text:
\section{Introduction}

\textcolor{cyan}{}High-treewidth graphical models typically yield
distributions where exact inference is intractable. To cope with this,
one often makes an approximation based on a tractable model. For example,
given some intractable distribution $q$, mean-field inference \cite{AMeanFieldTheoryLearningAlgorithm}
attempts to minimize $KL(p||q)$ over $p\in\text{TRACT}$, where $\text{TRACT}$
is the set of fully-factorized distributions. Similarly, structured
mean-field minimizes $ $the KL-divergence, but allows $\text{TRACT}$
to be the set of distributions that obey some tree \cite{ExploitingTractableSubstructures}
or a non-overlapping clustered \cite{AGeneralizedMeanField} structure.
In different ways, loopy belief propagation \cite{Yedidia05constructingfree}
and tree-reweighted belief propagation \cite{Wainwright_anewclass}
also make use of tree-based approximations, while Globerson and Jaakkola
\cite{ApproximateInferenceUsingPlanarGraph} provide an approximate
inference method based on exact inference in planar graphs with zero
field.\textcolor{cyan}{}

In this paper, we explore an alternative notion of a ``tractable''
model. These are ``fast mixing'' models, or distributions that,
while they may be high-treewidth, have parameter-space conditions
guaranteeing that Gibbs sampling will quickly converge to the stationary
distribution. While the precise form of the parameter space conditions
is slightly technical (Sections \ref{sec:Background}-\ref{sec:Mixing-Time-Bounds}),
informally, it is simply that interaction strengths between neighboring
variables are not too strong.

In the context of the Ising model, we attempt to use these models
in the most basic way possible-- by taking an arbitrary (slow-mixing)
set of parameters, projecting onto the fast-mixing set, using four
different divergences. First, we show how to project in the Euclidean
norm, by iteratively thresholding a singular value decomposition (Theorem
\ref{thm:projection_theorem-main}). Secondly, we experiment with
projecting using the ``zero-avoiding'' divergence $KL(q||p)$. Since
this requires taking (intractable) expectations with respect to $q$,
it is of only theoretical interest. \textcolor{cyan}{} Third, we
suggest a novel ``piecewise'' approximation of the KL divergence,
where one drops edges from both $q$ and $p$ until a low-treewidth
graph remains where the exact KL divergence can be calculated. Experimentally,
this does not perform as well as the true KL-divergence, but is easy
to evaluate. Fourth, we consider the ``zero forcing'' divergence
$KL(q||p)$. Since this requires expectations with respect to $p$,
which is constrained to be fast-mixing, it can be approximated by
Gibbs sampling, and the divergence can be minimized through stochastic
approximation. This can be seen as a generalization of mean-field
where the set of approximating distributions is expanded from fully-factorized
to fast-mixing.

\section{Background\label{sec:Background}}

The literature on mixing times in Markov chains is extensive, including
a recent textbook \cite{MarkovChainsAndMixingTimes}. The presentation
in the rest of this section is based on that of Dyer et al. \cite{MatrixNormsAndRapidMixingForSpinSystems}.

Given a distribution $p(x)$, one will often wish to draw samples
from it. While in certain cases (e.g. the Normal distribution) one
can obtain exact samples, for Markov random fields (MRFs), one must
generally resort to iterative Markov chain Monte Carlo (MCMC) methods
that obtain a sample asymptotically. In this paper, we consider the
classic Gibbs sampling method \cite{StochasticRelaxationGibbsDistributions},
where one starts with some configuration $x$, and repeatedly picks
a node $i$, and samples $x_{i}$ from $p(x_{i}|x_{-i})$. Under mild
conditions, this can be shown to sample from a distribution that converges
to $p$ as $t\rightarrow\infty$.

It is common to use more sophisticated methods such as block Gibbs
sampling, the Swendsen-Wang algorithm \cite{SwendsenWang}, or tree
sampling \cite{FromFieldsToTrees}. In principle, each algorithm could
have unique parameter-space conditions under which it is fast mixing.
Here, we focus on the univariate case for simplicity and because fast
mixing of univariate Gibbs is sufficient for fast mixing of some other
methods \cite{ExtraUpdates}.
\begin{defn}
Given two finite distributions $p$ and $q$, the \textbf{total variation
distance} $||\cdot||_{TV}$ is
\[
||p(X)-q(X)||_{TV}=\frac{1}{2}\sum_{x}|p(X=x)-q(X=x)|.
\]

\end{defn}
We need a property of a distribution that can guarantee fast mixing.
The \textbf{dependency} $R_{ij}$ of $x_{i}$ on $x_{j}$ is defined
by considering two configurations $x$ and $x'$, and measuring how
much the conditional distribution of $x_{i}$ can vary when $x_{k}=x'_{k}$
for all $k\not=j$.
\begin{defn}
Given a distribution $p$, the dependency matrix $R$ is defined by

\[
R_{ij}=\max_{x,x':x_{-j}=x_{-j}'}||p(X_{i}|x_{-i})-p(X_{i}|x_{-i}')||_{TV}.
\]

\end{defn}
Given some threshold $\epsilon$, the \textbf{mixing time} is the
number of iterations needed to guarantee that the total variation
distance of the Gibbs chain to the stationary distribution is less
than $\epsilon$.
\begin{defn}
Suppose that $\{X^{t}\}$ denotes the sequence of random variables
corresponding to running Gibbs sampling on some distribution $p$.
The mixing time $\tau(\epsilon)$ is the minimum time $t$ such that
the total variation distance between $X^{t}$ and the stationary distribution
is at most $\epsilon$. That is,
\begin{align*}
\tau(\epsilon)= & \min\{t:d(t)<\epsilon\},\\
d(t)= & \max_{x}||\mathbb{P}(X^{t}|X^{0}=x)-p(X)||_{TV}.
\end{align*}

\end{defn}
Unfortunately, the mixing time can be extremely long, which makes
the use of Gibbs sampling delicate in practice. For example, for the
two-dimensional Ising model with zero field and uniform interactions,
it is known that mixing time is polynomial (in the size of the grid)
when the interaction strengths are below a threshold $\beta_{c}$,
and exponential for stronger interactions \cite{CriticalIsingMixing}.
For more general distributions, such tight bounds are not generally
known, but one can still derive sufficient conditions for fast mixing.
The main result we will use is the following \cite{ASimpleConditionImplyingRapidMixing}.
\begin{thm}
\label{thm:mixing-time-theorem}Consider the dependency matrix $R$
corresponding to some distribution $p(X_{1},...,X_{n})$. For Gibbs
sampling with random updates, if $||R||_{2}<1,$ the mixing time is
bounded by

\[
\tau(\epsilon)\leq\frac{n}{1-||R||_{2}}\ln\left(\frac{n}{\epsilon}\right).
\]

\end{thm}
Roughly speaking, if the spectral norm (maximum singular value) of
$R$ is less than one, rapid mixing will occur. A similar result holds
in the case of systematic scan updates \cite{MatrixNormsAndRapidMixingForSpinSystems,ASimpleConditionImplyingRapidMixing}.

Some of the classic ways of establishing fast mixing can be seen as
special cases of this. For example, the Dobrushin criterion is that
$||R||_{1}<1$, which can be easier to verify in many cases, since
$||R||_{1}=\max_{j}\sum_{i}|R_{ij}|$ does not require the computation
of singular values. However, for symmetric matrices, it can be shown
that $||R||_{2}\leq||R||_{1}$, meaning the above result is tighter.

\section{Mixing Time Bounds\label{sec:Mixing-Time-Bounds}}

For variables $x_{i}\in\{-1,+1\},$ an Ising model is of the form
\[
p(x)=\exp\left(\sum_{i,j}\beta_{ij}x_{i}x_{j}+\sum_{i}\alpha_{i}x_{i}-A(\beta,\alpha)\right),
\]
where $\beta_{ij}$ is the interaction strength between variables
$i$ and $j$, $\alpha_{i}$ is the ``field'' for variable $i$,
and $A$ ensures normalization. This can be seen as a member of the
exponential family $p(x)=\exp\left(\theta\cdot f(x)-A(\theta)\right),$
where $f(x)=\{x_{i}x_{j}\forall(i,j)\}\cup\{x_{i}\forall i\}$ and
$\theta$ contains both $\beta$ and $\alpha$.
\begin{lem}
\label{lem:ising-dependency}For an Ising model, the dependency matrix
is bounded by
\[
R_{ij}\leq\tanh|\beta_{ij}|\leq|\beta_{ij}|
\]

\end{lem}
Hayes \cite{ASimpleConditionImplyingRapidMixing} proves this for
the case of constant $\beta$ and zero-field, but simple modifications
to the proof can give this result.

Thus, to summarize, an Ising model can be guaranteed to be fast mixing
if the spectral norm of the absolute value of interactions terms is
less than one.

