{\theta}$, so estimators of $\boldsymbol{\theta}^T$, or equivalently
$\Delta \boldsymbol{\theta} = \boldsymbol{\theta}^T-\boldsymbol{\theta}$,  can be constructed out of them.

\subsection{First-order estimator}

Before proceeding, we note that
\begin{equation}
  ^{11}W_{ij} = \left<\mathcal{L}_{,i} \mathcal{L}_{,j} \right> = -\left<\mathcal{L}_{,ij} \right> = F_{ij}
\end{equation}
is the Fisher matrix (where we have used Equation \ref{eq:4} for $n=2$).

Our first-order estimator comes from inspecting Equation \ref{eq:rax}
for the case when  $\Delta \boldsymbol{\theta}$ is sufficiently small that the series can be
truncated at the first order. We can write down the ansatz
\begin{equation}
  \boldsymbol{E}_1 = (^{11}\mathbf{W})^{-1}\, {^1\mathbf{U}} = F^{-1}_{ij} \mathcal{L}_{,j}.
\label{eq:e1}
\end{equation}
Plugging this solution back into Equation \ref{eq:rax} and remembering that
$^{mn}\mathbf{W}$ is not a function of $\boldsymbol{D}$ gives
\begin{eqnarray}
  \left< \boldsymbol{E}_1 \right>_{\bithetat} &=& (^{11}\mathbf{W})^{-1} \left<^1\mathbf{U} \right>_{\bithetat} \\
  &=& \Delta \boldsymbol{\theta}_1  + \frac{1}{2} \left(\mathbf{F}^{-1}\right)\, ^{12}\mathbf{W} \Delta \boldsymbol{\theta} ^2 + \ldots
\label{eq:e1_thetat}
\end{eqnarray}
This estimator is thus unbiased to quadratic order in
$\Delta \boldsymbol{\theta}$. Note that since $\boldsymbol{\theta}$ is known (i.e. it is the
assumed fiducial model), we can simply add it to $\boldsymbol{E}_1$ to convert an
estimator of $\Delta \boldsymbol{\theta}$ to an estimator of $\bithetat$.
The variance of the estimator is given by
\begin{equation}
  {\rm Var}(\boldsymbol{E}_1) = \mathbf{F}^{-1} + \mathbf{F}^{-1} \mathbf{F}^{-1} \Delta \boldsymbol{\theta} \left<^1\mathbf{U} ^1\mathbf{U} ^1\mathbf{U} \right> +\ldots,
\end{equation}
where the  contraction of indices goes as $\left[\mathbf{F}^{-1} \mathbf{F}^{-1} \Delta \boldsymbol{\theta}
  \left<^1\mathbf{U} ^1\mathbf{U} ^1\mathbf{U} \right>\right]_{ij} = F^{-1}_{ik} F^{-1}_{jl}
\Delta \boldsymbol{\theta}_m \left<^1U_k ^1U_l ^1U_m \right>$.  Thus, given the
Cramer-Rao bound, we have shown that this estimator is unbiased to
quadratic order in $\biDtheta$ and optimal to first order in $\biDtheta$.

\subsection{Higher-order estimators}

To construct higher-order estimators, we need to use  higher order
$\mathbf{U}$s. A quantity of the form
\begin{equation}
  \boldsymbol{E}_o = \sum_{m=1}^o ({^m\mathbf{A}}) ({^m\mathbf{U}}),
\end{equation}
where $^m\mathbf{A}$ is a $m+1$ index object (indices of the parameter derivatives,
i.e., see Eq. \ref{eq:u1}, etc.),
will have the mean given by
\begin{equation}
  \left<\boldsymbol{E}_o\right>_{\bithetat} =  \sum_{n=1}^{\infty} \frac{1}{n!}
  \left( \sum_{m=1}^o ({^m\mathbf{A}}) ({^{mn}\mathbf{W}})  \right) \biDtheta^n
\end{equation}

For a given order $o$, the weights $\mathbf{A}$ can be arranged so that the
pre-factor to $\Delta\boldsymbol{\theta}$ is unity and the prefactor to $\delta\boldsymbol{\theta}^2$ and
higher are zero up to order $o$. For a concrete example see Section
\ref{sec:shear-estimation} and Appendix \ref{sec:general-3rm-rd}. One
should note that higher order estimators, in general, have higher
variance with respect to the first-order estimator, however, they are
less biased.

Finally, we note that while this construction uniquely specifies one
possible estimator unbiased to a given order, it is clearly not
unique, since one could imagine constructing estimators that are
non-linear in $\mathbf{U}$ quantities and which might, in general, perform
better or worse than this one. We leave investigation of these
questions to future work.

\subsection{A note on iterations}

Since the first-order estimator is accurate to $\Delta\boldsymbol{\theta}$, one might be
tempted to simply iterate: start with a first-order estimator, move by
$\Delta\boldsymbol{\theta}$, do another iteration there, etc. Note, that such a process
will in general take you to the maximum likelihood point, since the
first-order estimator resembles a Newton-Raphson step.

It is known that maximum likelihood is not, in general, an unbiased
estimator (although it often happens to be, e.g. for mean and variance
of a Gaussian likelihood). We provide a concrete example in Appendix
\ref{sec:biasml}.  So, why does an iterative process not produce an
unbiased estimate?  The subtlety lies in the fact that the above
derivation assumes that the fiducial $\boldsymbol{\theta}$ was chosen without
knowing about the data.  Any iterative process  necessarily breaks
this assumption. Thus, to estimate the mean of an estimator after
several iterations, one would need to average not only over possible
realizations of the data, but also over all possible ``paths'' in the
theory space that a certain iterative process might take.  So, in
general, one should use a higher-order estimator to improve on the
accuracy of the first-order estimator, instead of iterating.

Of course, we expect that the bias due to iteration will be small when
the signal-to-noise is high, so that this will not matter in practice
in those cases.

