Portraits from Samuel Fosso's series "African Spirits" are among the art works in MOMA's "Unfinished Conversations" show.
In 2012, the artist and filmmaker John Akomfrah released a video installation, "The Unfinished Conversation," an art documentary and tribute to the Jamaican-born cultural theorist Stuart Hall, who died two years later, at the age of eighty-two. The Museum of Modern Art is presenting the video as the cornerstone of its sprawling interdisciplinary exhibition "Unfinished Conversations," which incorporates recently acquired works from the museum's collection and represents a diverse array of artists, including the Egyptian installation artist Iman Issa, the American multimedia artist Paul Chan, and the British-Ghanaian painter Lynette Yiadom-Boakye. Hall, who wrote how "museum collections have a deep and consistent symbiotic relationship with the environment of the world as well as the communities that created them," is the critical godfather of the exhibition, an attempt by MOMA to indirectly reckon with its exclusionary past.
MOMA has a history of excluding not only black artists but also women. Louise Lawler's recent photography retrospective interrupts a lineup of male artists and, as Peter Schjeldahl pointed out, offers a subtle yet trenchant institutional critique. (In general, museum photography curation, at MOMA and elsewhere, tends to be much more inclusive than painting and sculpture curation.) Modernists who are both black and women, like Alma Thomas, Faith Ringgold, and Senga Nengudi, have never had retrospectives at the museum. "Unfinished Conversations" is neither a corrective nor a direct response to this history and reputation, but the exhibition makes strides. It includes a great deal of work by artists of varying racial, ethnic, and class backgrounds, and avoids trumpeting the familiar self-congratulatory liberal politics—a tuneless, corporatized gospel of diversity. Instead, the curators have managed to emphasize the work itself over the identities of the artists and, crucially, without using this approach as a justification for excluding work by women and artists of color.
What "Unfinished Conversations" does offer is a complex and interracial politics that neither condemns black radicalism as segregationist nor lionizes it as purely righteous. During a tour of the exhibition, the curators Thomas Lax and Lucy Gallun explained that the exhibition involved an unusual amount of collaboration between multiple departments at the museum, and Lax, who came to MOMA from the Studio Museum in Harlem, spoke passionately of the "power of attending to culture," especially the undervalued and overlooked cultures of marginalized groups, as the intrinsic "lining" of all curatorial work. Both Akomfrah and Hall's understandings of blackness comprise a kind of smooth space, where blackness does not necessarily stand in opposition to whiteness but where the illusion of that opposition—specifically, that black and white people cannot, or should not, stand on equal footing together in the same society—is continuously unravelled. The "Unfinished Conversation" video both insists on the lives of black people and advocates for the futures of black people without prescribing what those lives or futures might be.
Certain artists in the exhibition—Anna Boghiguian, Erik van Lieshout, Andrea Bowers, and Samuel Fosso—express more straightforward political narratives through their work. In a series of large black-and-white portraits, Fosso poses as black political figures like Angela Davis, Malcolm X, and Patrice Lumumba. Bowers's large cardboard-and-marker drawing "A Menace to Liberty" features a towering figure bearing a flag labelled "Patriotism" and standing over a fallen figure with a flag labelled "Liberty." Above the scene, a block-lettered heading reads "Mother Earth." But the rest of the exhibition is less clearly representative. Yiadom-Boakye paints figures that appear to be models who sat for her, but, in fact, they're inventions. Zadie Smith, in her recent profile of Yiadom-Boakye, referencing the artist Chris Ofili's response to the work, points out that the paintings "say little explicitly, but you hear much." Instead, we, the viewers, are meant to fill up the remaining space between the eyes and ourselves.
Kara Walker's three-panel tableau drawing "40 Acres of Mules" offers a more troubling kind of complicity, with the naked bodies of black women exposed and crucified, but also sexually dominant among a tide of mules, white men (including a Klansman and a Confederate flag), and a few black men, too. Ever since her "Sugar Sphinx" went up at the old Domino Sugar Factory, in Williamsburg, Brooklyn, in 2014, critics have discussed Walker's art in terms of the behavior of those who encounter it, and sometimes the work itself gets lost. The exhibition disrupts this preoccupation by situating the art works unevenly among each other, with one installation flowing directly into another. Not only do museumgoers encounter the art—contemplating, gawking, snapping selfies—the art works encounter each other. They are not statuesque but dynamic in open space, never crushed under the weight of viewership.
Ultimately, the exhibition doesn't offer a flatly optimistic integrationist space, but it does ask the very question that Stuart Hall once posed: "How then can the particular and the universal, the claims of both difference and equality, be recognized?" "Multicultural" now has a negative connotation as the forebear of "diversity," signalling a haphazard, quota-filling mixing bowl of identities for appearance's sake. But for Hall the word indicated a commitment to inclusion and openness, to "liv[ing] with difference"—to smooth space. These ideas are born of the specific situation in Britain that Hall encountered as a Jamaican expat, and then as a citizen. "Unfinished Conversations" proposes untidy, unresolved ideas and images as valid forms of expression and engagement, and invites artists of different ethnicities, nationalities, races, and genders to participate in such "conversations" not only from the position of their politicized identities but also as investigators, as artists.
"The Unfinished Conversation," the film, which is shown in a black-walled theatre, on three large screens, the outer two of which are angled slightly inward, is enormously affecting. It acknowledges systemic violence and exclusion before gesturing toward the act of rebuilding, which, according to Hall, requires not only full societal participation but also a kind of humanist idealism in which participants believe a conversation can and should be had in the first place.
The screening room was the last one I walked into after a nearly dizzying survey of thirty-seven art works. When I entered, the video was arriving at its halfway point, and one of the panels showed a procession of white people marching in the streets—of London or of Birmingham, where Akomfrah shot much of the footage, I didn't know. On another panel, a black jazz musician played the upright bass. Later scenes included archival footage from the Vietnam War—a young woman amputee is carried through a hospital, helicopters roar through the sky, a mother desperately holds onto her children. Hall's voice fills the room with a booming precision that isn't so much oratory or academic as it is urgent; his words seemed to transform the dark-walled screening room from passive to active space. At the end, a photograph of Hall appears with the caption "For Stuart Hall, with deep gratitude and respect." Akomfrah's video does not represent a catalogue of black identity, black suffering, or even blackness itself, but it is a triumph of imagination—an effort to use a history of obstacles and obstructions in order to see beyond them. As the video started up again, I moved to a seat at the front and watched two more times, then walked slowly back through the gallery and absentmindedly around the gift shop, stocked with extra-large Robert Rauschenberg exhibition books, before finding my way out of the museum and into the world.
Cassie da Costa is a member of The New Yorker's editorial staff.
Landed in a war zone between creating art and objectifying it, between belonging to the art world and resenting it, she capers in the crossfire.
At a MOMA exhibit of abstract works by women, from 1942 to 1969, the drama is in the solitary struggles against steep odds.


text:
Q: Django EnumField for Python3 and DRF support I need EnumField for my Django models. I have tried django-enumfield. But it's not supporting with Django 1.10. I also tried this django-enumfields. But this is not supported with Django REST Framework.
My use case,
# models.py
role = EnumField(UserRoleEnum, default=UserRoleEnum.STUDENT)

# enums.py
class UserRoleEnum(Enum):
  STUDENT = 0
  TEACHER = 1
  ADMIN = 2

A: Note: After 3 days. No answer. :(
I used a like that,
enums.py
class MyEnum:
    ADMIN = 0
    NORMAL = 1

USER_ENUM_CHOICES = (
    (0, 'ADMIN'),
    (0, 'NORMAL')
)

models.py
class MyModel(models.Model):
    role = models.CharField(choices=MyEnum.USER_ENUM_CHOICES, max_length=1, ...)