\section{Projection\label{sec:Projection}}

In this section, we imagine that we have some set of parameters $\theta$,
not necessarily fast mixing, and would like to obtain another set
of parameters $\psi$ which are as close as possible to $\theta$,
but guaranteed to be fast mixing. This section derives a projection
in the Euclidean norm, while Section \ref{sec:Divergences} will build
on this to consider other divergence measures.

We will use the following standard result that states that given a
matrix $A$, the closest matrix with a maximum spectral norm can be
obtained by thresholding the singular values.\textcolor{cyan}{}
\begin{thm}
\label{thm:dense-projection}If $A$ has a singular value decomposition
$A=USV^{T}$, and $||\cdot||_{F}$ denotes the Frobenius norm, then
$B=\underset{B:||B||_{2}\leq c}{\arg\text{}\min}||A-B||_{F}$ can
be obtained as $B=US'V^{T},$ where $S_{ii}^{'}=\min(S_{ii},c^{2}).$
\end{thm}
We denote this projection by $B=\Pi_{c}[A]$. This is close to providing
an algorithm for obtaining the closest set of Ising model parameters
that obey a given spectral norm constraint. However, there are two
issues. First, in general, even if $A$ is sparse, the projected matrix
$B$ will be dense, meaning that projecting will destroy a sparse
graph structure. Second, this result constrains the spectral norm
of $B$ itself, rather than $R=|B|$, which is what needs to be controlled.
The theorem below provides a dual method that fixed these issues.

Here, we take some matrix $Z$ that corresponds to the graph structure,
by setting $Z_{ij}=0$ if $(i,j)$ is an edge, and $Z_{ij}=1$ otherwise.
Then, enforcing that $B$ obeys the graph structure is equivalent
to enforcing that $Z_{ij}B_{ij}=0$ for all $(i,j)$. Thus, finding
the closest set of parameters $B$ is equivalent to solving

\vspace{-20pt}
\begin{eqnarray}
\min_{B,D} &  & ||A-B||_{F}\text{ subject to }||D||_{2}\leq c,\,\, Z_{ij}D_{ij}=0,\,\, D=|B|.\label{eq:sparse-projection-minimization-singleline}
\end{eqnarray}

We find it convenient to solve this minimization by performing some
manipulations, and deriving a dual. The proof of this theorem is provided
in the appendix. To accomplish the maximization of $g$ over $M$
and $ $$\Lambda$, we use LBFGS-B \cite{ALimitedMemoryAlgorithmFroBoundConstrainedOptimization},
with bound constraints used to enforce that $M\geq0$.

The following theorem uses the ``triple dot product'' notation of
$A\cdot B\cdot C=\sum_{ij}A_{ij}B_{ij}C_{ij}$.
\begin{thm}
\label{thm:projection_theorem-main}Define $R=|A|$. The minimization
in Eq. \ref{eq:sparse-projection-minimization-singleline} is equivalent
to the problem of $\max_{M\geq0,\Lambda}g(\Lambda,M)$, where the
objective and gradient of $g$ are, for $D(\Lambda,M)=\Pi_{c}[R+M-\Lambda\odot Z],$

\begin{align}
g(\Lambda,M) & =\frac{1}{2}||D(\Lambda,M)-R||_{F}^{2}+\Lambda\cdot Z\cdot D(\Lambda,M) - M \cdot D(\Lambda,M) \label{eq:sparse-projection-dual-obj-1-1}\\
\frac{dg}{d\Lambda} & =Z\odot D(\Lambda,M)\label{eq:sparse-projection-dual-grad-1-1}\\
\frac{dg}{dM} & = -D(\Lambda,M).\label{eq:sparse-projection-dual-grad-2-1}
\end{align}

\end{thm}

\section{Divergences\label{sec:Divergences}}

Again, we would like to find a parameter vector $\psi$ that is close
to a given vector $\theta$, but is guaranteed to be fast mixing,
but with several notions of ``closeness'' that vary in terms of
accuracy and computational convenience. Formally, if $\Psi$ is the
set of parameters that we can guarantee to be fast mixing, and $D(\theta,\psi)$
is a divergence between $\theta$ and $\psi$, then we would like
to solve
\begin{equation}
\arg\min_{\psi\in\Psi}D(\theta,\psi).\label{eq:general_projection}
\end{equation}

As we will see, in selecting $D$ there appears to be something of
a trade-off between the quality of the approximation, and the ease
of computing the projection in Eq. \ref{eq:general_projection}.

In this section, we work with the generic exponential family representation
\vspace{-10pt}

\[
p(x;\theta)=\exp(\theta\cdot f(x)-A(\theta)).
\]
We use $\mu$ to denote the mean value of $f$. By a standard result,
this is equal to the gradient of $A$, i.e.

\[
\mu(\theta)=\sum_{x}p(x;\theta)f(x)=\nabla A(\theta).
\]
\vspace{-15pt}

\subsection{Euclidean Distance}

The simplest divergence is simply the $l_{2}$ distance between the
parameter vectors, $D(\theta,\psi)=||\theta-\psi||_{2}$. For the
Ising model, Theorem \ref{thm:projection_theorem-main} provides a
method to compute the projection $\arg\min_{\psi\in\Psi}||\theta-\psi||_{2}.$
While simple, this has no obvious probabilistic interpretation, and
other divergences perform better in the experiments below.

However, it also forms the basis of our projected gradient descent
strategy for computing the projection in Eq. \ref{eq:general_projection}
under more general divergences $D$. Specifically, we will do this
by iterating
\begin{enumerate}
\item $\psi'\leftarrow\psi-\lambda\frac{d}{d\psi}D(\theta,\psi)$
\item $\psi\leftarrow\arg\min_{\psi\in\Psi}||\psi'-\psi||_{2}$
\end{enumerate}
for some step-size $\lambda$. In some cases, $dD/d\psi$ can be calculated
exactly, and this is simply projected gradient descent. In other cases,
one needs to estimate $dD/d\psi$ by sampling from $\psi$. As discussed
below, we do this by maintaining a ``pool'' of samples. In each
iteration, a few Markov chain steps are applied with the current parameters,
and then the gradient is estimated using them. Since the gradients
estimated at each time-step are dependent, this can be seen as an
instance of Ergodic Mirror Descent \cite{ErgodicMirrorDescent}. This
guarantees convergence if the number of Markov chain steps, and the
step-size $\lambda$ are both functions of the total number of optimization
iterations.

\subsection{KL-Divergence}

Perhaps the most natural divergence to use would be the ``inclusive''
KL-divergence
\begin{equation}
D(\theta,\psi)=KL(\theta||\psi)=\sum_{x}p(x;\theta)\log\frac{p(x;\theta)}{p(x;\psi)}.\label{eq:zero-avoiding-KL}
\end{equation}

This has the ``zero-avoiding'' property \cite{DivergenceMeasuresAndMessagePassing}
that $\psi$ will tend to assign some probability to all configurations
that $\theta$ assigns nonzero probability to. It is easy to show
that the derivative is
\begin{equation}
\frac{dD(\theta,\psi)}{d\psi}=\mu(\psi)-\mu(\theta),\label{eq:KL-divergence-gradient}
\end{equation}
where $\mu_{\theta}=\mathbb{E}_{\theta}[f(X)].$ Unfortunately, this
requires inference with respect to both the parameter vectors $\theta$
and $\psi$. Since $\psi$ will be enforced to be fast-mixing during
optimization, one could approximate $\mu(\psi)$ by sampling. However,
$\theta$ is presumed to be slow-mixing, making $\mu(\theta)$ difficult
to compute. Thus, this divergence is only practical on low-treewidth
``toy'' graphs.

\subsection{Piecewise KL-Divergences}

Inspired by the piecewise likelihood \cite{PiecewiseTraining} and
likelihood approximations based on mixtures of trees \cite{SpanningTreeApproximations},
we seek tractable approximations of the KL-divergence based on tractable
subgraphs. Our motivation is the the following: if $\theta$ and $\psi$
define the same distribution, then if a certain set of edges are removed
from both, they should continue to define the same distribution%
\footnote{Technically, here, we assume that the exponential family is minimal.
However, in the case of an over-complete exponential family, enforcing
this will simply ensure that $\theta$ and $\psi$ use the same reparameterization.%
}. Thus, given some graph $T$, we define the ``projection'' $\theta(T)$
onto the tree such by setting all edge parameters to zero if not part
of $T$. Then, given a set of graphs $T$, the piecewise KL-divergence
is
\[
D(\theta,\psi)=\max_{T}KL(\theta(T)||\psi(T)).
\]
Computing the derivative of this divergence is not hard-- one simply
computes the KL-divergence for each graph, and uses the gradient as
in Eq. \ref{eq:KL-divergence-gradient} for the maximizing graph.

There is some flexibility of selecting the graphs $T$. In the simplest
case, one could simply select a set of trees (assuring that each edge
is covered by one tree), which makes it easy to compute the KL-divergence
on each tree using the sum-product algorithm. We will also experiment
with selecting low-treewidth graphs, where exact inference can take
place using the junction tree algorithm.