\section{Optimal quadratic estimator}
\label{sec:optim-quadr-estim}

For completeness, we begin by applying the above formalism to a common
inference problem.  To construct an optimal quadratic estimator
\cite{1998PhRvD..57.2117B,1998ApJ...503..492S,2000ApJ...533...19B},
we start with the data vector $\boldsymbol{D}_i$, with zero mean
($\left<\boldsymbol{D}\right>=0$), whose covariance can be modeled as
\begin{equation}
  \mathbf{C} = \left<\boldsymbol{D}\biD^T\right> =  \mathbf{N}+\theta_i \mathbf{S}_i.
\label{eq:coqe}
\end{equation}
Here $\theta_i$ are some parameters describing the two-point function
of the data, i.e. power spectrum or correlation function bins,
$\mathbf{S}_i$ is the response of the covariance to a change in the value of
$\theta_i$, and $\mathbf{N}$ is assumed to be a known ``noise'' matrix.

Ignoring constant terms, the log-likelihood can be written as
\begin{equation}
\mathcal{L} = -\frac{1}{2} \log \det \mathbf{C} - \frac{1}{2} \boldsymbol{D}^T \mathbf{C}^{-1}\boldsymbol{D}.
\end{equation}
In our notation, we have
\begin{equation}
  ^1U_i = -\frac{1}{2} {\rm Tr}\left(\mathbf{C}^{-1}\mathbf{S}_i\right) + \frac{1}{2} {\rm Tr} \left(\boldsymbol{D}^T\mathbf{C}^{-1}\mathbf{S}_i\mathbf{C}^{-1}\boldsymbol{D}\right).
\end{equation}
A brief calculation gives
\begin{equation}
  \left<^1U_i\right>_{\bithetat} = \frac{1}{2} {\rm Tr} \left(\mathbf{C}^{-1}\mathbf{S}_i
    \mathbf{C}^{-1}\mathbf{S}_j\right) \Delta \theta_j
\end{equation}
where we have used $\mathbf{C}(\boldsymbol{\theta}^T) =  \left<\boldsymbol{D}\biD^T\right>_{\bithetat} =  \mathbf{N}+\boldsymbol{\theta}^T_i \mathbf{S}_i = \mathbf{C}(\boldsymbol{\theta}) + \Delta \theta_i \mathbf{S}_i$,
and hence
\begin{eqnarray}
\left<{\rm Tr} \left(\boldsymbol{D}^T\mathbf{C}(\boldsymbol{\theta})^{-1}\mathbf{S}_i\mathbf{C}(\boldsymbol{\theta})^{-1}\boldsymbol{D}\right)\right>_{\bithetat} = \\
{\rm Tr} \left(\mathbf{C}(\boldsymbol{\theta})^{-1}\mathbf{S}_i\mathbf{C}(\boldsymbol{\theta})^{-1}(\mathbf{C}(\boldsymbol{\theta}) + \Delta \theta_j \mathbf{S}_j)\right).
\end{eqnarray}
It follows that
\begin{eqnarray}
  ^{11}\mathbf{W} &=& F_{ij} = \frac{1}{2}\mathrm{Tr} \left(\mathbf{C}^{-1} \mathbf{S}_{i} \mathbf{C}^{-1} \mathbf{S}_{j} \right)\\
  ^{n1}\mathbf{W} &=& 0 \,\  \mathrm{for}\,\   n>1.
\end{eqnarray}
Plugging these into Equation \eqref{eq:e1}, we recover the standard optimal
quadratic estimator
\begin{equation}
  \boldsymbol{E}_1 = \frac{1}{2}\left[\mathbf{F}^{-1}\right]_{ij} \left[ \boldsymbol{D}^T\mathbf{C}^{-1}\mathbf{S}_j\mathbf{C}^{-1}\boldsymbol{D} - b_j \right],
\label{eq:estx}
\end{equation}
where $b_i={\rm Tr}\left(\mathbf{C}^{-1}\mathbf{S}_i\right)$.
We have therefore recovered the standard optimal quadratic
estimator and at the same time shown that it is unbiased at all
orders.  The fact that $^{n1}\mathbf{W} = 0$ for $n>1$ implies that this
estimator is unbiased at all orders. Additionally, it can be shown
that this estimator is unbiased regardless of the assumption of a
Gaussian likelihood by calculating the expectation value of the above
equation.  However, this is not directly connected to the framework
here.  (Again, we note that the expectation value proving that the
standard quadratic estimator is unbiased assumes that the covariance
matrix that appears in it does not depend on the data,
but this assumption is invalidated by iteration.)

These beautiful properties are, of course, crucially dependent on the
theory covariance matrix being linear in theory parameters in Equation
\eqref{eq:coqe}. Fortunately, this is the case in the standard for
measurement of the power spectrum and its linear cousins such as
correlation function. If this is not the case, one can always Taylor
expand around fiducial model and the derivation is then the same with
$\mathbf{N}$ replaced with $\mathbf{N}+\mathbf{C}_{\rm fid.}$, but the
estimator is then only valid within the accuracy of this
approximation.