Note: For simple case it's not a bad idea.

redpajama_set_name: RedPajamaStackExchange


text:
All healthandbeauty archives from October 30, 2017 - HELLO!
Emma Willis reveals the breast cancer symptoms every woman needs to know
Emma Willis has encouraged women to check their breasts regularly in a new campaign video for Breast Cancer Care. The Voice UK presenter, who is a supporter of the charity, said she hopes to raise awareness of the lesser-known signs of breast cancer as part of the 'Not Just a Lump' campaign. RELATED: How to check your breasts - see our Facebook Live video "Like many women I thought a lump was the only warning sign of breast cancer. Now I've had my eyes opened to all the other things to look out for too – like a change in skin texture
Good Morning Britain's Laura Tobin returns home after giving birth prematurely in September
Good Morning Britain weather presenter Laura Tobin has revealed she has finally been able to take her baby daughter home after prematurely giving birth in September. The new mother, who is on maternity leave, spoke live on GMB on Monday morning, giving Susanna Reid and stand-in anchor Richard Madeley an update on her baby's progress - and revealed that she named her Charlotte. "She is getting on so well," she said. "We are finally home as a family - she was in hospital for quite a while, and she gave me quite a fright when she arrived very early.
Lauren Conrad debuts new short haircut
Lauren Conrad went to the hairdressers over the weekend, and had her ponytail cut off – quite literally. The fashion designer, who is renowned for her long, blonde hair, took to Instagram stories to show a stylist snipping into her ponytail to take the chop. In a second image, Lauren showcased the dramatic results, which shows her hair resting just below her shoulders. STORY: Lauren Conrad shares gorgeous photo of sleeping son Liam Lauren Conrad took a trip to the hairdressers Lauren, who has previously starred in The Hills, is no stranger to experimenting with her appearance, and has been inspiring her
Amanda Seyfried reveals her beauty secret for great hair
Amanda Seyfried has opened up about her beauty regimine, and has revealed that she avoids drying her hair with a hairdryer to keep her tresses in good condition. The actress is known for always looking picture perfect - whether she's posing on the red carpet or heading to a farmer's market. Her long, blonde hair is the envy of women worldwide and, in an interview with Harper's Bazaar magazine, Amanda admitted she keeps her locks healthy by steering clear of electrical hair tools. "I like having clean skin and taking care of my hair," Amanda said. "My secret is to avoid drying
Katie Holmes is glowing as she steps out following hair transformation
Katie Holmes has been treated to a personalised skincare serum which made her skin look more radiant that ever. The Dawson's Creek star had a custom formula created for her by popular beauty brand Kiehls, and she couldn't wait to show off the results. Taking to her Instagram account, the mother-of-one shared a photo of her holding up various serums, while her face looked gorgeous with a glowy, sun-kissed base. She captioned the photo: "I'm so excited to try my new personalized skincare serum! Thank you @kiehls for my glowing custom formula, mixed by me at home #ApothecaryPreparations." Fans were quick
Halloween Instagram Beauty Trends You Need to Try

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
\label{sec:intro}

Assessment of model adequacy is always fundamental in statistics -- this basic realization has given rise to a huge literature on testing goodness of model fit.
However, compared to the classical literature, the Bayesian literature on model adequacy test is much scarce.
A comprehensive overview of the existing approaches is provided in \ctn{Vehtari12}. Two relatively prominent existing formal and general
approaches in this direction are those of \ctn{Gelman96} and \ctn{Bayarri00}. The former relies on posterior predictive $P$-value associated with a discrepancy measure
that is a function of the data as well as the parameters. The latter criticize this approach on account of `double use of the data' and come up with two alternative
$P$-values, demonstrating their advantages over the posterior predictive $P$-value. Indeed, double use of the data prevents the posterior predictive $P$-value to have
uniform distribution on $[0,1]$, while the $P$-values of \ctn{Bayarri00} at least asymptotically has the desired uniform distribution on $[0,1]$.

\ctn{Bhattacharya13} introduced a different approach to Bayesian model assessment in `inverse problems', where the model is built with response
variables and covariates, but unlike `forward problems', the interest is to predict unobserved covariates using the rest of the data, not response variables
from the covariates and the remaining data. The palaeoclimate reconstruction problem provided the necessary motivation, where
`modern data' consisting of multivariate counts of species and observed climate values, and fossil assemblage data on the same species, deposited in lake sediments
over thousands of years, are available. The interest is to reconstruct the past climate values corresponding to the fossil assemblages using the available data.
Here, the species composition is modeled as a function of climate, since variations in climate is responsible for variations in species composition, but not vice versa.
The inverse nature of the problem is evident since the interest lies in prediction of the past climate variables, not the species composition.
Since the past climates are the unobserved (unknown) covariates, it is natural to consider a prior distribution for such unknown quantities.

The motivating example arises in quantitative palaeoclimate reconstruction where `modern data' consisting of multivariate counts of species
are available along with the observed climate values. Also available are fossil assemblages of the same species, but deposited in lake sediments
for past thousands of years. This is the fossil species data. However, the past climates corresponding to the fossil species data are unknown, and it is of interest
to predict the past climates given the modern data and the fossil species data. Roughly, the species composition are regarded as functions
of climate variables, since in general ecological terms, variations in climate drives variations in species, but not vice versa.
However, since the interest lies in prediction of climate variables,  the inverse nature of the problem is clear. The past climates, which must be regarded as random variables,
may also be interpreted as {\it unobserved covariates}. It is thus natural to put a prior probability distribution on the unobserved covariates.

Broadly, the model assessment method of \ctn{Bhattacharya13} is based on the simple idea that the model fits the data if the posterior distribution
of the random variables corresponding to the covariates capture the observed values of the covariates. Now note that the true (observed) values will not be known in reality,
which is why the training data (the modern data in the palaeoclimate problem, for instance) with observed covariates has been considered by \ctn{Bhattacharya13}.
Assuming that the covariates are unobserved, one can predict these values in terms of the posterior distribution of the random quantities standing for the (assumed) missing
covariates. \ctn{Bhattacharya13} demonstrate that it makes more sense to consider leave-one-out cross-validation (LOO-CV) of the covariates
particularly when some of the model parameters
are given improper prior. From the traditional statistical perspective, LOO-CV is also a very natural method in model assessment.
We henceforth concentrate on the LOO-CV approach proposed by \ctn{Bhattacharya13}. Briefly, based on the LOO-CV posteriors of the covariates, some appropriate
`inverse reference distribution' (IRD) is constructed. This IRD can be viewed as a distribution of some appropriate statistic associated with the unobserved covariates.
If the distribution captures the observed statistic associated with the observed covariates, then the model is said to fit the data. Otherwise, the model does not fit the data.
\ctn{Bhattacharya13} provide a Bayesian decision theoretic justification of the key idea and show that the relevant IRD based posterior probability analogue of the
aforementioned $P$-values have the uniform distribution on $[0,1]$.
Furthermore, ample simulation studies and successful applications to several real, palaeoclimate models and data sets reported in \ctn{Bhattacharya13},
\ctn{Bhattacharya06} and \ctn{Sabya13}, vindicate the practicality and usefulness of the IRD approach.

The rest of our paper is structured as follows. The general premise of our inverse regression model, LOO-CV and the IRD approach are described in Section \ref{sec:prelims}.
General consistency issues of the same are discussed in Section \ref{sec:discussion_consistency_IRD}.
We propose an appropriate prior for $\tilde x_i$ and investigate its properties in Section \ref{sec:prior}, and in Section \ref{sec:consistency_loo_cv}
prove consistency of the LOO-CV posteriors under reasonably mild conditions.
Relating consistency of the LOO-CV posteriors, we prove consistency of the IRD approach in Section \ref{sec:consistency_IRD}.
In Section \ref{sec:discussion_results} we provide a discussion on the issues and applicability of our asymptotic theory in various inverse regression contexts
and in Section \ref{sec:simstudy}, we illustrate our asymptotic theory with simulation studies. Finally, we make concluding remarks in Section \ref{sec:conclusion}.

\section{Preliminaries and general setup}
\label{sec:prelims}

We consider experiment with $n$ covariate observations $x_1, x_2, \ldots, x_n $ along with responses
$\{y_{ij} :1\leq i\leq n, 1\leq j\leq m\}$.
In other words, the experiment considered here will allow us to have $m$ samples of responses $\{y_{i1}, y_{i2}, \ldots, y_{im}\}$ against covariate observations
$x_{i}$, for $i=1, 2,\ldots, n$. Both $x_i$ and $y_{ij}$ are allowed to be multidimensional.
In this article, we consider large sample scenario where both $m, n\rightarrow \infty$.

For $i=1,\ldots,n$ and $j=1,\ldots,m$, consider the following general model setup:
conditionally on $x_i$ and $\theta$,
\begin{eqnarray}\label{model}
\begin{aligned}
	& y_{ij} \sim f_{\theta}\left(x_i\right),
\end{aligned}
\end{eqnarray}
independently.
In (\ref{model}), $f_\theta$ is a known distribution depending upon
(a set of) parameters $\theta\in\Theta$, where $\Theta$ is the parameter space, which may be infinite-dimensional.
For the sake of generality, we shall consider $\theta=(\eta,\xi)$, where $\eta$ is a function of the covariates, which we more explicitly denote as $\eta(x)$,
where $x\in\mathcal X$, $\mathcal X$ being the space of covariates. The part $\xi$ of $\eta$ will be assumed to consist of other parameters, such as the unknown
error variance.