\subsection{Reversed KL-Divergence}

We also consider the ``zero-forcing'' KL-divergence
\[
D(\theta,\psi)=KL(\psi||\theta)=\sum_{x}p(x;\psi)\log\frac{p(x;\psi)}{p(x;\theta)}.
\]

\begin{thm}
The divergence $D(\theta,\psi)=KL(\psi||\theta)$ has the gradient
\[
\frac{d}{d\psi}D(\theta,\psi)=\sum_{x}p(x;\psi)(\psi-\theta)\cdot f(x)\left(f(x)-\mu(\psi)\right).
\]
\vspace{-10pt}

\end{thm}
Arguably, using this divergence is inferior to the ``zero-avoiding''
KL-divergence. For example, since the parameters $\psi$ may fail
to put significant probability at configurations where $\theta$ does,
using importance sampling to reweight samples from $\psi$ to estimate
expectations with respect to $\theta$ could have high variance Further,
it can be non-convex with respect to $\psi$. Nevertheless, it often
work well in practice. Minimizing this divergence under the constraint
that the dependency matrix $R$ corresponding to $\psi$ have a limited
spectral norm is closely related to naive mean-field, which can be
seen as a degenerate case where one constrains $R$ to have zero norm.

This is easier to work with than the ``zero-avoiding'' KL-divergence
in Eq. \ref{eq:zero-avoiding-KL} since it involves taking expectations
with respect to $\psi$, rather than $\theta$: since $\psi$ is enforced
to be fast-mixing, these expectations can be approximated by sampling.
Specifically, suppose that one has generated a set of samples $x^{1},...,x^{K}$
using the current parameters $\psi$. Then, one can first approximate
the marginals by $\hat{\mu}=\frac{1}{K}\sum_{k=1}^{K}f(x^{k}),$ and
then approximate the gradient by
\begin{equation}
\hat{g}=\frac{1}{K}\sum_{k=1}^{K}\left((\psi-\theta)\cdot f(x^{k})\right)\left(f(x^{k})-\hat{\mu}\right).\label{eq:SGD_gradient_estimate}
\end{equation}
\vspace{-10pt}

It is a standard result that if two estimators are unbiased and independent,
the product of the two estimators will also be unbiased. Thus, if
one used separate sets of perfect samples to estimate $\hat{\mu}$
and $\hat{g}$, then $\hat{g}$ would be an unbiased estimator of
$dD/d\psi$. In practice, of course, we generate the samples by Gibbs
sampling, so they are not quite perfect. We find in practice that
using the same set of samples twice makes makes little difference,
and do so in the experiments.
\begin{figure}[H]
\begin{centering}
\hfill{}\includegraphics[width=0.445\textwidth]{esacc_plot/grid_mixed}\hfill{}\includegraphics[width=0.445\textwidth]{esacc_plot/grid_attractive}\hfill{}
\par\end{centering}

\begin{centering}
\hfill{}\includegraphics[width=0.445\textwidth]{esacc_plot/rg_a1_0. 3}\hfill{}\includegraphics[width=0.445\textwidth]{esacc_plot/rg_1_0. 3}\hfill{}\vspace{-10pt}

\par\end{centering}

\caption{The mean error of estimated univariate marginals on 8x8 grids (top
row) and low-density random graphs (bottom row), comparing 30k iterations
of Gibbs sampling after projection to variational methods. To approximate
the computational effort of projection (Table \ref{tab:Running-Times}),
sampling on the original parameters with 250k iterations is also included
as a lower curve. (Full results in appendix.)}
\end{figure}

\section{Experiments}

Our experimental evaluation follows that of Hazan and Shashua \cite{ConvergentMessagePassingAlgorithms}
in evaluating the accuracy of the methods using the Ising model in
various configurations. In the experiments, we approximate randomly
generated Ising models with rapid-mixing distributions using the projection
algorithms described previously. Then, the marginals of rapid-mixing
approximate distribution are compared against those of the target
distributions by running a Gibbs chain on each. We calculate the mean
absolute distance of the marginals as the accuracy measure, with the
marginals computed via the exact junction-tree algorithm.

We evaluate projecting under the Euclidean distance (Section 5.1),
the piecewise divergence (Section 5.3)\textcolor{black}{, and the
zero-forcing KL-divergence $KL(\psi||\theta)$} (Section 5.4). On
small graphs, it is possible to minimize the zero-avoiding KL-divergence
$KL(\theta||\psi)$ by computing marginals using the junction-tree
algorithm. However, as minimizing this KL-divergence leads to exact
marginal estimates, it doesn't provide a useful measure of marginal
accuracy. Our methods are compared with four other inference algorithms,
namely loopy belief-propagation (LBP), Tree-reweighted belief-propagation
(TRW), Naive mean-field (MF), and Gibbs sampling on the original parameters.

LBP, MF and TRW are among the most widely applied variational methods
for approximate inference. The MF algorithm uses a fully factorized
distribution as the tractable family, and can be viewed as an extreme
case of\textcolor{black}{{} minimizing the zero forcing KL-divergence
$KL(\psi||\theta)$} under the constraint of zero spectral norm. The
tractable family that it uses guarantees ``instant'' mixing but
is much more restrictive. Theoretically, Gibbs sampling on the original
parameters will produce highly accurate marginals if run long enough.
However, this can take exponentially long and convergence is generally
hard to diagnose \cite{Cowles96markovchain}\textcolor{cyan}{.} In
contrast, Gibbs sampling on the rapid-mixing approximation is guaranteed
to converge rapidly but will result in less accurate marginals asymptotically.
Thus, we also include time-accuracy comparisons between these two
strategies in the experiments.

\subsection{Configurations}

Two types of graph topologies are used: two-dimensional $8\times8$
grids and random graphs with $10$ nodes. Each edge is independently
present with probability $p_{e}\in\{0.3,0.5,0.7\}$. Node parameters
$\theta_{i}$ are uniformly drawn from unif$(-d_{n},d_{n})$ and we
fix the field strength to $d_{n}=1.0$. Edge parameters $\theta_{ij}$
are uniformly drawn from unif$(-d_{e},d_{e})$ or unif$(0,d_{e})$
to obtain mixed or attractive interactions respectively. We generate
graphs with different interaction strength $d_{e}=0,0.5,\dots,4$.
All results are averaged over 50 random trials.

To calculate piecewise divergences, it remains to specify the set
of subgraphs $T$. It can be any tractable subgraph of the original
distribution. For the grids, one straightforward choice is to use
the horizontal and the vertical chains as subgraphs. We also test
with chains of treewidth 2. For random graphs, we use\textcolor{black}{{}
the sets of random spanning trees which can cover every edge of the
original graphs as the set of subgraphs}.

\textcolor{black}{A stochastic gradient descent algorithm is applied
to minimize the zero forcing KL-divergence $KL(\psi||\theta)$. In
this algorithm, a ``pool'' of samples is repeatedly used to estimate
gradients as in Eq. \ref{eq:SGD_gradient_estimate}. After each parameter
update, each sample is updated by a single Gibbs step, consisting
of one pass over all variables. The performance of this algorithm
can be affected by several parameters, including the gradient search
step size, the size of the sample pool, the number of Gibbs updates,
and the number of total iterations. (This algorithm can be seen as
an instance of Ergodic Mirror Descent \cite{ErgodicMirrorDescent}.)
Without intensive tuning of these parameters, we choose a constant
step size of $0.1$, sample pool size of 500 and 60 total iterations,
which performed reasonably well in practice. }

For each original or approximate distribution, a single chain of Gibbs
sampling is run on the final parameters, and marginals are estimated
from the samples drawn.\textcolor{black}{{} Each Gibbs iteration is
one pass of systematical scan over the variables in fixed order. Note
that this does not take into account the computational effort deployed
during projection, which ranges from 30,000 total Gibbs iterations
with repeated Euclidean projection $(KL(\psi||\theta)$) to none at
all (Original parameters). It has been our experience that more aggressive
parameters can lead to this procedure being more accurate than Gibbs
in a comparison of total computational effort, but such a scheduling
tends to also reduce the accuracy of the final parameters, making
results more difficult to interpret.}

In Section 3.2, we show that for Ising models, a sufficient condition
for rapid-mixing is the spectral norm of pairwise weight matrix is
less than 1.0. However, we find in practice using a spectral norm
bound of $2.5$ instead of $1.0$ can still preserve the rapid-mixing
property and gives better approximation to the original distributions.
(See Section \ref{sec:Discussion} for a discussion.)

\begin{figure}[t]
\begin{centering}
\hfill{}\includegraphics[width=0.445\textwidth]{tmacc_grid_plot/tmacc_1. 00_4. 00}\hfill{}\includegraphics[width=0.445\textwidth]{tmacc_grid_plot/tmacc_a_1. 00_4. 00}\hfill{}
\par\end{centering}

\begin{centering}
\hfill{}\includegraphics[width=0.445\textwidth]{tmacc_rg_plot/tmacc_rnd_0. 30_1. 00_3. 00}\hfill{}\includegraphics[width=0.445\textwidth]{tmacc_rg_plot/tmacc_rnd_a_0. 30_1. 00_3. 00}\hfill{}\vspace{-10pt}