While this result is not new, it is important to put this into
context. Traditionally, quadratic estimators are often cast as a
Newton-Raphson step towards higher likelihood (see
e.g. \cite{Dodelson-Cosmology-2003}), but here one must remember that,
if the goal is simply function maximization, the true second
derivative may not give the best performance. Numerical work has shown
that performing a Newton-Raphson step with the true second derivative
instead of the Fisher matrix can be an order of magnitude slower in
convergence to the maximum (e.g., when starting power spectrum
parameters are far below the true value).  This is because the true
second derivative and the Fisher matrix are increasingly different as
we move away from the true position in parameter space.  Since the
Fisher matrix estimate is unbiased, one might expect that anything
that deviates from the Fisher estimate must be suboptimal with slower
convergence (strictly speaking, being unbiased does not guarantee
faster convergence if the scatter around the mean is larger but in
practice we do not expect this to happen).  We note however, that even
though an estimate is unbiased when starting with a model that is a
very poor match to the true model, the uncertainties based on a Fisher
matrix will nevertheless be grossly misestimated.

\begin{figure*}
  \centering
  \includegraphics[width=0.24\linewidth]{ders0-crop.pdf}
  \includegraphics[width=0.24\linewidth]{ders1-crop.pdf}
  \includegraphics[width=0.24\linewidth]{ders2-crop.pdf}
  \includegraphics[width=0.24\linewidth]{ders3-crop.pdf}

  \caption{The $i$-th derivative of the likelihood with respect to
    $g_1$ for the posterior distribution at zero shear, where i=0,1,2,3 for
    the toy model described in the text. The x and y axis are the
    measured ellipticities for $e_1$ and $e_2$ respectively, and the color bar saturates
    positively at red and negatively at blue.}
  \label{fig:1}
\end{figure*}

\begin{figure}
\includegraphics[width=\linewidth]{fig2.pdf}
\caption{The relative biases in the recovered $g_1$ as a function of
  the input $g_1$, with input $g_2$ held at zero. For the $E_1$ and
  $E_3$ estimators, the error was calculated from the variance in estimates, while
  for the $E_{AB}$ estimator, it was assumed to be given by the inverse of the second derivative
  of the posterior.}
\label{fig:2}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{fig3.pdf}
\caption{The error of estimators relative to the Fisher matrix
  prediction at zero shear. For the $E_1$ and $E_3$ estimators, the error was
  calculated from the variance in estimates, while for the $E_{AB}$ estimator, it
  was assumed to be given by the inverse of the second derivative of the posterior.}
\label{fig:3}
\end{figure}

\section{Shear estimation}
\label{sec:shear-estimation}

To apply the formalism above to the problem of shear estimation, we take
as a starting point work in \cite{2014MNRAS.438.1880B}.  We describe the
likelihood for shear, $L(\ibg)$, through its derivatives
at zero shear as:
\begin{eqnarray}
P &=& L(\boldsymbol{D}|\ibg=0) \\
\mathbf{Q} &=& \nabla_{\ibg} L(\boldsymbol{D}|\ibg) |_{\ibg=0} \\
\mathbf{R} &=& \nabla_{\ibg} \nabla_{\ibg} L(\boldsymbol{D}|\ibg) |_{\ibg=0} \\
\mathbf{S} &=& \nabla_{\ibg} \nabla_{\ibg} \nabla_{\ibg} L(\boldsymbol{D}|\ibg) |_{\ibg=0}
\end{eqnarray}
BA14 expand to second order, but we generalize to third. Note that
theory parameters here are the two components of shear, and we will use
$\ibg$ and $\boldsymbol{\theta}$ interchangeably below.  Derivatives of log
likelihood (at zero shear) are thus given by
\begin{eqnarray}
  \mathcal{L}_{,i} &=& \frac{Q_i}{P} \\
  \mathcal{L}_{,ij} &=& \frac{R_{ij}}{P} - \frac{Q_iQ_j}{P^2} \\
  \mathcal{L}_{,ijk} &=& \frac{S_{ijk}}{P} -  \left( \frac{R_{ij}Q_k}{P^2} + {\rm
        cyc}\right) + 2\frac{Q_i Q_j Q_k}{P^3},
\end{eqnarray}
and the $U$ quantities are given simply by
\begin{eqnarray}
  ^1U_i  &=& \frac{Q_i}{P} \\
  ^2U_{ij}  &=& \frac{R_{ij}}{P} \\
  ^3U_{ijk}  &=& \frac{S_{ijk}}{P}.
\end{eqnarray}

BA14 advocate calculating the above quantities for
each galaxy.  If all galaxies have the same shear, the total  probability
can be calculated by summing derivatives of the log likelihood. For a
sufficient number of galaxies, the likelihood collapses to a Gaussian
and the shear can be estimated as
\begin{equation}
  \boldsymbol{E}_{BA} = -\left( \sum \mathcal{L}_{,ij}\right)^{-1} \left( \sum \mathcal{L}_{,j} \right)
\end{equation}
For a sufficiently large number of galaxies $N_g$, the sum of second
derivatives will approach
\begin{eqnarray}
  \sum_1^{N_g} \mathcal{L}_{,i} &\rightarrow& N_g \left<
    \mathcal{L}_{,i}\right>_{\bithetat} \\
  \sum_1^{N_g} \mathcal{L}_{,ij} &\rightarrow& N_g \left<
    \mathcal{L}_{,ij}\right>_{\bithetat}
\end{eqnarray}
Summing the first and second derivatives of the log likelihood is akin to
averaging over the true distribution.
Therefore, in the limit of an infinite number of galaxies, the
estimator will give
\begin{equation}
  \left< \boldsymbol{E}_{BA} \right>_{\bithetat} = -\left (\left< \mathcal{L}_{,ij}(\boldsymbol{\theta})\right>_{\bithetat}\right)^{-1} \left< \mathcal{L}_{,j} \right>_\bithetat
\end{equation}
Note that this is subtly different from our estimator, which uses the
Fisher matrix, $F_{ij} = -\left< \mathcal{L}_{,ij}(\boldsymbol{\theta})\right>_{\boldsymbol{\theta}}$, which is the mean of the second derivative of the log likelihood \emph{assuming zero shear}:
\begin{equation}
  \left< \boldsymbol{E}_1 \right>_{\bithetat} = -\left (\left< \mathcal{L}_{,ij}(\boldsymbol{\theta})\right>_{\boldsymbol{\theta}}\right)^{-1} \left< \mathcal{L}_{,j} \right>_\bithetat
\end{equation}