\subsection{Examples of the above model setup}
\label{subsec:model_examples}
\begin{itemize}
\item[(i)] $y_{ij}\sim Bernoulli(p_i)$, where $p_i=H\left(\eta(x_i)\right)$, where $H$ is some appropriate link function and $\eta$ is some
function with known or unknown form. For known, suitably parameterized form, the model is parametric. If the form of $\eta$ is unknown,
one may model it by a Gaussian process, assuming adequate smoothness of the function.
\item[(ii)] $y_{ij}\sim Poisson(\lambda_i)$, where $\lambda_i=H\left(\eta(x_i)\right)$, where $H$ is some appropriate link function and $\eta$ is
some function with known (parametric) or unknown (nonparametric) form. Again, in case of unknown form of $\eta$,
the Gaussian process can be used as a suitable model under sufficient smoothness
assumptions.
\item[(iii)] $y_{ij}=\eta(x_i)+\epsilon_{ij}$, where $\eta$ is a parametric or nonparametric function and
$\epsilon_{ij}$ are $iid$ Gaussian errors. In particular, $\eta(x_i)$ may be a linear regression function, that is, $\eta(x_i)=\beta'x_i$, where
$\beta$ is a vector of unknown parameters.
Non-linear forms of $\eta$ are also permitted.
Also, $\eta$ may be a reasonably smooth function of unknown form, modeled by some appropriate Gaussian process.
\end{itemize}

\subsection{The Bayesian inverse LOO-CV setup and the IRD approach}
\label{subsec:loo_cv}

In the Bayesian inverse LOO-CV setup, for $i\geq 1$, we successively leave out $x_i$ from the data set, and attempt to predict the same
using the rest of the dataset, in the form of the posterior $\pi(\tilde x_i|\mathbf X_{n,-i},\mathbf Y_{nm})$, where $\mathbf Y_{nm}=\left\{y_{ij}:i=1,\ldots,n;j=1,\ldots,m\right\}$,
$\mathbf X_n=\{x_i:i=1,\ldots,n\}$ and $\mathbf X_{n,-i}=\mathbf X_n\backslash\{x_i\}$, and $\tilde x_i$ is the random quantity corresponding to the left out $x_i$.

In this article, we are interested in proving that $\pi(\tilde x_i\in U^c_i|\mathbf X_{n,-i},\mathbf Y_{nm})\rightarrow 0$ almost surely as $m,n\rightarrow\infty$, where $U_i$
is any neighborhood of $x_i$. Here, for any set $A$, $A^c$ denotes the complement of $A$.

Note that the $i$-th LOO-CV posterior is given by
\begin{equation}
\pi(\tilde x_i|\mathbf X_{n,-i},\mathbf Y_{nm})=\int_{\Theta}\pi(\tilde x_i|\theta,\mathbf y_i)d\pi(\theta|\mathbf X_{n,-i},\mathbf Y_{nm}).
\label{eq:loo_cv}
\end{equation}

In the IRD approach, we consider the distribution of any suitable statistic $T(\tilde\mathbf X_n)$, where the distribution of
$\tilde\mathbf X_n=\left\{\tilde x_1,\ldots,\tilde x_n\right\}$
is induced by the respective LOO-CV posteriors of the form (\ref{eq:loo_cv}). The distribution of $T(\tilde\mathbf X_n)$ is referred to as the IRD
in \ctn{Bhattacharya13}. Now consider the observed statistic $T(\mathbf X_n)$. In a nutshell, if $T(\mathbf X_n)$ falls within the desired $100(1-\alpha)\%$ ($0<\alpha<1$)
of the IRD, then the model is said to fit the data;
otherwise, the model does not fit the data. Typical examples of $T(\mathbf X_n)$, which turned out to be useful in the palaeoclimate modeling context
are (see \ctn{Sabya13}) are:
\begin{eqnarray}
T_1(\mathbf X_n) &=& \sum_{i=1}^n\frac{(x_i - E_{\pi}(\tilde x_i))^2}{V_{\pi}(\tilde x_i)}
\label{eq:discrepancy1}\\
T_2(\mathbf X_n) &=& \sum_{i=1}^n\frac{|x_i - E_{\pi}(\tilde x_i)|}{\sqrt{V_{\pi}(\tilde x_i)}}
\label{eq:discrepancy2}\\
T_3(\mathbf X_n) &=& x_i
\label{eq:discrepancy3}
\end{eqnarray}
To obtain $T(\tilde\mathbf X_n)$ corresponding to $T(\mathbf X_n)$ above, we only need to replace $x_i$ with $\tilde x_i$ in (\ref{eq:discrepancy1}) -- (\ref{eq:discrepancy3}).
In the above, $E_{\pi}$ and $V_{\pi}$ denote the expectation and the variance, respectively, with respect to the LOO-CV posteriors.
The statistic $T_3(\tilde\mathbf X_n)$ is $\tilde x_i$ itself, so that the posterior of $T_3(\tilde\mathbf X_n)$ is nothing but the $i$-th LOO-CV posterior. Such a statistic can
be important when there is particular interest in $x_i$, for instance, if one suspects outlyingness of $x_i$. An example of such an issue is considered in
\ctn{Bhatta07}.

\section{Discussion regarding consistency of the LOO-CV and the IRD approach}
\label{sec:discussion_consistency_IRD}

The question now arises if the IRD approach is at all consistent. That is, whether by increasing $n$ and $m$, the distribution of $T(\tilde\mathbf x)$ will increasingly
concentrate around $T(\mathbf x)$. A sufficient condition for this to hold is consistency of the $i$-th LOO-CV posterior at $x_i$, for $i\geq 1$.
From (\ref{eq:loo_cv}) it is clear that consistency of $\pi(\theta|\mathbf X_{n,-i},\mathbf Y_{nm})$ at the truth $\theta_0$ is required for this purpose, but even if
$\theta$ in $\pi(\tilde x_i|\theta,\mathbf y_i)$ is replaced with $\theta_0$, consistency of (\ref{eq:loo_cv}) at $x_i$ does not hold for arbitrary priors on $\tilde x_i$,
and for fixed $m\geq 1$. This has been demonstrated in \ctn{Chatterjee17} with the help of a simple Poisson regression with mean $\theta x_i$, where both $\theta$ and $x_i$
are positive quantities. Special priors on $\tilde x_i$ is needed, along with the setup with $m\rightarrow\infty$, to achieve desired consistency of
the LOO-CV posterior of $\tilde x_i$ at $x_i$. In Section \ref{sec:prior} we propose such an appropriate prior form and establish some requisite properties
of the prior and $\pi(\tilde x_i|\theta,\mathbf y_i)$. With such prior and with conditions that ensure consistency of $\pi(\theta|\mathbf X_{n,-i},\mathbf Y_{nm})$ at $\theta_0$,
we establish consistency of the LOO-CV posteriors in Section \ref{sec:consistency_loo_cv}.

Indeed, in the setups that we consider, for any $m\geq 1$,
$\pi(\theta|\mathbf X_n,\mathbf Y_{nm})$ is consistent at the true value $\theta_0$.
That is, for any neighbourhood $V$ of $\theta_0$, for given $m\geq 1$, $\pi(\theta\in V|\mathbf X_n,\mathbf Y_{nm})\rightarrow 1$ almost surely, as $n\rightarrow\infty$.
Assuming complete separable metric space $\Theta$, this is again equivalent to weak convergence of $\pi(\theta|\mathbf X_n,\mathbf Y_{nm})$ to $\delta_{\theta_0}$,
as $n\rightarrow\infty$, for $m\geq 1$, for almost all data sequences (see, for example, \ctn{Ghosh03}, \ctn{Ghosal17}).

In our situations, we assume that the conditions of \ctn{Shalizi09} hold for $m\geq 1$, which would ensure consistency of
$\pi(\theta|\mathbf X_n,\mathbf Y_{nm})$ is consistent at the true value $\theta_0$. The advantages of Shalizi's results include great generality of the model and prior
including dependent setups, and reasonably easy to verify conditions. The results crucially hinge on verification of the asymptotic equipartition property.
In Section \ref{subsec:shalizi_briefing} we provide an overview of the main assumptions and result of Shalizi. The full details of the seven assumptions
($S1$)--($S7$) of Shalizi are provided in the Appendix.
In Section \ref{subsec:weak_conv} we show that Shalizi's result leads to weak convergence of the posterior of $\theta$ to the point mass at $\theta_0$, which will play
an useful role in our proof of consistency of the LOO-CV posteriors.