\par\end{centering}

\caption{Example plots of the accuracy of obtained marginals vs. the number
of samples. Top: Grid graphs. Bottom: Low-Density Random graphs. (Full
results in appendix.)}
\end{figure}
\begin{table}
\setlength\tabcolsep{4pt}

\begin{tabular}{|c|cc|cc|cc|}
\cline{2-7}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{{\small{Grid, Strength 1.5}}} & \multicolumn{2}{c|}{{\small{Grid, Strength 3}}} & \multicolumn{2}{c|}{{\small{Random Graph, Strength 3.}}}\tabularnewline
\multicolumn{1}{c|}{} & {\small{Gibbs Steps}} & {\small{SVDs}} & {\small{Gibbs Steps}} & {\small{SVDs}} & {\small{Gibbs Steps}} & {\small{SVDs}}\tabularnewline
\hline
{\small{30,000 Gibbs steps}} & {\small{30k / 0.17s}} &  & {\small{30k / 0.17s}} &  & {\small{30k / 0.04s}} & \tabularnewline
{\small{250,000 Gibbs steps}} & {\small{250k / 1.4s}} &  & {\small{250k / 1.4s}} &  & {\small{250k / 0.33s}} & \tabularnewline
{\small{Euclidean Projection}} &  & {\small{22 / 0.04s}} &  & {\small{78 / 0.15s}} &  & {\small{17 / .0002s}}\tabularnewline
{\small{Piecewise-1 Projection}} &  & {\small{322 / 0.61s}} &  & {\small{547 / 1.0s}} &  & {\small{408 / 0.047s}}\tabularnewline
{\small{KL Projection}} & {\small{30k / 0.17s}} & {\small{265 / 0.55s}} & {\small{30k / 0.17s}} & {\small{471 / 0.94s}} & {\small{30k / 0.04s}} & {\small{300 / 0.037s}}\tabularnewline
\hline
\end{tabular}

\caption{Running Times on various attractive graphs, showing the number of
Gibbs passes and Singular Value Decompositions, as well as the amount
of computation time. The random graph is based on an edge density
of 0.7. Mean-Field, Loopy BP, and TRW take less than 0.01s.\label{tab:Running-Times}}
\end{table}

\section{Discussion\label{sec:Discussion}}

Inference in high-treewidth graphical models is intractable, which
has motivated several classes of approximations based on tractable
families. In this paper, we have proposed a new notion of ``tractability'',
insisting not that a graph has a fast algorithm for exact inference,
but only that it obeys parameter-space conditions ensuring that Gibbs
sampling will converge rapidly to the stationary distribution. For
the case of Ising models, we use a simple condition that can guarantee
rapid mixing, namely that the spectral norm of the matrix of interaction
strengths is less than one.

Given an intractable set of parameters, we consider using this approximate
family by ``projecting'' the intractable distribution onto it under
several divergences. First, we consider the Euclidean distance of
parameters, and derive a dual algorithm to solve the projection, based
on an iterative thresholding of the singular value decomposition.
Next, we extend this to more probabilistic divergences. Firstly, we
consider a novel ``piecewise'' divergence, based on computing the
exact KL-divergnce on several low-treewidth subgraphs. Secondly, we
consider projecting onto the KL-divergence. This requires a stochastic
approximation approach where one repeatedly generates samples from
the model, and projects in the Euclidean norm after taking a gradient
step.

We compare experimentally to Gibbs sampling on the original parameters,
along with several standard variational methods. The proposed methods
are more accurate than variational approximations. Given enough time,
Gibbs sampling using the original parameters will always be more accurate,
but with finite time, projecting onto the fast-mixing set to generally
gives better results.

Future work might extend this approach to general Markov random fields.
This will require two technical challenges. First, one must find a
bound on the dependency matrix for general MRFs, and secondly, an
algorithm is needed to project onto the fast-mixing set defined by
this bound. Fast-mixing distributions might also be used for learning.
E.g., if one is doing maximum likelihood learning using MCMC to estimate
the likelihood gradient, it would be natural to constrain the parameters
to a fast mixing set.

One weakness of the proposed approach is the apparent looseness of
the spectral norm bound. For the two dimensional Ising model with
no univariate terms, and a constant interaction strength $\beta$,
there is a well-known threshold $\beta_{c}=\frac{1}{2}\ln(1+\sqrt{2})\approx.4407$,
obtained using more advanced techniques than the spectral norm \cite{CriticalIsingMixing}.
Roughly, for $\beta<\beta_{c}$, mixing is known to occur quickly
(polynomial in the grid size) while for $\beta>\beta_{c}$, mixing
is exponential. On the other hand, the spectral bound norm will be
equal to one for $\beta=.25$, meaning the bound is too conservative
in this case by a factor of $\beta_{c}/.25\approx1.76$. A tighter
bound on when rapid mixing will occur would be more informative.

\newpage{}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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Last year, Velden was a GREAT trip for me and everyone at Team Intertops! Three of us had deep runs but fun was had by all! I finished 22nd and one of our players even hit the final table for 6th place, cashing for almost 30k! But even without such success for our players, the trip itself would have been a win! We had a beautiful location, great event, delicious (but weird lol) food and some amazing company! It was a blast! Exciting poker and an exciting atmosphere! Velden is a vacation hot spot and there's always something fun going on, both in the Poker room and out! Be sure not to miss out! Get your seat in tomorrow's satellite final and stay tuned for the next!! Join us and grab your share of estimated prize money of more than 700,000 Euro (about $770,000)! The European Poker Championship 2017 takes place from July 19th to 30th (main event: July 27th to 30th) Until then here's the first part of last years adventure!! Enjoy! and I hope to see you at this years event!
« $4,500 Prize Package guaranteed!
On to the next one! The WSOP may be over but the poker never stops! We have arrived in Velden, Austria for the another amazing tournament series! Myself and the rest of Team Intertops are in this beautiful location to take a shot at the massive 2700 euro Poker EM main event! Intertops had satellites running for a while a couple months ago for this awesome trip, and lucky few of us are here to attempt to make the poker dream a reality. The idea of scooping a win here for insane amounts of money for only a cheap satellite entry online is "Living the Dream" to be sure! Intertops has done it again, not only did they pick a great tourney in a great spot, they over deliver on taking care of their players. From helping with rides from the airport to getting us settled at the hotel/casino to taking us all out on a boat ride to an amazing dinner! Its been incredible and its only day two! lol No ones even played poker yet!
After writing my last blog post on lesson learned from the WSOP, I knew I needed to take a step back and get some perspective. I will admit, at first the idea of traveling this far and playing poker again so soon after the WSOP wrapped up was not looking so great to me. The life of playing poker professionally can be brutal sometimes. The airport lines, the long flights, hotel living, the costs and just the straight grind that is tournament poker wears on you quickly. But as soon as I arrived here, I knew this opportunity could not have come at a better time. The country is beautiful, the location is amazing, and the people I've met already are great! It's already been the best reminder for why I wanted to do this for a living! These are the experiences that make all the losses, bad beats and close calls worth it. The job could allows us to travel across the globe to play a GAME…It puts things in perspective really quickly, especially with all the tragedies that seem to be happening on a daily basis in the world.
Wish us all luck, as we start our main event here today! And I know from my previous experience in Seefeld that the competition will be fierce. At least the insanely aggressive play will not be a surprise this time around! lol It should be a really challenging tournament to say the least, but I know will all will enjoy it! Maybe Team Intertops can come away with a five way chop!? (Nahhhh no chops! but I'll let them have 2-5th at least! lol) We are all hoping for a deep run and to come home with the championship trophy, but all we can do is play our best and enjoy the ride! Everyone here has already won in my book! I'll be back soon with another post updating our day 1 status.