\subsection{Toy model}
To test the above ideas, we use the same toy model that was used
in BA14.
\newcommand{\boldsymbol{e}}{\boldsymbol{e}}
We draw a source ellipticity from an isotropic unlensed
distribution with probability distribution given by
\begin{equation}
  P(|\bie^i|) \propto (1-|\bie^i|^2)^2 \exp\left(-\frac{|\bie^i|^2}{2\sigma^2_p} \right)
\label{eq:toy1}
\end{equation}
for the magnitude of the ellipticity and a random orientation.
The effect of shear is most easily expressed
if we cast the intrinsic  ellipticity and shear as complex vectors
$\bie^i=e_1^i+ie_2^i$ and $\ibg = g_1+ig_2$. Then the sheared
ellipticity vector is given by
\begin{equation}
  \bie^s = \frac{\bie^i-\ibg}{1-\ibg^*\bie^i}.
\label{eq:toy2}
\end{equation}
Finally, we add random Gaussian noise to obtain the observed
ellipticity $\bie^o$:
\begin{equation}
  \bie^o = \bie^s + \boldsymbol{\epsilon},
\label{eq:toy3}
\end{equation}
where each component of $\boldsymbol{\epsilon}$ is drawn from a truncated Gaussian
with variance $\sigma_n$ ensuring that $|\boldsymbol{e}^o|<1$ (in practice random
realizations of noise are added to $\boldsymbol{e}^s$ until $|\boldsymbol{e}^o|<1$ is satisfied).
In this work we limit ourselves to the example of $\sigma_p=0.3$ and
$\sigma_n=0.05$.

\subsection{Third-order estimator}

It is clear that at least in the case of this particular problem,
symmetry ensures that the second order correction to the estimator
vanishes if one expands around zero shear. There are several ways to
see this. First, given that shear is a spin-2 quantity, the lowest
order scalar one can make is $|\ibg|^2$ and therefore, one expects the
lowest-order correction to an estimate of $\ibg$ to scale as $\ibg
|\ibg^2|$, which is third order in $\ibg$. Second, if one only estimates
$g_1$, it is natural to expect that the correction to $g_1$ must be
the same and of opposite sign to the correction to $-g_1$ --
estimation of shear must be symmetric with respect to mirroring over
the origin. Therefore, it cannot receive a $g_1^2$ correction, and the
lowest order correction to the estimator must scale as $g_1^3$.  Note that
in Equation \ref{eq:e1_thetat}, this means that $^{12}\mathbf{W} = 0$.

Therefore, we construct a third-order estimator from quantities $^1\mathbf{U}$
and $^3\mathbf{U}$. Again, because of the symmetry of the problem, we construct
it assuming the problem is one dimensional, i.e. we are attempting to
recover the $g_1$ component. In that case all $\mathbf{W}$ quantities are
scalar.

Starting with the system of equations:
\begin{eqnarray}
  \left<^1\mathbf{U}\right> &=& ^{11}\mathbf{W} \Delta\boldsymbol{\theta} +\frac{^{13}\mathbf{W}}{6}\Delta\boldsymbol{\theta}^3 + \ldots, \\
  \left<^3\mathbf{U}\right> &=& ^{31}\mathbf{W} \Delta\boldsymbol{\theta} +\frac{^{33}\mathbf{W}}{6}\Delta\boldsymbol{\theta}^3 + \ldots,
\end{eqnarray}
it is not difficult to show that, ignoring higher order terms,
\begin{equation}
  \frac{ ^{33}\mathbf{W} \left<^1\mathbf{U}\right>  - ^{31}\mathbf{W}
    \left<^3\mathbf{U}\right>}{^{11}\mathbf{W}^{33}\mathbf{W} - ^{13}\mathbf{W} ^{31}\mathbf{W}} = \Delta\boldsymbol{\theta}
\end{equation}

Hence, we can write an ansatz:
\begin{equation}
 \boldsymbol{E}_3 =   \frac{ ^{33}\mathbf{W} ^1\mathbf{U}  - ^{31}\mathbf{W} ^3\mathbf{U}}{^{11}\mathbf{W}^{33}\mathbf{W} - ^{13}\mathbf{W} ^{31}\mathbf{W}}
\label{eq:e3}
\end{equation}
Since $\mathbf{W}$ quantities do not depend on data, $\left<\boldsymbol{E}_3 \right>
=\Delta\boldsymbol{\theta}$ and hence this is our third order estimator.  For more
realistic cases, the rotational symmetry might be broken due to
systematic and instrumental effects and for completeness we show how
to build a complete 3$^{\rm rd}$ order estimator in Appendix
\ref{sec:general-3rm-rd}.

\subsection{Results for toy model}
For this toy example, we can calculate the likelihood and its
derivatives simply by brute force Monte Carlo - we can draw a large
enough number of samples from the parent distribution such that the
gridded values of sampled $\boldsymbol{e}$ become a good approximation for the
probability distribution. The derivatives are then calculated by
finite difference methods from gridded likelihoods. Note that this
short-cut is unlikely to work in a more realistic setting due to
the higher dimensionality of the problem.

In Figure \ref{fig:1}, we plot the $i$-th derivative of the likelihood
with respect to $g_1$, that is quantities $P$, $Q_1$, $R_{11}$,
$S_{111}$, showing how the posterior distribution of ellipticities responds
to shear at each order.

In Figure \ref{fig:2}, we show results for the three estimators
discussed in this text. As expected, the $\boldsymbol{E}_{BA}$ and $\boldsymbol{E}_1$ estimators
show a quadratic increase in bias as a function of shear, which is
mostly removed by the $\boldsymbol{E}_3$ estimator. In this particular case, our
$\boldsymbol{E}_1$ estimator seems to be performing somewhat better than the
original $\boldsymbol{E}_{BA}$ estimator, although it is not clear whether this
will translate to similar gains in more realistic scenarios. However,
the $\boldsymbol{E}_3$ estimator is designed to be more accurate and performs with
a 0.1\% relative precision all the way to shears of 0.2, at which
point we are well out of the validity of the small shear approximation, and
flexion effects \cite{2005ApJ...619..741G} become important, which are
not captured in this toy model.