\subsection{A briefing of Shalizi's approach}
\label{subsec:shalizi_briefing}
Let $\mathbf Y_n=(Y_1,\ldots,Y_n)^T$, and let $f_{\theta}(\mathbf Y_n)$ and $f_{\theta_0}(\mathbf Y_n)$ denote the observed and the true likelihoods respectively, under the given value of the parameter $\theta$
and the true parameter $\theta_0$. We assume that $\theta\in\Theta$, where $\Theta$ is the (often infinite-dimensional) parameter space. However, it is not required to
assume that $\theta_0\in\Theta$, thus allowing misspecification. This is the general situation; however, as already mentioned, we do not consider misspecification
for our purpose.
The key ingredient associated with Shalizi's approach to proving convergence of the posterior distribution of $\theta$ is to show that the
asymptotic equipartition property holds.
To elucidate, let us consider the following likelihood ratio:
\begin{equation*}
R_n(\theta)=\frac{f_{\theta}(\mathbf Y_n)}{f_{\theta_0}(\mathbf Y_n)}.
\end{equation*}
Then, to say that for each $\theta\in\Theta$, the generalized or relative asymptotic equipartition property holds, we mean
\begin{equation}
\underset{n\rightarrow\infty}{\lim}~\frac{1}{n}\log R_n(\theta)=-h(\theta),
\label{eq:equipartition}
\end{equation}
almost surely, where
$h(\theta)$ is the KL-divergence rate given by
\begin{equation}
h(\theta)=\underset{n\rightarrow\infty}{\lim}~\frac{1}{n}E_{\theta_0}\left(\log\frac{f_{\theta_0}(\mathbf Y_n)}{f_{\theta}(\mathbf Y_n)}\right),
\label{eq:S3}
\end{equation}
provided that it exists (possibly being infinite), where $E_{\theta_0}$ denotes expectation with respect to the true model.
Let
\begin{align}
h\left(A\right)&=\underset{\theta\in A}{\mbox{ess~inf}}~h(\theta);\notag\\
J(\theta)&=h(\theta)-h(\Theta);\notag\\
J(A)&=\underset{\theta\in A}{\mbox{ess~inf}}~J(\theta).\notag
\end{align}
Thus, $h(A)$ can be roughly interpreted as the minimum KL-divergence between the postulated and the true model over the set $A$. If $h(\Theta)>0$, this indicates
model misspecification.
For $A\subset\Theta$, $h(A)>h(\Theta)$, so that $J(A)>0$.

As regards the prior, it is required to construct an appropriate sequence of sieves $\mathcal G_n$ such that $\mathcal G_n\rightarrow\Theta$ and $\pi(\mathcal G^c_n)\leq\alpha\exp(-\beta n)$,
for some $\alpha>0$.

With the above notions, verification of (\ref{eq:equipartition}) along with several other technical conditions ensure that for any $A\subseteq\Theta$ such that $\pi(A)>0$,
\begin{equation}
\underset{n\rightarrow\infty}{\lim}~\pi(A|\mathbf Y_n)=0,
\label{eq:post_conv1}
\end{equation}
almost surely, provided that $h(A)>h(\Theta)$.
\begin{comment}
and
\begin{equation}
\underset{n\rightarrow\infty}{\lim}~\frac{1}{n}\log\pi(A|\mathbf Y_n)=-J(A),
\label{eq:post_conv2}
\end{equation}
where $\pi(\cdot|\mathbf Y_n)$ denotes the posterior distribution of $\theta$ given $\mathbf Y_n$.
With respect to (\ref{eq:post_conv1}) note that $h(A)>h(\Theta)$ implies positive KL-divergence in $A$, even if $h(\Theta)=0$. In other words, $A$ is the set in which the postulated model
fails to capture the true model in terms of the KL-divergence. Hence, expectedly, the posterior probability of that set converges to zero. The result (\ref{eq:post_conv2})
asserts that the rate at which the posterior probability of $A$ converges to zero is about $\exp(-nJ(A))$. From the above results it is clear that the posterior concentrates
on sets of the form $N_{\epsilon}=\left\{\theta:h(\theta)\leq h(\Theta)+\epsilon\right\}$, for any $\epsilon>0$.

As regards the rate of posterior convergence, let $N_{\epsilon_n}=\left\{\theta:h(\theta)\leq h(\Theta)+\epsilon_n\right\}$, where $\epsilon_n\rightarrow 0$ such that $n\epsilon_n\rightarrow\infty$.
Then under an additional technical assumption it holds, almost surely, that
\begin{equation}
\underset{n\rightarrow\infty}{\lim}~\pi\left(N_{\epsilon_n}|\mathbf Y_n\right)=1.
\label{eq:conv_rate}
\end{equation}

Moreover, it was shown by Shalizi that the squares of the Hellinger and the total variation distances between the posterior predictive distribution
and the best possible predictive distribution under the truth, are asymptotically almost surely bounded above by $h(\Theta)$ and $4h(\Theta)$, respectively. In other words, if $h(\Theta)=0$,
then this entails very accurate approximation of the true predictive distribution by the posterior predictive distribution.
\end{comment}

\subsection{Weak convergence of Shalizi's result}
\label{subsec:weak_conv}

From (\ref{eq:post_conv1}) it follows that for any $\epsilon>0$,
\begin{equation}
\underset{n\rightarrow\infty}{\lim}~\pi(\mathbb N^c_{\epsilon}|\mathbf Y_n)=0,
\label{eq:cons1}
\end{equation}
where $\mathbb N_{\epsilon}=\left\{\theta:h(\theta)\leq h\left(\Theta\right)+\epsilon\right\}$.
In our case, we shall not consider misspecification, as we are interested in ensuring posterior consistency. Thus, we have $h\left(\Theta\right)=0$ in our context.
Now observe that $h(\theta)$ given by (\ref{eq:S3}) is not a proper KL-divergence between two distributions. Thus the question arises if
(\ref{eq:cons1}) suffices for posterior consistency, and hence weak convergence of the posterior to $\delta_{\theta_0}$. Lemma \ref{lemma:lemma1} below settles this question
in the affirmative.
\begin{lemma}
\label{lemma:lemma1}
Given any neighborhood $U$ of $\theta_0$, the set $\mathbb N_{\epsilon}$ is contained in $U$ for sufficiently small $\epsilon$.
\end{lemma}
\begin{proof}
It is sufficient to prove that $h(\theta)>0$ if and only if $\theta\neq\theta_0$.
Note that
$E_{\theta_0}\left(\log\frac{f_{\theta_0}(\mathbf Y_n)}{f_{\theta}(\mathbf Y_n)}\right)$ is a proper KL-divergence and hence is non-decreasing with $n$ (see \ctn{Erven14}). Hence if
$\theta\neq\theta_0$, then there exists $\varepsilon>0$ such that $E_{\theta_0}\left(\log\frac{f_{\theta_0}(\mathbf Y_n)}{f_{\theta}(\mathbf Y_n)}\right)>\varepsilon$ for all $n\geq 1$.
Hence, $h(\theta)$ given by (\ref{eq:S3}) is larger than $\varepsilon$ if $\theta\neq\theta_0$.
Of course, if $h(\theta)>0$, we must have $\theta\neq\theta_0$, since otherwise, $E_{\theta_0}\left(\log\frac{f_{\theta_0}(\mathbf Y_n)}{f_{\theta}(\mathbf Y_n)}\right)=0$
for all $n$, which would imply $h(\theta)=0$.
This proves the lemma.
\end{proof}

It follows from Lemma \ref{lemma:lemma1} that for any neighborhood $U$ of $\theta_0$, $\pi(U|\mathbf Y_n)\rightarrow 1$, almost surely, as $n\rightarrow\infty$.
Thus, $\pi(\cdot|\mathbf Y_n)\stackrel{w}{\longrightarrow} \delta_{\theta_0}(\cdot)$, almost surely, as $n\rightarrow\infty$, where $``\stackrel{w}{\longrightarrow}"$
denotes weak convergence.

\section{Prior for $\tilde x_i$}
\label{sec:prior}
We consider the following prior for $\tilde x_i$: given $\theta$,
\begin{equation}
	\tilde x_i\sim Uniform\left(B_{im}(\theta)\right),
\label{eq:prior_x}
\end{equation}
where
\begin{equation}
	B_{im}(\theta)=\left(\left\{x:\eta(x)\in \left[\bar y_i-\frac{cs_i}{\sqrt{m}},\bar y_i+\frac{cs_i}{\sqrt{m}}\right]\right\}\right),
\label{eq:set1}
\end{equation}
In (\ref{eq:set1}), $\bar y_i=\frac{1}{m}\sum_{j=1}^my_{ij}$ and $s^2_i=\frac{1}{m-1}\sum_{j=1}^m(y_{ij}-\bar y_i)^2$, and $c\geq 1$ is some constant.
We denote this prior by $\pi(\tilde x_i|\eta)$. Lemma \ref{lemma:prior_continuity} shows that the density or any probability
associated with $\pi(\tilde x_i|\eta)$ is continuous with respect to $\eta$.