text:
Matt Patricia, Detroit Lions will not face discipline for sexual assault allegation
The Sports Xchange
New England Patriots head coach Bill Belichick (L) and defensive coordinator Matt Patricia (R) watch the Denver Broncos offense in the third quarter during the AFC Championship Game on January 24, 2016 at Sports Authority Field at Mile High in Denver. File photo by Gary C. Caskey/UPI | License Photo
The NFL will not discipline the Detroit Lions or Matt Patricia after a sexual assault allegation from 1996 surfaced against the head coach, the league announced Monday.
ESPN's Adam Schefter cited league sources as saying the NFL met with the Lions and Patricia last week.
"Our office reviewed the matter with the Lions and Mr. Patricia, and ensured the club engaged in appropriate and thorough hiring practices and that the Coach did not mislead the team during the interview process," league spokesperson Brian McCarthy said in a statement to NFL Network's Tom Pelissero.
"We determined that the Lions handled the interview process in a thoughtful and comprehensive manner and fully and appropriately compiled with all applicable employment laws. As we learned from both Mr. Patricia and the Lions, the matter was not part of his employment interview process for job opportunities outside of or within the NFL. He was under no legal or other requirement to raise this issue.
RELATED New England Patriots' McDaniels offers support for Detroit Lions coach Patricia
"This is not subject to the personal conduct policy as the allegations occurred well before his affiliation with the NFL."
Patricia, now 43, was accused of one count of sexually assaulting a woman in South Padre Island, Texas, during spring break in 1996. He denied the accusation -- at the time and in a press conference on May 10.
"I was innocent then and I am innocent now," Patricia said during a May 10 news conference. "Let me be clear, my priorities remain the same: To move forward and strive to be the best coach, teacher and man that I can possibly be."
RELATED Detroit Lions head coach Matt Patricia denies 1996 assault allegation
Lions owner Martha Ford, team president Rod Wood and general manager Bob Quinn released a joint statement in support of Patricia.
The indictment never went to trial and the case was dismissed in January 1997, after the woman who made the accusation was "unable to testify and can not give a date certain when she will be available," according to the dismissal paperwork, per ESPN. "Victim does not feel she can face the pressures or stress of a trial. Victim may request that the case be refiled at a later date."
Patricia was hired by the Lions on Feb. 5, one day after the New England Patriots fell to the Philadelphia Eagles in Super Bowl LII. Patricia previously served as defensive coordinator of the Patriots.
RELATED Lions coach Patricia addresses 1996 charge
He replaced Jim Caldwell, who was fired on Jan. 11 after posting a 36-28 record in four seasons with the club.
NFL // 7 hours ago
Jan. 20 (UPI) -- Tennessee Titans defensive backs coach Kerry Coombs is returning to Ohio State as defensive coordinator.
NFL // 10 hours ago
Jan. 20 (UPI) -- The Houston Texans are moving on from defensive coordinator Romeo Crennel.
Jan. 20 (UPI) -- The Kansas City Chiefs opened as slight betting favorites to beat the San Francisco 49ers in Super Bowl LIV.
NFL // 1 day ago
Raheem Mostert leads 49ers over Packers in NFC title game
Jan. 19 (UPI) -- The San Francisco 49ers used a record-setting performance from Raheem Mostert to beat the Green Bay Packers 37-20 on Sunday in the NFC Championship Game.
49ers' Tevin Coleman carted off with shoulder injury vs. Packers
Jan. 19 (UPI) -- San Francisco 49ers running back Tevin Coleman was knocked out of Sunday's NFC Championship Game with a shoulder injury.
Chargers' Philip Rivers 'permanently' moving from San Diego to Florida
Jan. 19 (UPI) -- Quarterback Philip Rivers has "permanently" moved from San Diego, leaving his future with the Los Angeles Chargers in question.
Chiefs beat Titans to reach first Super Bowl in 50 years
Jan. 19 (UPI) -- The Kansas City Chiefs punched their ticket to the Super Bowl for the first time in 50 years with a 35-24 win over the Tennessee Titans in Sunday's AFC Championship Game.
Chiefs' Patrick Mahomes shows off wheels with electric 27-yard touchdown run
Jan. 19 (UPI) -- Kansas City star quarterback Patrick Mahomes gave the Chiefs their first lead of Sunday's AFC Championship Game against the Tennessee Titans with a spectacular 27-yard touchdown run.
Titans offensive lineman Dennis Kelly sets NFL record with big-man touchdown
Jan. 19 (UPI) -- Tennessee Titans offensive lineman Dennis Kelly set an NFL record with a short touchdown catch in Sunday's AFC Championship Game against the Kansas City Chiefs.
NFL // 3 days ago
NFL playoffs preview: How to watch, betting odds for conference championships
Jan. 17 (UPI) -- Here's how to watch and the betting odds for Sunday's AFC and NFC championship games.
Australian Open: Cori Gauff beats Venus Williams in straight sets

redpajama_set_name: RedPajamaCommonCrawl


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text:
Portland Book Review Home
Reviews (post-April 2020)
Archived Reviews (pre-April 2020)
Book Roundups
The Millionaire and the Mummies: Theodore Davis's Gilded Age in the Valley of the Kings
by Site Owner | Jan 29, 2014 | Archived Reviews (pre-April 2020), Biographies & Memoirs | 0 comments
The Man Behind the Mummies
By John M. Adams
St. Martin's Press, $26.99, 363 pages
If one has seen an Egyptian exhibit in a museum, the person to whom credit and gratitude should be given is Theodore Davis. In the early years of the 20th century, he financed and even oversaw many of the excavations in the Valley of the Kings in Luxor, Egypt. In The Millionaire and the Mummies, John M. Adams describes how a young son of a widowed mother grew into being one of the wealthiest men in the United States, how his curiosity and cunning took him from Michigan to Iowa, to New York and on to Egypt, and how his choices in the Valley of the Kings shaped archaeology.
"Davis's personality was the product of his experience, and he would always retain a bit of a rough-edged, frontier persona. His intellectual curiosity and continual self-education overcame his lack of formal schooling, but although he became an extremely sophisticated individual he was never what eastern society considered polished. He was 'eccentric' and 'brusque,' and known for 'wisecracks and tactless sayings.'"
Adams writes with clarity about Davis's choices and the results of those choices. He intersperses what could be long narratives of scientific digs in Egypt with stories of how Davis came to be the financier that he was in his 70s. The book weaves a fascinating tale of a man whose name is little known outside the Egyptian archaeological world, and weaves it well.
Reviewed by Mary-Lynne Monroe
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Poetry & Short Stories
Spirituality & Inspiration
Copyright 2010-2021 City Book Review


Fluke's T6 electrical testers have taken top honors in the Tools and Instruments category of the AHR 2018 Innovation Awards. The winners were selected by a panel of third-party ASHRAE member judges who evaluated all award entries based on innovative design, creativity, application, value, and market impact.
The Fluke® T6 electrical testers with FieldSense Technology take simultaneous voltage and current measurements without test leads to troubleshoot electrical systems safer, faster, and more efficiently.
FieldSense Technology eliminates the need to make contact with electrical conductors with test leads or alligator clips, significantly reducing the potential for arc flash while saving time and minimizing potential errors.
FieldSense Technology uses an open-fork voltage sensing technology that involves transducing and calculating a known signal to derive measurements for the source voltage. This is done by designing the device to generate a reference signal of known amplitude and frequency. Then, when grounded, the resulting composite waveform is detected by an electronic sensor built into the tester. After amplification, processing, and digital calculations, voltage and frequency measurements are derived.
The Fluke® T6 measures voltage up to 1000V through the open fork without test leads or exposed conductors, simultaneously measures and displays voltage and current, and measures resistance up to 100.0 kΩ. It can also measure 4/0 wires up with current up to 200 A.


The main purpose of this study is to develop an Android-based Event Manager.
To assess the students' satisfaction level towards usage of the app.
School. Since one of the school problems faced by the students is difficulty in time management, the application can be a useful tool to be used by the school and introduce it to students. This could help the school through increasing productivity rate among students aimed at producing high quality pioneers.
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Students. The product of this study may help students monitor of all their tasks and to develop time management to which they can use as their basis to effectively perform and accomplish their tasks without difficulties. This may also assist them to develop time management skills among themselves and to help them to be organized and promote productivity.
Future Researchers. This study may serve as a reference material for the anticipated researchers that wish to develop and conduct similar study.
Develop an Android-based Event Manager.
Run from the latest version up to the earliest version of android.
Keeping daily tasks align, one step to adding task and time and removal of the task after it is done.
Cover all about the time management of the participants.
To determine the satisfaction level of the participants after trial of the app.
To help students to effectively establish time management and improve time management skills in terms of using the app.