In Figure \ref{fig:3}, we show the error (square root of variance) for
the three estimators discussed here, normalized to the Fisher matrix
prediction at zero shear. As we can see, both $\boldsymbol{E}_{BA}$ and $\boldsymbol{E}_1$
converge to the Fisher matrix prediction at zero shear, but $\boldsymbol{E}_3$ is marginally
noisier. The effect is small, sub 1\%, but clearly detectable. For
higher shear, the $\boldsymbol{E}_1$ and $\boldsymbol{E}_3$ estimators begin to become slightly less noisy
than the zero-shear Fisher prediction. Note that this does not violate
the Cramer-Rao bound, since the bound only holds if the true shear is zero.

Finally, we demonstrate explicitly that our estimator can measure
correlations. To that end, we draw pairs of galaxies with shear
$\ibg_a$ and $\ibg_b$, which we randomly choose to follow
\begin{equation}
    \left<\ibg_a \ibg_a^T\right> = \left< \ibg_b \ibg_b^T \right> = \left( \begin{array}{cc}
0.05^2 & 0 \\
 0  & 0.05^2 \\
\end{array} \right)
\end{equation}
and
\begin{equation}
    \left<\ibg_a \ibg_b^T \right> = \left( \begin{array}{cc}
0.00125 & 0.00075 \\
 0.00075  & 0.00125 \\
\end{array} \right).
\end{equation}
These pairs of galaxies are modeled using Equations \ref{eq:toy1},
\ref{eq:toy2}, and \ref{eq:toy3} with $\sigma_p=0.3$, $\sigma_n=0.05$
to obtain observed values and then with the $E_3$ estimator to obtain
an estimate.  These estimates where then used to obtain the
correlations: $\left<\tilde{\ibg}_a \tilde{\ibg}_b^T \right>_{11} =
0.00125319\pm2.8 \times 10^{-6}$ and $\left<\tilde{\ibg}_a
  \tilde{\ibg}_b^T \right>_{12} = 0.007552\pm2.8\times 10^{-6}$,
consistent with the input values and sub-percent level accurate. Of
course, this exercise \emph{had} to work, so it is really just a
sanity check.

\section{Conclusions}

In this paper, we have derived a general framework for generating
unbiased estimators. The framework is general and can be used wherever
we are measuring a quantity which is perturbatively close to the
assumed model. We have shown that the inverse of Fisher matrix
multiplied by the first derivative vector is a general formula for a
first order unbiased estimator.  In special cases such as an optimal
quadratic estimator, the estimator is unbiased at all orders.  We have
applied our framework to the problem of estimating weak lensing shear
and constructed a first and third-order estimator.

In the realm of the toy problem of BA14, our third-order estimator is
unbiased for all relevant shear magnitudes with a negligible increase
in the estimator variance compared to the Fisher prediction at zero
shear. In typical weak-lensing analyses, shears are small enough that
the first-order estimator may be sufficient. However, there are two
cases where third order correction might matter. First, when measuring
the cosmic shear power spectrum, an error term proportional to $g^3$
will ``renormalize'' to give a correction to the measured shear power
spectrum proportional to $\left<\left|\ibg\right|^2\right>P_{gg}$,
where $P_{gg}$ is the true shear power spectrum. This is of the same
order of magnitude as the overall LSST error
\cite{2009arXiv0912.0201L}.  Second, in regions of high-shear, such as
those around clusters of galaxies, the third-order estimator will be
useful, simply because shear are large-enough that the third order
correction matters.  The formalism presented here can trivially be
extended to the flexion measurement, and it should correctly account
for the correlation between shear and flexion. We refrain from making
more quantitative statements since it is not clear how realistic the
toy model is.

More importantly, we have constructed an estimator which performs as
well as the BA14 estimator, but also returns shear estimates for
individual galaxies, which makes it usable in direct measurements of
the $n$-point function of the shear field.

We also note that to some extent the main problem with shear
measurements is not the underlying framework, which is the focus of
this paper, but the bias arising from inadequate modeling of the
properties of unlensed galaxies, and it might turn out that these
problems are best solved using very phenomenological approaches as
those discussed in e.g. \cite{2007MNRAS.382..315M,2013arXiv1303.4739R}.

Putting this estimator into practice might be more complicated. In
particular, in its current incarnation, it gives the same weight to
all galaxies, while we know that this will not hold in
reality. The correct way to solve this problem is to separate galaxies
into sub-classes in a way that does not correlate (or negligibly
correlates) with the underlying shear. A separate estimator can be
constructed for each class, and the Fisher matrix is the appropriate
weight. We leave testing of this framework in more realistic settings for
the future work.

\acknowledgments The authors thank Gary Bernstein and Erin Sheldon for
useful conversations.  M.M. is supported by an SBU-BNL Research
Initiatives Seed Grant: Award Number 37298, Project Number
1111593. AS is supported by the DOE Early Career award.

redpajama_set_name: RedPajamaArXiv


Forest River, Inc. "Forest River" is recalling certain 2018-2019 Forest River Berkshire recreational vehicles and StarTrans PS2 transit buses and 2019 Glaval Legacy transit buses. The brake caliper mounting bolts may have been insufficiently tightened.
Forest River will notify owners, and Daimler Trucks North America "DTNA" authorized service facilities will check and tighten the brake caliper mounting bolts, free of charge. The recall began November 21, 2018. Owners may contact Forest River customer service at 1-574-642-3112. Forest River's number for this recall is 35 and 5C-0897.