\subsection{Illustrations}
\label{subsec:illustrations_prior}
\begin{itemize}
	\item[(i)] $y_{ij}\sim Poisson(\theta x_i)$, where $\theta>0$ and $x_i>0$ for all $i$. Here, under the prior $\pi(\tilde x_i|\theta)$,
		$\tilde x_i$ has uniform distribution on the set
		$B_{im}(\theta)=\left\{x>0:\frac{\bar y_i-\frac{cs_i}{\sqrt{m}}}{\theta}\leq x\leq \frac{\bar y_i+\frac{cs_i}{\sqrt{m}}}{\theta}\right\}$.
	\item[(ii)] $y_{ij}\sim Poisson(\lambda_i)$, where $\lambda_i=\lambda(x_i)$, with $\lambda(x)=H(\eta(x))$. Here $H$ is a known, one-to-one,
		continuously differentiable function and $\eta(\cdot)$ is an unknown function modeled by Gaussian process.
		Here, the prior for $\tilde x_i$ is the uniform distribution on $$B_{im}(\eta)=\left\{x:\eta(x)\in
		H^{-1}\left\{\left[\bar y_i-\frac{cs_i}{\sqrt{m}},\bar y_i+\frac{cs_i}{\sqrt{m}}\right]\right\}\right\}.$$
	\item[(iii)] $y_{ij}\sim Bernoulli(p_i)$, where $p_i=\lambda(x_i)$, with $\lambda(x)=H(\eta(x))$. Here $H$ is a known, increasing,
		continuously differentiable, cumulative distribution function and $\eta(\cdot)$ is an unknown function modeled by some appropriate Gaussian process.
		Here, the prior for $\tilde x_i$ is the uniform distribution on $B_{im}(\eta)=\left\{x:\eta(x)\in
		H^{-1}\left\{\left[\bar y_i-\frac{cs_i}{\sqrt{m}},\bar y_i+\frac{cs_i}{\sqrt{m}}\right]\right\}\right\}$.
	\item[(iv)] $y_{ij}=\eta(x_i)+\epsilon_{ij}$, where $\eta(\cdot)$ is an unknown function modeled by some appropriate Gaussian process,
		and $\epsilon_{ij}$ are $iid$ zero-mean Gaussian noise with variance $\sigma^2$.
		Here, the prior for $\tilde x_i$ is the uniform distribution on $B_{im}(\eta)=\left\{x:\eta(x)\in
		\left[\bar y_i-\frac{cs_i}{\sqrt{m}},\bar y_i+\frac{cs_i}{\sqrt{m}}\right]\right\}$.
\end{itemize}

\subsection{Some properties of the prior}
\label{subsec:prior_properties}
Our proposed prior for $\tilde x_i$ possesses several useful properties necessary for our asymptotic theory. These are formally provided
in the lemmas below.
\begin{lemma}
\label{lemma:prior_continuity}
	The prior density $\pi(\tilde x_i|\eta)$ or any probability associated with $\pi(\tilde x_i|\eta)$ is continuous with respect to $\eta$.
\end{lemma}
\begin{proof}
Let $\left\{\eta_k:k=1,2,\ldots\right\}$ be a sequence of functions such that $\|\eta_k-\eta\|\rightarrow 0$, as $k\rightarrow\infty$,
	where $\|\cdot\|$ denotes the sup norm. It then follows that for any set $A$,
	$$\left\{x:\eta_k(x)\in A\right\}\cap B_{im}(\eta)\rightarrow \left\{x:\eta(x)\in A\right\}\cap B_{im}(\eta),~\mbox{as}~k\rightarrow\infty.$$
Hence, as $k\rightarrow\infty$,
	$$Leb\left(\left\{x:\eta_k(x)\in A\right\}\cap B_{im}(\eta)\right)\rightarrow Leb\left(\left\{x:\eta(x)\in A\right\}\cap B_{im}(\eta)\right),$$
where, for any set $A$, $Leb(A)$ denotes the Lebesgue measure of $A$.
This proves the lemma.
\end{proof}
If the density of $\mathbf y_i$ given $\tilde x_i$ and $\theta$, which we denote by $f(\mathbf y_i|\theta,\tilde x_i)$,
is continuous in $\theta$ and $\Theta$ is bounded then it would follow from
Lemma \ref{lemma:prior_continuity} and the dominated convergence theorem that $\pi(\tilde x_i|\theta,\mathbf y_i)$ and its associated probabilities are also continuous
in $\theta$. Below we formally present the result as Lemma \ref{lemma:post_continuity}.

\begin{lemma}
\label{lemma:post_continuity}
	If $f(\mathbf y_i|\theta,\tilde x_i)$ is continuous in $\theta$ and $\Theta$ is bounded, then the density $\pi(\tilde x_i|\theta,\mathbf y_i)$
	or any probability associated with $\pi(\tilde x_i|\theta,\mathbf y_i)$ is continuous with respect to $\theta$.
\end{lemma}
However, we usually can not assume a compact parameter space. For example, such compactness assumption is invalid for Gaussian process priors for $\theta$. But in most
situations, continuity of the density of $\pi(\tilde x_i|\theta,\mathbf y_i)$ and its associated probabilities with respect to $\theta$ hold even without the compactness
assumption, provided $f(\mathbf y_i|\theta,\tilde x_i)$ is continuous in $\theta$. We thus make the following realistic assumption:
\begin{as}
$\pi(\tilde x_i|\theta,\mathbf y_i)$ is continuous in $\theta$.
\label{as:as1}
\end{as}
The following result holds due to Assumption \ref{as:as1} and Scheffe's theorem (see, for example, \ctn{Schervish95}).
\begin{lemma}
\label{lemma:postprob_continuity}
	If Assumption \ref{as:as1} holds, then any probability associated with $\pi(\tilde x_i|\theta,\mathbf y_i)$ is continuous in $\theta$.
\end{lemma}

\section{Consistency of the LOO-CV posteriors}
\label{sec:consistency_loo_cv}

For consistency of the LOO-CV posteriors given by (\ref{eq:loo_cv}), we first need to ensure weak convergence of $\pi(\theta|\mathbf X_{n,-i},\mathbf Y_{nm})$ almost surely
to $\delta_{\theta_0}$, as $n\rightarrow\infty$, for $m\geq 1$. This holds if and only if $\pi(\theta|\mathbf X_n,\mathbf Y_{nm})$ is consistent at $\theta_0$. This can be seen by
noting that the $i$-th factor of $\log R_n(\theta)$, obtained by integrating out $\tilde x_i$, does not play any role
in by (\ref{eq:equipartition}) and (\ref{eq:S3}), so that these limits remain the same as in the case of $\pi(\theta|\mathbf X_n,\mathbf Y_{nm})$. The other conditions of Shalizi
also remain the same for both the posteriors $\pi(\theta|\mathbf X_n,\mathbf Y_{nm})$ and $\pi(\theta|\mathbf X_{n,-i},\mathbf Y_{nm})$.

Hence, assuming that conditions (S1)--(S7) of Shalizi are verified for $\pi(\theta|\mathbf X_n,\mathbf Y_{nm})$, for fixed $m$, it follows that
$\pi(\cdot|\mathbf X_{n,-i},\mathbf Y_{nm})\stackrel{w}{\longrightarrow}\delta_{\theta_0}(\cdot)$, almost surely, as $n\rightarrow\infty$.