text:
Blog | Featured Story
SNAP Matters: Stories Recorded with Support from the Robert Wood Johnson Foundation
In summer 2018, StoryCorps recorded 49 conversations as part of a project focused on changing the the narrative about the Supplemental Nutrition Assistance Program (SNAP), the program formerly known as food stamps. We recorded conversations between close to 100 participants, most of whom were SNAP beneficiaries, in partnership with 15 community-based organizations in Alabama, Texas, Michigan, and Kansas. Made possible with support from the Robert Wood Johnson Foundation, the project also aimed to emphasize the critical support SNAP provides for children and families across the country.
In Alabama, Texas, Michigan, and Kansas, StoryCorps' participants talked about what compelled them to apply for SNAP and how the program has improved their lives. In the stories we share below, you'll hear from participants who received SNAP benefits when they lost a job, required emergency surgery, or needed to purchase expensive, allergy-friendly food. SNAP enabled these folks to cover gaps in income and buy affordable, nutritious food at local markets, grocery stores, and farmers' markets.
Individuals with whom we recorded asserted that SNAP had inspired them to become food justice activists. SNAP beneficiaries described going on to earn multiple degrees and passing forward the benefit as mentors in their own communities. Farmers who accept SNAP spoke about finding fulfillment in serving communities in health crises and food deserts, areas in which it is difficult to buy affordable or good-quality fresh food. SNAP allies described changing their minds about the value of the program when they connected with customers who lived plate to plate.
Below, you'll also hear from participants who shared stories of confrontations — at the grocery store and at the Health department — in which members of their own communities questioned their motives in accessing these benefits. These folks' experiences demonstrate that negative stereotypes long associated with food stamps persist with SNAP. Participants described the difficulty of accepting what is still perceived as a handout rather than a "hand-up."
You can listen to some of the stories we recorded below, and learn more here at the home of the project, the State of Obesity.
Jeremy Huffman and Adam Ingrao
"It's not just about access to food; it's about access to hope."
Fellow veterans and friends talk about transitioning from the military to their roles as farmers and healthy food advocates supporting families who participate in SNAP.
Emily and Tim Brown
"One thing I feel I've learned in life over the long run is that you never know what a day may bring."
Husband and wife Tim and Emily Brown recall how SNAP helped their family get through a difficult time after Tim lost his job.
Jennifer Wells-Marshall and Helen Jones
"I worked every single day of the week, but it wasn't enough."
Dr. Jennifer Wells-Marshall tells her friend and colleague, Helen Jones, about when she received SNAP benefits for a period of time when her daughter was young before going on to get her Ph.D.
Max Gage and Catherine Gage
"It's been huge to actually see firsthand what food does to us and our health, to see how it can turn your life around."
Catherine and her 16-year-old son Max talk about the importance of having access to food and how SNAP helps their family get the nutrition they need.
Kolia Souza and Brian Johnson
"Having those benefits was my way out and my way to make a better life for myself and my daughters."
Kolia Souza reflects on how SNAP enabled her to get back on her feet after leaving an abusive relationship.
Andrika Harmon and Kristi Gay
"I had to turn around to her and explain, look, this is what is going on in my life."
Andrika Harmon talks with Kristi Gay, her nurse home visitor, about how SNAP helps her buy healthy food to support her young family while she is working and finishing college.
Newer Post	Older Post
Stories for Martin Luther King Jr. Day
A collection of conversations in memory of Dr. Martin Luther King Jr., and stories of those who carry the legacy of the civil rights movement with them.
StoryCorps on "60 Minutes"
We hope you'll tune into "60 Minutes" on CBS this Sunday, January 9, 2022. StoryCorps—a nonprofit dedicated to recording, preserving, and sharing the stories of Americans of all backgrounds and beliefs—is honored to be the subject of a feature segment on the premier newsmagazine program.
Introducing Our New CEO: Sandra M. Clark
We are thrilled to announce Sandra M. Clark as StoryCorps' new CEO beginning February 2022. Read up on her distinguished career, and why she is the perfect leader for our mission to honor stories across America.
"My StoryCorps experience" - A letter from Jason Reynolds
If you're following StoryCorps, it's because you understand the importance of stories. You know that they cultivate our wisdom, decrease our fear of the other, and inspire us in every possible way.


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Now KJS is using new online system to process the manuscripts called OJS and it will assists with every stage of the refereed publishing process, from submission through to online publication and indexing.
So you are invited to Register as "Author" and submit your manuscript into the online system by clicking here.
As a continuation of Kuwait Journal of Science and Engineering (KJSE, ISSN: 1024-8684), which has been in publication by the Academic Publication Council of Kuwait University since 1974, we are now publishing Kuwait Journal of Science (KJS) as an independent new Journal catering exclusively to a wide community of professionals in various disciplines of science.
Kuwait Journal of Science (KJS) is indexed and abstracted by major publishing houses such as Chemical Abstract, Science Citation Index, Current contents, Mathematics Abstract, Micribiological Abstracts etc. KJS publishes peer-review articles in various fields of Science including Mathematics, Computer Science, Physics, Statistics, Biology, Chemistry and Earth & Environmental Sciences. In addition, it also aims to bring the results of scientific research carried out under a variety of intellectual traditions and organizations to the attention of specialized scholarly readership. As such, the publisher expects the submission of original manuscripts which contain analysis and solutions about important theoretical, empirical and normative issues.
The Journal's Online system assists with every stage of the refereed publishing process, from submissions through to online publication and indexing. Through its management systems, its finely grained indexing of research, and the context it provides for research, Online System seeks to improve both the scholarly and public quality of refereed research.
The Journal welcomes submission of scholarly work of science from professionals that meet the stipulated criteria with regard to significance, value and scientific excellence. We look forward to receiving your scholarly work for publication in KJS.


text:
Q: The multi-part identifier could not be bound - SET clause I've searched for this answer, but most seem to come from the table being updated rather than the table having the information being taken from.
The first B.DED1 in the SET is the one throwing the multi-part identifier. Yet the one in the WHERE clause works just fine.
UPDATE #output
SET col1 = B.DED1
SELECT *
FROM #output A
 INNER JOIN TableName B
 ON A.Key = B.Key
WHERE B.DED1 = 'col1'

A: UPDATE A
SET col1 = B.DED1
FROM [#output] A
 INNER JOIN TableName B
 ON A.Key = B.Key
WHERE B.DED1 = 'col1'

A: Change your query as following:
UPDATE O SET col1 = B.DED1
FROM [#output] O
INNER JOIN TableName B
ON O.Key = B.Key
WHERE B.DED1 = 'col1'

redpajama_set_name: RedPajamaStackExchange


It's safe to say majority of us have had a nightmare of a holiday. Here's some of our best sent in by you!
A celebration of all mothers and their most popular parental phrases. Join Steph and Jake on Mother's Day for an explicit free family orientated episode!
Bad Dates part 2; We've all had them. Join Steph and Jake on their first podcast episode about their/your bad dates.
Bad Dates part 1; We've all had them. Join Steph and Jake on their first podcast episode about their/your bad dates.


Stoddard County Marriage records transcribed from County Court House records. Genealogy databases, links, resources and more.
These marriages are scattered years donated by a very generous reader: Peggy Wilson.
Transcribed from Court House records.