Venus Business Communications completed its transition to M247 following acquisition in March this year. The transition now means more profitability for resellers and more productivity for M247 customers.
M247's acquisition of Venus Business Communications cements the Group's nationwide commitment to deliver a world class network throughout the UK, Europe and Internationally. M247 is committed to delivering more for all its stakeholders, so resellers and partners can now access and deliver an expanded product and service portfolio to its customers. Headed up by newly appointed Channel Director, Will Brooks, M247 has a partnership programme with three categories; reseller, referral and wholesale partners. This programme promises to enable channel partners to differentiate their offering, improve customer solutions, build margins and improve their order-to-revenue time. The M247 mission to empower business productivity extends to its partners, with support every step of the way to grow their businesses.
Following the transition and expansion M247 now employs 230 people across six sites including Manchester, Newcastle, London and Bucharest.
Don't be held back by poor connectivity anymore.


Within the framework of "People's Brand" project of the Assembly of people of Kazakhstan in Shymkent, representatives of various ethnic groups engaged in entrepreneurship share their experience in business on a regular basis. This time, successful entrepreneurs, who are members of the city's Turkish ethno-cultural association, held a master class for young people and talked about the secrets of success in doing business.
Opening the meeting, the head of "Kogamdyk kelisim" MPI in Shymkent Erbol Baikonakov dwelt on the importance of the project, noting that today ethnocultural associations not only hold cultural events, but also make a worthy contribution to the development of economic potential of the region.
Firkan LLP is a well-known company uniting a network of supermarkets and confectionery products. To date, more than a thousand people work here. Idris Askarov, Commercial Director of the company, spoke about how he made the way from the first day of the company's establishment to the present day. Currently, while doing business, he also noted the importance of acquiring new knowledge and skills.
In turn, the head of the Department of Entrepreneurship and Industrial-Innovative Development of the city of Shymkent Usen Isayev focused on state support of business, development of business plans and promotion of business ideas, financing of business initiatives.
The meeting also provided necessary consultations on small, medium and large businesses. It was discussed that along with knowledge in business and achievement of concrete results, it is necessary to apply new technologies and have a system of innovative development.


Learn more about Paulette Selmi PhD, a(n) Psychologist in Sterling, IL.
Finding a Illinois Psychologist has never been easier. Just peruse the directory below to find a Psychologist in Sterling, Illinois (IL). Are you a Psychologist? Join our directory of Psychologists in Sterling today and get connected with cancer patients, survivors and supporters who might need your assistance.


Spend a week (or two) with the whole family at a beach house in Bethany Beach and experience the best in seaside living. Long, lazy days with your toes in the sand roll into free concerts, movies on the beach and more in the evenings. From the beach to the boardwalk to the bay, there are so many family-friendly things to do in Bethany Beach, you'll be wondering about booking for week number three.


New paving or long lasting repairs. Crack repairs and resurfacing.
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Work staff is experienced. You'll like the way it looks.
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An expert SoCal contractor has the knowledge to plan and construct you a new one or just repair the one you have.
They can build with all kinds of materials, including concrete, asphalt or interlocking pavers or tile.
It will not matter if your project is big or small. And it doesn't matter if you're a homeowner or a business property owner.
When you are interested in a new and inexpensive driveway, asphalt could be a good choice, especially if your drive is long.
Some homeowners get their blacktop driveways resurfaced to keep them looking good. Sealing also helps them last longer too.
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A concrete installation expert is able to do new concrete paving and curbing.
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Maintenance may include sealing the surface or filling up holes or cracks. In unusual situations, the foundation moves due to erosion or improper preparation.
Repairs and maintenance will help your pavement last longer. Since they aren't expensive to do, it's usually a smart idea to have them completed when needed.
If your driveway needs a lot of work, maybe it's better to build a new one.
When it's time to exchange your existing one, there are several materials, designs and finishes to consider if you want to.
Your paving contractor has seen numerous designs and can help you with that. They also understand which materials would work best in your circumstances and our typical hot weather.
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Get more info for your home in the neighborhoods around Passons Blvd, Rosemead Blvd and along the west side of I-605.


text:
Ben Bold
Brits lack financial savvy and obsessed with instant gratification
OMD and News UK's report finds that consumers are assuaging Brexit worries with luxury purchases.
The Future of Money: exploring UK's attitude to finance
A worryingly large proportion of UK consumers are financially illiterate, steeped in debt and overly obsessed with short-term gratification, according to a study by OMD UK and News UK.
The findings are the result of the agency and media owner's The Future of Money report, which also showed that as political disarray over Brexit heightens, concern about the the UK's departure from the European Union is escalating (up from among 31% of Brits in June 2018 to 50% in December). Accordingly, concern over the economy shot up from 29% to 41% of Brits over the same period.
The study found that 29% of Brits have "little or no financial knowledge" (rising to 44% among 16- to 24-year-olds), while 82% with credit card debt also have unsecured debt elsewhere.
Consumers appear to be more enamoured with short-term gains – buying holidays, gadgets and experiences – at the expense of longer-term financial stability. While 74% said they save on a regular basis, only 53% have a savings account, denoting a "piggy bank" saving mentality.
An attitude that favours gratification is a boon for many brands, which can still expect healthy levels of consumer spending on goods and services – at least at present. But the report's authors suggest that marketers be sensitive to consumer unease and consider carefully how a culture of short-termism might impact long-term prospects.
The research, which is part of OMD's Future of Britain initiative, is designed to offer insights into the "real Britain". It examines how Brits plan, spend, prioritise and save, and the ramifications of their behaviour for brands.
Laura Fenton, OMD UK's managing director, said: "The concerns around our level of financial literacy should be taken seriously and there are opportunities for advertisers to become educators and assist consumers through their financial journeys."
Mark Barry, News UK's agency development partner, added: "With the research revealing that only 21% of parents actively engage with their children on financial subjects – such as saving, debt and borrowing – there is certainly a bigger role that parents and peers can play to educate them better.
"However, it's also important for brands to understand the role they play, offering value at the right moments. Context is important when it comes to driving education around finance and news brands deliver the trusted environment that can help brands to really flourish."
The partnership is the first of its type for OMD and News UK. The report was compiled on the basis of quantitative research conducted among 2,700 UK adults, across a demographically representative sample. Qualitative research was conducted among focus groups in the West Midlands and London, as well as via OMD UK's YourVoice research community.
A full version of the report is available here.
Advertising Marketing Media OMD Group
Quality news sites get 'halo effect' on ad viewability
Mobile internet to account for a third of global media consumption, report finds
Advertisers 'missing opportunity' from growth in spend on personal passions
Marketers' confidence drops to six-year low as budget growth slows
The future: technology in retail
Browse Agency Jobs