For any neighborhood $U_i$ of $x_i$, note that the probability $\pi(\tilde x_i\in U^c_i|\theta,\mathbf y_i)$ is continuous in $\theta$
due to Lemma \ref{lemma:postprob_continuity}.
Moreover, since it is a probability, it is bounded. Hence, by the Portmanteau theorem, using (\ref{eq:loo_cv}) and consistency of
$\pi\left(\theta|\mathbf X_{n,-i},\mathbf Y_{nm}\right)$ it holds almost surely that
\begin{align}
	\pi(\tilde x_i\in U^c_i|\mathbf X_{n,-i},\mathbf Y_{nm})&=\int_{\Theta}\pi(\tilde x_i\in U^c_i|\theta,\mathbf y_i)d\pi(\theta|\mathbf X_{n,-i},\mathbf Y_{nm})\notag\\
	&\stackrel{a.s.}{\longrightarrow}
	\pi(\tilde x_i\in U^c_i|\theta_0,\mathbf y_i),~\mbox{as}~n\rightarrow\infty,~\mbox{for any}~m\geq 1.
\label{eq:loo_cons1}
\end{align}
We formalize this result as the following theorem.
\begin{theorem}
	\label{theorem:loo_cons1}
	Assume conditions (S1)--(S7) of Shalizi. Then for $i\geq 1$, under the prior (\ref{eq:prior_x}) and
	Assumption \ref{as:as1}, (\ref{eq:loo_cons1}) holds almost surely, for any $m\geq 1$,
	for any neighborhood $U_i$ of $x_i$.
\end{theorem}

Let us now make the following extra assumptions:
\begin{as}
$f(\mathbf y_i|\theta_0,\tilde x_i)$ is continuous in $\tilde x_i$.
\label{as:as2}
\end{as}

\begin{as}
$\eta_0$ is a one-to-one function.
\label{as:as3}
\end{as}

With these assumptions, we have the following result.
\begin{theorem}
\label{theorem:theorem1}
	Under the prior (\ref{eq:prior_x}) and Assumptions \ref{as:as2} and \ref{as:as3}, for any neighborhood $U_i$ of $x_i$, for any $i\geq 1$,
\begin{equation}
	\pi(\tilde x_i\in U^c_i|\theta_0,\mathbf y_i)\stackrel{a.s.}{\longrightarrow}0,~\mbox{as}~m\rightarrow\infty.
	\label{eq:conv2}
\end{equation}
\end{theorem}
\begin{proof}
Note that
\begin{align}
\pi(\tilde x_i\in U^c_i|\theta_0,\mathbf y_i)&
=\frac{\int_{U^c_i}\pi(\tilde x_i|\theta_0)f(\mathbf y_i|\theta_0,\tilde x_i)d\tilde x_i}
	{\int_{U^c_i}\pi(\tilde x_i|\theta_0)f(\mathbf y_i|\theta_0,\tilde x_i)d\tilde x_i+\int_{U_i}\pi(\tilde x_i|\theta_0)f(\mathbf y_i|\theta_0,\tilde x_i)d\tilde x_i}.
	\label{eq:conv3}
\end{align}
Let us consider $\int_{U^c_i}\pi(\tilde x_i|\theta_0)f(\mathbf y_i|\theta_0,\tilde x_i)d\tilde x_i$ of (\ref{eq:conv3}).
Since the support of $\tilde x_i$ is compact, Assumption \ref{as:as2} ensures that $f(\mathbf y_i|\theta_0,\tilde x_i)$ is bounded.
Hence,
\begin{align}
\int_{U^c_i}\pi(\tilde x_i|\theta_0)f(\mathbf y_i|\theta_0,\tilde x_i)d\tilde x_i\leq
K\int_{U^c_i}\pi(\tilde x_i|\theta_0)d\tilde x_i=K\pi(\tilde x_i\in U^c_i|\theta_0),
\label{eq:conv4}
\end{align}
	for some positive constant $K$. Now note that $\pi(\tilde x_i\in U^c_i|\theta_0)=\pi(\tilde x_i\in U^c_i\cap B_{im}(\theta_0)|\theta_0)$, and
	Assumption \ref{as:as3} ensures that $B_{im}(\theta_0)\rightarrow\{x_i\}$ almost surely, as $m\rightarrow\infty$, for all $i\geq 1$.
	It follows that there exists $m_0\geq 1$ such that $U^c_i\cap B_{im}(\theta_0)=\emptyset$, for $m\geq m_0$. Hence,
	$\pi(\tilde x_i\in U^c_i\cap B_{im}(\theta_0)|\theta_0)\rightarrow 0$, as
$m\rightarrow\infty$. This implies, in conjunction with (\ref{eq:conv4}) and (\ref{eq:conv3}), that (\ref{eq:conv2}) holds.

\end{proof}
Combining Theorems \ref{theorem:loo_cons1} and \ref{theorem:theorem1} yields the following main result.
\begin{theorem}
\label{theorem:theorem2}
	Assume conditions (S1)--(S7) of Shalizi. Then with the prior (\ref{eq:prior_x}), under further Assumptions \ref{as:as1} -- \ref{as:as3}, for $i\geq 1$,
	\begin{equation}
		\underset{m\rightarrow\infty}{\lim}\underset{n\rightarrow\infty}{\lim}~\pi(\tilde x_i\in U^c_i|\mathbf X_{n,-i},\mathbf Y_{nm})=0,~\mbox{almost surely},
	\end{equation}
for any neighborhood $U_i$ of $x_i$.
\end{theorem}

\section{Consistency of the IRD approach}
\label{sec:consistency_IRD}
Due to practical usefulness, we consider consistency of IRD associated with (\ref{eq:discrepancy1}) -- (\ref{eq:discrepancy3}). Among these, the IRD associated with $T_3$
is just the $i$-th LOO-CV posterior, which is consistent by Theorem \ref{theorem:theorem2}. For $T_1$ and $T_2$, we consider slight modification by
dividing the right hand sides of (\ref{eq:discrepancy1}) and (\ref{eq:discrepancy2}) by $n$, and adding some small quantity $\varepsilon>0$ to $V_{\pi}(\tilde x_i)$.
These adjustments are not significant for practical applications, but seems to be necessary for our asymptotic theory.
With these, we provide the consistency result and its for the IRD corresponding to $T_1$; that corresponding to $T_2$ would follow in the same way.

\begin{theorem}
\label{theorem:consistency_T1}
Assume conditions (S1)--(S7) of Shalizi, and the prior (\ref{eq:prior_x}). Also let Assumptions \ref{as:as1} -- \ref{as:as3} hold, for $i\geq 1$,
Define for some $\varepsilon>0$, the following:
$$T_1(\tilde\mathbf X_n) =\frac{1}{n} \sum_{i=1}^n\frac{(\tilde x_i - E_{\pi}(\tilde x_i))^2}{V_{\pi}(\tilde x_i)+\varepsilon}$$
and $$T_1(\mathbf X_n) =\frac{1}{n} \sum_{i=1}^n\frac{(x_i - E_{\pi}(\tilde x_i))^2}{V_{\pi}(\tilde x_i)+\varepsilon}.$$
Then
\begin{equation}
	\left|T_1(\tilde\mathbf X_n)-T_1(\mathbf X_n)\right|\stackrel{P}{\longrightarrow}0,~\mbox{as}~m\rightarrow\infty,~n\rightarrow\infty,~\mbox{almost surely}.
	\label{eq:consistency_T1}
\end{equation}
In the above, $``\stackrel{P}{\longrightarrow}"$ denotes convergence in probability.
\end{theorem}
\begin{proof}
The assumptions of this theorem ensures consistency of the LOO-CV posteriors due to Theorem \ref{theorem:theorem2}. This again is equivalent to
almost sure weak convergence of the $i$-th cross-validation posterior to $\delta_{\{x_i\}}$, for $i\geq 1$. This is again equivalent to convergence
in (cross-validation posterior) distribution of $\tilde x_i$, to the degenerate quantity $x_i$, almost surely. Due to degeneracy, this is again equivalent to convergence
in probability, almost surely.

For notational clarity we denote $\tilde x_i$ by $\tilde x^{nm}_i$, whose LOO-CV posterior is $\pi(\cdot|\mathbf X_{n,-i},\mathbf Y_{nm})$.
Let also $\tilde\mathbf X^{nm}=\{\tilde x^{nm}_1,\ldots,\tilde x^{nm}_n\}$, so that we now denote $T_1(\tilde\mathbf X_n)$ by $T_1(\tilde\mathbf X^{nm})$.
It follows from the above arguments that for $i\geq 1$,
\begin{equation}
\tilde x^{nm}_i\stackrel{P}{\longrightarrow}\tilde x_i,~\mbox{almost surely},~\mbox{as}~m\rightarrow\infty,~n\rightarrow\infty.
\label{eq:cons_T1_2}
\end{equation}
Now consider $T_1(\tilde\mathbf X^{nm})-T_1(\mathbf X_n)$, which is an average of $n$ terms, the $i$-th term being
\begin{equation}
	z^{nm}_i=\frac{(\tilde x^{nm}_i - E_{\pi}(\tilde x^{nm}_i))^2-(x_i - E_{\pi}(\tilde x^{nm}_i))^2}{V_{\pi}(\tilde x^{nm}_i)+\varepsilon}.
\label{eq:cons_T1_3}
\end{equation}
Due to bounded support of $\tilde x^{nm}_i$ and (\ref{eq:cons_T1_2}), uniform integrability entails $E_{\pi}(\tilde x_i)\rightarrow x_i$
and $V_{\pi}(\tilde x_i)\rightarrow 0$, almost surely. The latter two results ensure, along with (\ref{eq:cons_T1_2}), that for $i\geq 1$,
\begin{equation}
z^{nm}_i\stackrel{P}{\longrightarrow}0,~\mbox{as}~m\rightarrow\infty,~n\rightarrow\infty,~\mbox{almost surely}.
\label{eq:cons_T1_4}
\end{equation}
Now note that if $z^{nm}_i$ were non-random, then $z^{nm}_i\rightarrow 0$, as $m\rightarrow\infty$, $n\rightarrow\infty$, would imply
$\frac{1}{n}\sum_{i=1}^nz^{nm}_i\rightarrow 0$ as $m\rightarrow\infty$, $n\rightarrow\infty$.
Hence, by Theorem 7.15 of \ctn{Schervish95} (page 398), it follows that
$$T_1(\tilde\mathbf X^{nm})-T_1(\mathbf X_n)\stackrel{P}{\longrightarrow}0,~\mbox{as}~m\rightarrow\infty,~n\rightarrow\infty,~\mbox{almost surely}.$$
In other words, (\ref{eq:consistency_T1})) holds.
\end{proof}