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Section Agriculture & Land Civil Society & Foreign Aid Economy, Market & Finance Energy & Environment Industry Infrastructure, Communication & Technology Law & Justice Millennium Development Goals (MDGs) Public Finance Public Service Social & Human Development State & Politics Sustainable Development Goals (SDGs) Institution by Type Bagmati Province Bikas Udhyami Central/Federal Gandaki Province International Karnali Province Lumbini Province Metropolitan City Non Government Province Province 1 Province 2 Sub-Metropolitan City Sudurpashchim Province Specific Institution Agribusiness Promotion and Marketing Development Directorate Air Visual Armed Police Force Nepal Asian Development Bank Bikas Udhyami Camp Coordination and Camp Management (CCCM) Center for Global Development Central Bureau of Statistics Central Child Justice Committee Central Investigation Bureau Centre for Investigative Journalism Civil Aviation Authority of Nepal Commission for the Investigation of Abuse of Authority COVID-19 Crisis Management Center (CCMC) Credit Information Bureau of Nepal Crime Investigation Department Department of Agriculture Department of Archaeology Department of Commerce Department of Commerce, Supply & Consumers Protection Department of Co-operatives Department of Cottage and Small Industries Department of Customs Department of Education Department of Electricity Development Department of Environment Department of Food technology and Quality Control Department of Foreign Employment Department of Forest Research and Survey Department of Health Services Department of Hydrology and Meteorology Department of Immigration Department of Industry Department of Information and Broadcasting Department of Livestock Services Department of National Park and Wildlife Conservation Department of Plant Resources Department of Printing Department of Prison Management Department of Roads Department Of Supply Management And Protection Of Consumers Interest Department of Survey Department of Transport Management Department of Water Supply & Sewerage Management Development Initiatives Dharan Sub-Metropolitan City Economic Research Department Education Review Office Election Commission, Nepal Epidemiology and Disease Control Division Federal Parliament Federation of Nepali Journalists Financial Comptroller General Office Financial Comptroller Office of Province Food and Agriculture Organization of the United Nations Food Security Information Network Foreign Aid Coordination Division Friedrich-Ebert-Stiftung Global Partnership for Sustainable Development Google Health Emergency Operating Center Health Emergency Operation Center (HEOC) Heath Effects Institutes High Level Commission for Information Technology House of Representative Human Resource and Administration Department Human Rights Watch Inland Revenue Department Intergovernmental Science Policy Platform on Biodiversity and Ecosystem Services International Centre for Integrated Mountain Development (ICIMOD) International Experts International Food Policy Research Institute International Labour Organization International Monetary Fund International Organization for Migration International Telecommunication Union (ITU) Investment Board Nepal Kalimati Fruits and Vegetable Market Development Kathmandu Metropolitan City Khadadevi Rural Municipality Local Level Structuring Metropolitan Police Office Ministry of Agricultural Development Ministry of Agriculture and Livestock Development Ministry of Agriculture, Land Management and Cooperatives Ministry of Commerce Ministry of Commerce and Supplies Ministry of Communication and Information Technology Ministry of Cooperatives and Poverty Allevation Ministry of Culture, Tourism and Civil Aviation Ministry of Defence Ministry of Economic Affairs and Planning Ministry of Education Ministry of Education, Science and Technology Ministry of Energy, Water Resources and Irrigation Ministry of Federal Affairs and General Administration Ministry of Federal Affairs and Local Development Ministry of Finance Ministry of Foreign Affairs Ministry of Forest and Environment Ministry of General Administration Ministry of Health and Population Ministry of Home Affairs Ministry of Industry Ministry of Industry, Commerce and Supplies Ministry of Industry, Tourism, Forests and Environment Ministry of Information and Communication Ministry of Internal Affairs and Law Ministry of Irrigation Ministry of Labour and Employment Ministry of Labour, Employment and Social Security Ministry of Land Management, Agriculture and Cooperative Ministry of Land Management, Cooperatives and Poverty Alleviation Ministry of Land Reform and Management Ministry of Law, Justice and Parliamentary Affairs Ministry of Local Development Ministry of Peace and Reconstruction Ministry of Physical Infrastructure Development Ministry of Physical Infrastructure & Transport Ministry of Population and Environment Ministry of Science and Technology Ministry of Social Development Ministry of Supply Ministry of Urban Development Ministry of Water Supply and Sanitation Ministry of Women, Children and Senior Citizen Ministry of Women, Children and Social Welfare Ministry of Youth and Sports Narcotics Control Bureau NASA GOV National Assembly, Nepal National Child Rights Council National Foundation For Development of Indigenous Nationalities National Health Research Council National Human Rights Commission National Information Commission National Judicial Academy (NJA), Nepal National Natural Resources And Fiscal Commission National Planning Commission National Reconstruction Authority National Tea and Coffee Development Board National Tuberculosis Center National Vigilance Centre National Women Commission National Youth Council Nepal Administrative Staff College Nepal Disaster Risk Reduction Portal Nepal Electricity Authority Nepal Food Security Monitoring System Nepal Government Nepal in Data Nepal Investment Board Nepal Law Commission Nepal Oil Corporation Nepal Police Nepal Police Hospital Nepal Rastra Bank Nepal Telecommunication Authority Office of Attorney General Office of Auditor General Office of Chief Minister and Council of Ministers Office of the Chief Minister and Council of Ministers Office of the Company Registrar Office of the Investment Board Office of the Prime Minister and Council of Ministers Opensignal Organisation for Economic Co-operation and Development Others Oxfam International PARTNERSHIP IN STATISTICS FOR DEVELOPMENT IN THE 21ST CENTURY (PARIS 21) Penguin Publication Privatisation Cell Province Planning Commission Province Policy and Planning Commission Public Debt Management Department Public Service Commission Royal Statistical Society Save the Children Securities Board of Nepal Social Security Fund Social Welfare Council Springer Supreme Court of Nepal The Economist The Economist Intelligence Unit The International Food Policy Research Institute (IFPRI) The International Work Group for Indigenous Affairs Trade & Export Promotion Centre Transparency International United Nation Development Program United Nations United Nations Capital Development Fund (UNCDF) United Nations Educational,Scientific and Cultural Organization United Nations Environment Programme United Nations International Children's Emergency Fund (UNICEF) UNITED NATIONS OFFICE ON DRUGS AND CRIME United Nations Population Fund United State Agency for International Development (USAID) University Grants Commission (UGC) U.S. Embassy of Nepal Wellcome World Bank World Economic Forum World Food Programme World Health Organization World Hepatitis Alliance World Intellectual Property Organization World Travel and Tourism Council World Wide Fund for Nature Language Awadhi Bhojpuri English English & Nepali Kirati- Kirawa (Bantawa) Limbu Magar Maithili Nepali Newari Spanish Tharu Paschimeli Tharu Purbeli Content Type AUDIO DOCS IMAGE PDF VIDEO XLX ZIP
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आ.व. २०७७/७८ मा विशेष अनुदान मार्फत् प्रदेश र स्थानीय तहबाट कार्यान्वयन गर्न छानिएका आयोजना/कार्यक्रमहरु र विनियोजित बजेटको विवरण
The "आ.व. २०७७/७८ मा विशेष अनुदान मार्फत् प्रदेश र स्थानीय तहबाट कार्यान्वयन गर्न छानिएका आयोजना/कार्यक्रमहरु र विनियोजित बजेटको विवरण" is published by National Planning Commission and can be found in the Publications of the Nepal in Data Portal. The information contained in this publication can be accessed via the Data Tab of the Nepal in Data Portal by selecting the section Public Finance. This publication is in Nepali and is published in pdf format.
राष्ट्रिय योजना आयोग - वार्षिक प्रतिवेदन आ. व. २०७६/०७७
The "राष्ट्रिय योजना आयोग - वार्षिक प्रतिवेदन आ. व. २०७६/०७७" is published by National Planning Commission and can be found in the Publications of the Nepal in Data Portal. The information contained in this publication can be accessed via the Data Tab of the Nepal in Data Portal by selecting the section Social & Human Development. This publication is in Nepali and is published in pdf format.
EXCEL - NATIONAL ECONOMIC CENSUS 2018 WARD PROFILE NO.2
The "EXCEL - NATIONAL ECONOMIC CENSUS 2018 WARD PROFILE NO.2" is published by Central Bureau of Statistics and can be found in the Publications of the Nepal in Data Portal. The information contained in this publication can be accessed via the Data Tab of the Nepal in Data Portal by selecting the section Economy, Market & Finance. This publication is in Nepali and is published in xlsx format.
NATIONAL ECONOMIC CENSUS 2018 WARD PROFILE NO.2
The "NATIONAL ECONOMIC CENSUS 2018 WARD PROFILE NO.2" is published by Central Bureau of Statistics and can be found in the Publications of the Nepal in Data Portal. The information contained in this publication can be accessed via the Data Tab of the Nepal in Data Portal by selecting the section Economy, Market & Finance. This publication is in English and is published in pdf format.
कोभिड - १९ ले अर्थतन्त्रमा पारेको प्रभावसम्बन्धी प्रथम पुनरावृत्ति सर्वेक्षणको प्रतिवेदन
The "कोभिड - १९ ले अर्थतन्त्रमा पारेको प्रभावसम्बन्धी प्रथम पुनरावृत्ति सर्वेक्षणको प्रतिवेदन" is published by Nepal Rastra Bank and can be found in the Publications of the Nepal in Data Portal. The information contained in this publication can be accessed via the Data Tab of the Nepal in Data Portal by selecting the section Economy, Market & Finance. This publication is in Nepali and is published in pdf format.
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Dress appropriately
If you're in Chicago, Omaha, Neb., Minneapolis, Detroit, Cleveland, Seattle or Portland, Ore., you'd better get your snow boots ready. According to Joe Bastardi, chief long-range meteorologist for AccuWeather.com, those cities will receive above-normal winter snowfall this year. Alaska residents, along with western and central Canada, will be shivering through severe cold. "The Canadian winter will be as harsh as last year's was gentle," Bastardi said.
Southern winter drought
The southern tier of the U.S. should expect above-normal temperatures and below-normal precipitation this winter, according to AccuWeather.com. But that's not necessarily a good thing — the dry weather could lead the region into a perilous period similar to the 1950s and 1960s, which led to drought conditions and water shortages. Southern California could also suffer from serious wildfire threats.
80 percent of the heat energy from greenhouse gases heating the Earth has gone straight to the ocean, according to a study published in "Journal of Climate." The deep water is heating up, too. The study found that the deep ocean — the water below 3,300 feet — is taking up about 16 percent of what the upper ocean is absorbing. If that heat went into the atmosphere, it would be warming the Earth at a rate of more than 5 degrees per decade.
Arctic report card
The Arctic region is heating up so much that it's severely affecting the local population and ecosystem, says a team of international scientists as reported in the Arctic Report Card. Greenland is experiencing record-setting high temperatures, which is melting the ice and glaciers. And the 2009-10 summer sea ice cover was the third-lowest since satellite monitoring started in 1979.
Warnings vs. advisories
According to the National Weather Service, a wind chill warning occurs when there is extreme wind chill of -30 degrees or colder and winds are greater than or equal to 10 mph. A wind chill advisory is when there is wind chill between -20 and -30 degrees and winds are greater than or equal to 10 mph.
Varied forecasts
A moderate to strong La Niña will be the main climate factor influencing most of the winter weather in the U.S. If you're in the Pacific Northwest, you should brace yourself for a colder and wetter winter than average, while people in the South and Southeast will have warmer and drier weather than average through February, according to the National Oceanic and Atmospheric Administration.