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text:
Women's Basketball: Three Pertinent Preseason Ponderings
After some rough sledding, the Commodores finally have a chance to make big strides this season, and it all starts with a winning pedigree.
Vanderbilt Athletics
Vanderbilt freshman Kendal Cheesman with head coach Shea Ralph this week. (Vanderbilt Athletics)
Jonah Barbin
The last year has been a turbulent one for the Vanderbilt women's basketball program. One might argue the turbulence has been present for the last five years, but there are a slew of reasons to be excited as a fan in 2021.
Vanderbilt enters this season with a brand new head coach, Shea Ralph, after dismissing Stephanie White following five lackluster seasons that all concluded with NCAA Tournament misses. The Commodores are also coming off a season incomplete, as they played just eight games in 2020 before deciding on Jan. 18 to suspend their season.
The team also returns only two seniors, both who saw minimal to no action last year due to injury and COVID-19 complications. If that wasn't enough adversity, throw in the fact that the Commodores will be tasked with competing in the best conference in college basketball—Vanderbilt has quite a daunting season on its plate. Big change could be just what Vanderbilt needs, but all of these moving parts surely leave many questions.
New captain on board: How will the Commodores change under the leadership of Shea Ralph?
Experience clearly won't be a concern with this hire as Ralph has been on the court practically her whole life. From her playing days at UConn, to being drafted by the WNBA's Utah Starzz, to making the University of Pittsburgh into a nationally ranked program, to being an assistant under one of the best coaches of all time in Geno Auriemma—Ralph has seen it all.
This hire indicates that Vanderbilt is determined to make a strong change to the culture of their program. If anyone has a winning pedigree, it's clearly Ralph.
"I believe sustainable excellence starts and ends with people," Ralph said in her introductory press conference. "I will look for key qualities that I know I can't coach, for example confidence, competitiveness, coachability, a tireless work ethic [and] being a great teammate, just to name a few."
This is just what fans of the Commodores have been waiting for: a coach who sets the tone early, holds players accountable and promotes a strong work ethic above all else. So, how will this winning pedigree translate? When is the turnaround coming that seems to be promised with her hiring?
"I would love for it to be sooner than later, but what I've learned over my career is when you build something the right way, it might take a little bit of time but it will last forever," Ralph said in response to that very question.
Clearly, this hire was an indication from both Vanderbilt and Ralph that they are in it for the long run. If, for some reason, the national championships, countless awards and years of hard work in winning programs aren't enough evidence of Ralph's readiness for this job, simply take it from her as she declared to the press, "I have spent my entire life preparing for this moment."
Best of the best: How will Vandy fare in the ultra-competitive SEC?
The SEC is known for having the fiercest overall competition in NCAA women's basketball. The conference is stacked with teams that very well could punch their ticket to a Final Four appearance, including the No. 1 ranked team in the country, South Carolina.
While the Commodores start their season with a fair amount of nonconference play, they will truly be put to the test in the SEC this year, playing solely conference games from Dec. 30 until the season's conclusion on Feb. 27. On top of this, Vanderbilt will have three different instances this season where they face top-25 SEC opponents in back-to-back games.
It's going to be a challenging stretch for the Commodores this winter, but definitely a telling one. After just experiencing an all too short and disjointed season of basketball, they will certainly get valuable experience under their belt with a full plate of action in the 2021-22 season. In the last five years under Stephanie White, the Commodores were a measly 13-52 against SEC competition, including losses in all three bouts of the kind last year. This trend obviously needs to be flipped if Vanderbilt has hopes of joining the echelon of elite basketball programs in the nation.
"It is hands-down, and I've thought for a long time, the best conference in the country," Ralph said of the SEC in her opening presser. "Day in and day out you're battling, and I love a good battle."
Stepping up to the challenge of the SEC will certainly be Ralph's main focus as we head into winter.
Changes, changes, changes: What will the squad look like minus the Commodores' three leading scorers?
On top of switching the head of the dragon, Vanderbilt will look quite different when it comes to the wings, too. That analogy made sense, right?
The biggest blow to the team came on Mar. 31, as sophomore star Koi Love entered the transfer portal, eventually ending up at No. 22 Arizona. Love has quite a resume, as she was ranked in the top 70 of her recruiting class, was named to the All-SEC Freshman team in her rookie campaign and led the Commodores in points (20.8) and rebounds (9.5) per game in their abridged COVID-19 season.
This is not the only loss the Commodores suffered. Their second and third leading scorers from the 2020-21 season, Chelsie Hall and Enna Pehadzic, both graduated. It'll be up to Ralph and co. to facilitate the blossoming of a new looking squad, presumably with seniors Jordyn Cambridge and Brinae Alexander at the helm.
"I am completely healthy. The rust is completely knocked off. I'm just ready to play in a game," Cambridge, who is coming off an ACL injury, told the media last week.
It is also crucial that juniors Yaubryon Chambers, Demi Washington and Brylee Bartman continue their development in order for the Commodores to take the next step. New additions to watch include two top-100 ranked freshman recruits Sacha Washington and De'Mauri Flournoy, as well as Dartmouth graduate Asha Taylor. It will certainly be a year of new faces for the Commodores all around, but again, that might be just what the doctor ordered.
The Commodores will begin their season on Nov. 9 when they host Gardner-Webb at Memorial Gymnasium.
Brinae Alexander
Brylee Bartman
Demi Washington
Jordyn Cambridge
Shea Ralph
Vanderbilt Women's Basketball
Jonah Barbin, Sports Copy Editor
Jonah Barbin ('25) is majoring in Human and Organizational Development and Cinema and Media Studies. In addition to writing about sports, you can catch him acting, scouring the fantasy football waiver wire, playing golf and fantasizing about what Odell Beckham Jr.'s career would have been if the Giants never traded him. You can reach him at [email protected]