\section{Discussion of the applicability of our asymptotic results in the inverse regression contexts}
\label{sec:discussion_results}
From the development of the asymptotic results it is clear that there are two separate aspects that ensures consistency of the LOO-CV posteriors.
The first is consistency of the posterior of the parameter(s) $\theta$, and then consistency of $\pi(\tilde x_i|\theta,\mathbf y_i)$. Once consistency
of the posterior of $\theta$ is ensured, our prior for $\tilde x_i$ then guarantees consistency of the posterior of $\tilde x_i$ at $x_i$.
For verify consistency of the posterior of $\theta$, we referred to the general conditions of Shalizi because of their wide applicability, including dependent
setups, and relatively easy verifiability of the conditions. Indeed, the seven conditions of Shalizi have been verified in the contexts of general stochastic process
(including Gaussian process) regression (\ctn{Chatterjee1}) with both Gaussian and double exponential errors,
binary and Poisson regression involving general stochastic process (including Gaussian process) and known link functions (\ctn{Chatterjee2})
Moreover, for finite-dimensional parametric problems, the conditions are much simpler to verify. Thus, the examples provided in Section \ref{subsec:illustrations_prior}
are relevant in this context, and the LOO-CV posteriors, and hence the IRD, are consistent. Furthermore, \ctn{Chandra1} and \ctn{Chandra2} establish the
conditions of Shalizi in an autoregressive regression context, even for the so-called ``large $p$, small $n$" paradigm. In such cases, our asymptotic results
for the LOO-CV posteriors and the IRD, will hold.

There is one minor point to touch upon regarding our requirement for ensuring consistency. In all the aforementioned works regarding verification of Shalizi's conditions,
$m=1$ was considered. For our asymptotic theory, we first require consistency of $\theta$ as $n\rightarrow\infty$, for fixed $m\geq 1$, and then take the limit as
$m\rightarrow\infty$. This is of course satisfied if consistency
holds for $m=1$, as for more information about $\theta$ brought in for larger values of $m$, consistency automatically continues to hold.
Indeed, for fixed $m\geq 1$, the limit as $n\rightarrow\infty$ does not depend upon $m$, as the posterior of $\theta$ converges weakly to
the point mass at $\theta_0$, almost surely. Thus, it is always sufficient to verify consistency of the posterior of $\theta$ for $m=1$.

\section{Simulation studies}
\label{sec:simstudy}
\subsection{Poisson parametric regression}
\label{subsec:poisson_parametric}
Let us first consider the case where $y_{ij}\sim Poisson(\theta x_i)$, as briefed in Section \ref{subsec:illustrations_prior} (i). Here we investigate consistency
of the posterior of $\tilde x_i$. We generate the data by simulating $\theta\sim Uniform(0,2)$, $x_i\sim Uniform(0,2)$, $i=1,\ldots,n$, and then by generating
$y_{ij}\sim Poisson(\theta x_i)$, for $j=1,\ldots,m$ and $i=1,\ldots,n$.
We set $\pi(\theta)=1$; $\theta>0$, for the prior for $\theta$.

Since numerical integration turned out to be unstable, we resort to Gibbs sampling from the posterior, noting that the full conditional
distributions of $\theta$ and $\tilde x_i$ are of the forms
\begin{align}
	[\theta|\tilde x_i,\mathbf X_{n,-i},\mathbf Y_{nm}]&\propto\theta^{\sum_{i=1}^n\sum_{j=1}^my_{ij}}\exp\left\{-m\theta\left(\tilde x_i+\sum_{j\neq i}x_j\right)\right\}
	I_{\left[\frac{\max\left\{0,\bar y_i-cs_i/\sqrt{m}\right\}}{\tilde x_i},\frac{\bar y_i+cs_i/\sqrt{m}}{\tilde x_i}\right]}(\theta);\notag\\
	[\tilde x_i|\theta,\mathbf X_{n,-i},\mathbf Y_{nm}]&\propto\tilde x^{m\bar y_i}_i\exp\left(-m\theta \tilde x_i\right)
	I_{\left[\frac{\max\left\{0,\bar y_i-cs_i/\sqrt{m}\right\}}{\theta},\frac{\bar y_i+cs_i/\sqrt{m}}{\theta}\right]}(\tilde x_i).\notag
\end{align}
It follows that $[\theta|\tilde x_i,\mathbf X_{n,-i},\mathbf Y_{nm}]$ has the gamma distribution with shape parameter $\sum_{i=1}^n\sum_{j=1}^my_{ij}+1$
and rate parameter $m\left(\tilde x_i+\sum_{j\neq i}x_j\right)$, truncated on
$\left[\frac{\max\left\{0,\bar y_i-cs_i/\sqrt{m}\right\}}{\tilde x_i},\frac{\bar y_i+cs_i/\sqrt{m}}{\tilde x_i}\right]$.
Similarly, $[\tilde x_i|\theta,\mathbf X_{n,-i},\mathbf Y_{nm}]$ has the gamma distribution with shape parameter $m\bar y_i+1$ and rate parameter $m\tilde x_i$, truncated on
$\left[\frac{\max\left\{0,\bar y_i-cs_i/\sqrt{m}\right\}}{\theta},\frac{\bar y_i+cs_i/\sqrt{m}}{\theta}\right]$.

For our investigation, we set $i=1$. That is, without loss of generality, we address consistency of the posterior of $\tilde x_1$ via simulation study.
As for the choice of $c$, we set $c=20$. This choice ensured
that the full conditional distributions have reasonably large support, for given values of $n$ and $m$. We run our Gibbs sampler for $11000$ iterations, and discard
the first $1000$ iterations as burn-in.

Figure \ref{fig:poisson_parametric} displays the posterior densities of $\tilde x_1$ for different values of $m$ and $n$;
here, for convenience of presentation, we have set $m=n$.
The vertical line denotes the true value $x_1$.
The diagram vividly depicts that the LOO-CV posterior of $\tilde x_1$ concentrates more and more around $x_1$ as $n$ and $m$ increase.
\begin{figure}
\begin{center}
\includegraphics[width=9cm,height=9cm]{simstudy_poisson/consistency_inv_poisson2-crop.pdf}
\end{center}
\caption{Demonstration of posterior consistency in inverse paremetric Poisson regression. The vertical line denotes the true value.}
\label{fig:poisson_parametric}
\end{figure}

\subsection{Poisson nonparametric regression}
\label{subsec:poisson_nonparametric}
We now consider the case where $y_{ij}\sim Poisson(\lambda(x_i))$, where $\lambda(x)=H(\eta(x))$, as briefed in Section \ref{subsec:illustrations_prior} (ii).
In particular, we let $H(\cdot)=\exp(\cdot)$ and $\eta(\cdot)$ be a Gaussian process with mean function $\mu(x)=\alpha+\beta x$ and covariance
$Cov\left(\eta(x_1),\eta(x_2)\right)=\sigma^2\exp\left\{-(x_1-x_2)^2\right\}$, where $\sigma$ is unknown. We assume that the true data-generating distribution
is $y_{ij}\sim Poisson(\lambda(x_i))$, with $\lambda(x)=\exp(\alpha_0+\beta_0(x))$. We generate the data by simulating $\alpha_0\sim Uniform(-1,1)$, $\beta_0\sim Uniform(-1,1)$
and $x_i\sim Uniform(-1,1)$; $i=1,\ldots,n$, and then finally simulating $y_{ij}\sim Poisson(\lambda(x_i))$; $j=1,\ldots,m$, $i=1,\ldots,n$.