After we recorded and edited our segment on storyboards, we found out that Bill Plympton had freshly posted his opinion on waterboarding… sorry, storyboarding! He disagreed entirely what I had to say, on every smallest point. Battle of Animated Minds.
You can check out his blog ScribbleJunkies if you need more inspiration.
Just don't forget to come back next Tuesday when we post the next segment on animated film production – making stop motion sets.
This entry was posted in The Work in Progress, Uncategorized and tagged animation, arts, Bill Plympton, bill plymton, film, rocks in my pockets, stop motion, storyboards, video. Bookmark the permalink.
8 Responses to Animated Film Production: Storyboards.
Every man has his vice.
Every woman – vice versa.
I board very little myself.
1) I spent 11 years directing and editing television commercials so I can instinctually visualize what need to do.
2) I find I simplify things better.
3) I rarely follow them when do them anyway.
4) it's not like I have a massive crew that needs to be intimately informed of every single detail.
Yes, Elliot, for a short film or commercial spot that is fair, but to keep all 90 minutes of the work-in-progress feature in your head at all time is sort of difficult, things get fragmented and you can never be sure how they look together as one film. So I sort of see Bill's point, I just can't follow his advise, because following a storyboard makes me so bored I collapse in a weeping pile on top of it.
I guess only you and me are brave enough to improvise a feature film? do you have boards for your feature?
I started out doing so out of a sense of duty but really didn't look at them afterwards.
I'm trying to do the thing with as few shots as possible so it's no so essential.
I've boarded the scenes that others are doing, of course.


I'm resurrecting this blog for posts related to Apex, as using my java.net blog for that purpose would be inappropriate. I'm not sure how often I'll post. When things of general interest crop up while I'm working on the Apex compiler, I suppose.
Today's topic came up as part of a customer support escalation. It's a tricky gotcha with the Apex getMap() function, used for getting all the fields for an sObject type (kind of like reflection for your database objects).
Don't pass the map around between classes.
In my example, I'm getting the AccountId field. This field was added in version 17. Let's say you have two classes, QueryBuilder and DescribeUtil.
Why? Because QueryBuilder is running in a world where the AccountId field doesn't exist yet! If the getMap() call were done in QueryBuilder, it wouldn't be in there.
So, do yourselves a favor and don't pass these maps around from class to class. The version mismatch is an easy thing to miss.


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Pop-Jazz
Najee Radio
Betcha Don't Know
Is It The Way Back (feat. Eric Roberson)
You, Me And Forever
Though he's not quite on the plane of Kenny G, Najee still ranks among the most popular of the post-Grover Washington, Jr. Smooth Jazz sax players. He hit on a commercially successful formula with his 1987 debut Najee's Theme and has stuck closely to it ever since. His recordings are full-scale productions, employing large casts of drum programmers, vocalists, and keyboardists to back up his vapory soprano sax and flute melodies. His 1995 recording Songs From the Key of Life was something of an exception, as it added a little more muscle as well as a few touches of improvisation and some relatively creative string and horn arrangements. In general though, he likes to keep things smooth, polished and predictable, and that's why he's sold so many records over the years.
George Howard
Grover Washington, Jr.


Would anyone have code to validate the banking account details provided by a respondent? At the end of the survey we ask for bank account and BIC-code for giving them an incentive.
The survey is fielded in US, Australia, Canada and several European countries.
The first step would require you to provide the correct banking detail formats for each country and apply a regular expression of some sort to validate the entry. Regular expressions control what is allowed to be entered in a numeric or text field.
There is a few people who are very handy with regular expressions on this forum who could assist if you provide format details.
For the European countries, IBAN is used.
In Australia, SWIFT is used, but I don't have the rules yet.
I'd be really cautious about gathering banking information for a couple reasons. First, financial data like that requires special handling to ensure that it stays safe. With the rampant phishing going on these days, I'd worry about ending up on the 6:00 news for even asking about it, let alone if one of your respondents has their account information stolen (whether your fault or not!). With the number of alternative payment methods available, it seems like sending an Amazon certificate or prepaid Visa card might be a better, safer alternative.
Second, I worry about what it might do to overall confidence in marketing research. If it looks like phishing, it might impact all of our ability to get respondents to take our surveys.
Hi Aaron, thanks for your comments. I had reflected similar concerns about confidentiality, GDPR regulations and phishing internally. But that's the way they like to do it now... I will stimulate to seek for another solution for other surveys!
Also make sure your survey is being fielded on a server with SSL active. Because it's sensitive, people may not reply unless they see the padlock showing the site is secure.


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About TACS
The Tennessee Association of Christian Schools (TACS) is a Christian school organization that exists to serve the Christian schools in Tennessee who align with the association's mission statement and subscribe to the statement of faith.
The parameters set by these guidelines encompass the heart and soul of TACS, and they serve to direct the association's many services and support helps. We focus on school improvement through services such as accreditation, certification, educational publications, teacher conventions and conferences, administrative seminars, student activities, and legislative protection, all of which are fashioned to directly or indirectly improve student performance. These and other services provide member schools with a variety of teacher and student helps that assist schools in achieving their academic and spiritual goals.
No association is better recognized and more respected by the Tennessee legislature and state department of education than TACS. Our association provided the leadership in 1976 to pass TCA 49-50-801, a bill that prohibits the state from imposing curriculum standards and teaching requirements for Christian schools. We worked with the state department of education in 1986 in drafting the guidelines that allow Christian school organizations in Tennessee to serve as a Category II agency.
We have achieved these and other milestones without compromising the association's Biblical distinctives. The efforts and support of a competent office staff, an elected executive committee, and the pastors and principals of our member schools have catapulted our association to the forefront of Christian education. It is because of this team effort that we are, and have been for years, recognized as one of, if not the top state Christian school association in America.
If you can embrace our statement of faith and subscribe to our mission statement, we invite you to seriously and prayerfully consider making application for membership. Call our office today; we stand ready to assist you in any way possible.
Randy Scallions, Executive Director
Purpose of TACS
TACS Staff
Feature Title
Feature content here.


Play Battlefield 3? Great, win some prizes!
TweakTown and Noctua are teaming up for the April Battlefield 3 contest. Noctua, market of premium grade CPU heatsinks, fans and thermal paste will give away three prize packages to the top three point scores spread across the two TweakTown Battlefield 3 game servers.
This contest is a little different than last month's contest that only took place on the Air Maps Server. TweakTown now has the Infantry Server up and running. The combined scores of both servers will be combined to determine the top three players by server score.
This is an amazing prize package donated by Noctua!
The fine print: Winners cannot have won a TweakTown sponsored Battlefield 3 Prize Package in the last 6 months. You will also need to post your in game name in this forum thread. This contest runs throughout April and scores will be calculated April 30th at 11:59 PM. Winners will be contacted by May 5th.


The luxurious Catamaran Ocean Spray offers one of the best performances in Galapagos Cruising. The elegant design includes a huge sundeck, Jacuzzi and spacious cabins, each with private balcony. Your Ocean Spray Galapagos cruise: a truly comfortable first-class vacation.
The speed, comfort, top guides and itinerary will make your Ocean Spray Galapagos cruise a perfect experience. Kayaks are also available for use.
PM: Cruise to Bartholomew island with its famous Pinnacle Rock. Enjouy stunning views from the summit of this xtinct volcano. In the surrounding landscape you can observe penguins, pioneer plants, lava cactus; snorkeling: penguins, reef sharks, rays, and great colorful fish. There is the possibility for snorkeling.
PM: Visit to the Prince Phillip's Steps and enjoy the great views from the cliff-face. The vegetation is mainly Palo Santo and you may spot Red-footed Boobies, Short-eared Lava Owls, Galapagos Swallows, and Galapagos Doves.
AM: Cruise to Espumilla Beach at Santiago Island. On this beach you can observe marine iguanas and Sally Lightfull Crabs that are hunted by herons. While snorkeling and swimming you can discover octopi, moray eel and shark.
Later on you visit Buccaneer Cove, a former pirates' haven and now nesting area for sea turtles. Observe marine birds and while snorkeling tropical fish, turtles, sting-rays.
PM: On your visit to Egas Port you can spot Galapagos Hawks and Lava Lizards. Follow the trail to the coastline and observe varied fauna in tide pools and grottos and sea lions sunbathing. While snorkeling there is the possibility to spot reef sharks, turtles, rays and nice underwater formations.
AM: Cruise to North Seymour Island that was formed by submarine lava and is characterized by arid vegetation. Observe Galapagos Sea Lions, Blue-footed Boobies and Magnificent Frigatebirds as well as land and marine iguanas. While snorkeling you can observe Rays, reef shark, fish, garden eels.
PM: Cruise to Santa Cruz and visit to the Highland of the island. Walk through a landscape with ferns, wide hill an volcanoes and observe the Galapagos Giant Tortoise in its natural habitat. You can also spot mockingbirds, Bahama ducklings, White-cheeked Pintail ducklings, Darwin finches.
AM: Visit one of the largest populations of sea lions on this rocks and coral reef island. You can also observe different shore birds.
PM: Transfer to the airport for your national flight to the continent.
International airfare to and from destination; Galapagos National park entrance fee US$100,- per Person (to be paid in cash at arrival at the airport in Galapagos); Transit Control Card US$20,- per Person (to be paid in cash at the airport in Quito or Guayaquil);	meals not described; cost of medical immunizations as needed; trip cancellation insurance or any other travel insurance; cost of optional activities; excess baggage charges; airport taxes (if any); alcoholic beverages or soft drinks; optional gratuities to your guides; items of personal nature.