text:
Home Industries Nonprofit United Way 2019 campaign brings in more than $56 million
United Way 2019 campaign brings in more than $56 million
Exceeds campaign goal
Campaign co-chairs Cecelia Gore, Blake Moret, George Oliver and Mike Flynn.
Last updated on December 17th, 2019 at 12:24 pm
United Way of Greater Milwaukee & Waukesha County's 2019 community campaign brought in $56.15 million, exceeding its target by $1 million.
The organization's past few campaigns have raised $56 million (2018); $60.4 million (2017); and $60.1 million (2016).
United Way launched this year's campaign in August with a goal of raising $55 million.
The campaign total was announced Wednesday evening at Discovery World in Milwaukee.
This year's campaign co-chairs include: Cecelia Gore, executive director of Brewers Community Foundation; Mike Flynn, president of First Business Bank's Milwaukee region; Blake Moret, chairman and chief executive officer of Rockwell Automation; and George Oliver, chairman and CEO of Johnson Controls.
More than 1,500 companies and organizations and 50,000 donors contributed to the campaign. It supports United Way's investments in more than 220 programs and more than 110 partner agencies in Milwaukee, Ozaukee, Washington and Waukesha counties
"We are so grateful for the hard work and dedication of our campaign co-chairs, our campaign cabinet, our agency program partners, and so many members of our community," said United Way president and chief executive officer Amy Lindner. "They are the reason we were so successful with our campaign in 2019."
The campaign's largest corporate supporters included:
Fiduciary Management/T&M Partners, LLC, $4.8 million
Johnson Controls, $3.9 million
Northwestern Mutual, $3.4 million
Advocate Aurora Health, $3 million
Baird, $3 million
Rockwell Automation, $2.8 million
BMO Harris Bank, $1.7 million
We Energies, $1.7 million
Froedtert Health, $1.1 million
ManpowerGroup, $1.1 million
Harley-Davidson Motor Company, $1 million
Lauren Anderson covers health care, nonprofits, education and insurance for BizTimes. Lauren previously reported on education for the Waukesha Freeman. She graduated from the University of Wisconsin-Madison, where she studied journalism, history and African studies. In her free time, Lauren enjoys spending time with family and friends and seeing live music wherever she can.


text:
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var tape = require( 'tape' );
var unsnakecase = require( './../lib/unsnakecase.js' );

// TESTS //

tape( 'main export is a function', function test( t ) {
	t.ok( true, __filename );
	t.equal( typeof unsnakecase, 'function', 'main export is a function' );
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tape( 'the function converts a snakecase name to a normal name', function test( t ) {
	var out = unsnakecase( 'william_j_clinton' );
	t.equal( out, 'William J Clinton', 'converts snakecase name' );
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I was born in Levelland, TX, but moved to the MidWest in middle school. I graduated from Portage HS, Portage, IN. My undergraduate degree is from Campbellsville University, Campbellsbville, KY with a Vocal Performance degree. I have a Master of Music degree from SWBTS in Fort Worth, TX.
My have taught in private, Christian & public schools in all levels of music education. My most recent places of teaching are in Dallas ISD (last 2 years) and McKinney ISD (4 years as MS choir teacher).
My husband, Bruce, & I have been married a LONG time :) We have 4 adult children & 2 small grandkids. Bruce is the new pastor of Immanuel Baptist here in Brownfield. We are relocating from Richardson, TX. Our kids live in 3 different states & our oldest son is deployed to Afghanistan.
Our yorkie mix dog, "Butterfly" is spoiled now that our kids are gone. Bruce is an organic gardener and we love to cook from scratch (when we have time!).


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text:
Daily Sundial • May 6, 2010 • https://sundial.csun.edu/26486/sports/lion-knockout-in-three-csun-falls-to-penn-state-in-the-ncaa-semifinal-in-straight-sets/
Lion knockout in three: CSUN falls to Penn State in the NCAA Semifinal in straight sets
Victor Corona
After a 17 year hiatus the Matadors made it back to the NCAA Final Four, only to lose in straight sets against Penn State in the semifinals on Thurday night at Maples Pavilion. Photo Credit: Brianna Hart Special to Daily Sundial
The Matadors first appearance in the NCAA National Semifinal in 17 years ended in three sets as Penn State dominated the Matadors 30-21, 30-23, 30-28 Thursday night at Maples Pavilion.
"We just knew what to expect from each one of their hitters," Penn State All-American Max Lipsitz said. "I feel like we were at the right place at the right time."
In a losing effort, All-American middle blocker Jacek Ratajczak led the Matadors with 15 kills, while senior outside Mike Gaudino added seven. Senior outside hitter Theo Edwards and redshirt freshman John Baker finished with five a piece.
Senior opposite hitter William Price and redshirt sophomore Joe Sunder led the Nittany Lions with 18 kills a piece.
Penn State took control of the match right from the opening serve, winning the first set in dominating fashion 30-21. Things did not get better for the Matadors in the second set, as the team fell behind 20-16 and were unable to mount a comeback in route to losing the frame 30-23. In the third set Northridge once again fell behind by as many as seven points but fell short with a late rally dropping the third set 30-28.
"As a coach I can sum this up. We looked like we hadn't been here in 17 years and they looked like they were here every year," Matador head coach Jeff Campbell said.