For our convenience, we reparameterize $\sigma^2$ as $\exp(\omega)$, where $-\infty<\omega<\infty$.
For the prior on the parameters, we set $\pi\left(\alpha,\beta,\omega\right)=1$, for $-\infty<\alpha,\beta,\omega<\infty$. Now note that
the prior for $\tilde x_i$, which is uniform on
$B_{im}(\eta)=\left\{x:\eta(x)\in H^{-1}\left\{\left[\bar y_i-\frac{c_1s_i}{\sqrt{m}},\bar y_i+\frac{c_2s_i}{\sqrt{m}}\right]\right\}\right\}$, does not have a closed form,
since the form of $\eta(x)$ is unknown. However, if $m$ is large, the interval
$H^{-1}\left\{\left[\bar y_i-\frac{c_1s_i}{\sqrt{m}},\bar y_i+\frac{c_2s_i}{\sqrt{m}}\right]\right\}$ is small, and $\eta(x)$ falling in this small interval
can be reasonably well-approximated by a straight line. Hence, we set $\eta(x)=\mu(x)=\alpha+\beta x$, for $\eta(x)$ falling in this interval.
In our case, it follows that $[\tilde x_i|\eta]\sim Uniform(a,b)$, where $a=\beta^{-1}\left(\log\left(\bar y_i-\frac{c_1s_i}{\sqrt{m}}\right)-\alpha\right)$
and $b=\beta^{-1}\left(\log\left(\bar y_i+\frac{c_2s_i}{\sqrt{m}}\right)-\alpha\right)$.

We set $c_1=1$ and $c_2=100$, for ensuring positive value of $\bar y_i-\frac{c_1s_i}{\sqrt{m}}$ (so that logarithm of this quantity is well-defined)
and a reasonably large support of the prior for $\tilde x_i$. As before, we set $i=1$, for our purpose, thus focussing on posterior consistency of $\tilde x_1$ only.

In this example, both numerical integration and Gibbs sampling are infeasible. Hence, we resort to Transformation based Markov Chain Monte Carlo (TMCMC)
(\ctn{Dutta13}) for simulating from the posterior. In particular, we use the additive transformation and update all the unknowns simultaneously, in a single block.
More specifically, at each iteration $t=1,2,\ldots$, we first generate $\epsilon\sim N(0,1)$, a standard normal variable. Then, letting
$\left(\tilde x^{(t)}_1,\alpha^{(t)},\beta^{(t)},\omega^{(t)},\eta^{(t)}(x_2),\ldots,\eta^{(t)}(x_n)\right)$
denote the values of the unknowns at the $t$-th iteration, at the $(t+1)$-th iteration we set $\tilde x_1=\tilde x^{(t)}_1\pm 0.5\epsilon$,
$\alpha=\alpha^{(t)}\pm 0.5\epsilon$, $\beta=\beta^{(t)}\pm 0.5\epsilon$, $\omega=\omega^{(t)}\pm 0.05\epsilon$, and $\eta(x_k)=\eta^{(t)}(x_k)\pm 0.00005\epsilon$;
$k=2,\ldots,n$. We accept these proposed values with an appropriate acceptance probability (see \ctn{Dutta13} for details), provided the prior conditions are satisfied.
This strategy has yielded reasonable mixing properties of the additive TMCMC algorithm, for all values of $n$ and $m$ chosen.
We run our additive TMCMC algorithm for $11000$ iterations, discarding the first $1000$ iterations as burn-in.

Figure \ref{fig:poisson_nonparametric} shows the posterior densities of $\tilde x_1$ for this nonparametric inverse regression problem for different values of $n$ and $m$.
Again, it is clearly evident that the posterior concentrates more and more around the true value $x_1$, as $n$ and $m$ are increased.
\begin{figure}
\begin{center}
\includegraphics[width=9cm,height=9cm]{simstudy_poisson_gp/consistency_inv_poisson_gp-crop.pdf}
\end{center}
\caption{Demonstration of posterior consistency in inverse nonparemetric Poisson regression. The vertical line denotes the true value.}
\label{fig:poisson_nonparametric}
\end{figure}

\section{Conclusion}
\label{sec:conclusion}
In this paper, we have proposed a prior for $\tilde x_i$ that seems to be natural for ensuring consistency of the LOO-CV posteriors, and hence
of the IRD approach. Crucially, we need $m$ observations corresponding to each $x_i$, and $m$ is taken to infinity for the asymptotic theory.
Note that for $m=1$, or for any finite $m$, consistency of the LOO-CV posterior of $\tilde x_i$ not achievable, even though consistency of the corresponding posterior of $\theta$
is attainable for any $m\geq 1$. This issue sets apart the problem of LOO-CV consistency from the usual parameter consistency.

An interesting issue is that, for forward Bayesian problems, the posterior predictive distribution of the $i$-th response $y_i$ does not tend to point mass at $y_i$,
even if the corresponding posterior of $\theta$ is consistent at $\theta_0$. The reason is that the distribution of $y_i$ given $\theta$ and $x_i$ is specified
as per the modeled likelihood, and does not admit any prior construction as in the inverse setup. Since the modeled response variable is always associated with positive
variability, even under the true model, the posterior predictive distribution of $y_i$ always has positive variance, and hence, can not be consistent at $y_i$.
From this perspective, even in forward problems, it perhaps makes sense to consider the IRD approach for model validation.
Indeed, our simulation studies demonstrate the effectiveness of the IRD approach to model validation compared to the forward approach.

As a final remark, we mention that for our prior on $\tilde x_i$ we required independence among $\{y_{i1},\ldots,y_{im}\}$, for the strong law of large numbers
to hold for $\bar y_i$ and $s^2_i$. However, independence is not strictly necessary, as the ergodic theorem can often be utilized for ensuring limits in the strong sense.

\begin{appendix}

\section*{Appendix}

\section{Preliminaries for ensuring posterior consistency under general setup}
\label{sec:shalizi}

Following \ctn{Shalizi09} we consider a probability space $(\Omega,\mathcal F, P)$,
and a sequence of random variables $y_1,y_2,\ldots$,
taking values in some measurable space $(\Xi,\mathcal Y)$, whose
infinite-dimensional distribution is $P$. Let $\mathbf Y_n=\{y_1,\ldots,y_n\}$. The natural filtration of this process is
$\sigma(\mathbf Y_n)$, the smallest $\sigma$-field with respect to which $\mathbf Y_n$ is measurable.

We denote the distributions of processes adapted to $\sigma(\mathbf Y_n)$
by $F_{\theta}$, where $\theta$ is associated with a measurable
space $(\Theta,\mathcal T)$, and is generally infinite-dimensional.
For the sake of convenience, we assume, as in \ctn{Shalizi09}, that $P$
and all the $F_{\theta}$ are dominated by a common reference measure, with respective
densities $f_{\theta_0}$ and $f_{\theta}$. The usual assumptions that $P\in\Theta$ or even $P$ lies in the support
of the prior on $\Theta$, are not required for Shalizi's result, rendering it very general indeed.

\subsection{Assumptions and theorems of Shalizi}
\label{subsec:assumptions_shalizi}

\begin{itemize}
\item[(S1)] Consider the following likelihood ratio:
\begin{equation*}
R_n(\theta)=\frac{f_{\theta}(\mathbf Y_n)}{f_{\theta_0}(\mathbf Y_n)}.
\end{equation*}
Assume that $R_n(\theta)$ is $\sigma(\mathbf Y_n)\times \mathcal T$-measurable for all $n>0$.
\end{itemize}

\begin{itemize}
\item[(S2)] For every $\theta\in\Theta$, the KL-divergence rate
\begin{equation*}
h(\theta)=\underset{n\rightarrow\infty}{\lim}~\frac{1}{n}E\left(\log\frac{f_{\theta_0}(\mathbf Y_n)}{f_{\theta}(\mathbf Y_n)}\right).
\end{equation*}
exists (possibly being infinite) and is $\mathcal T$-measurable.
\end{itemize}

\begin{itemize}
\item[(S3)] For each $\theta\in\Theta$, the generalized or relative asymptotic equipartition property holds, and so,
almost surely,
\begin{equation*}
\underset{n\rightarrow\infty}{\lim}~\frac{1}{n}\log R_n(\theta)=-h(\theta).
\end{equation*}
\end{itemize}

\begin{itemize}
\item[(S4)]
Let $I=\left\{\theta:h(\theta)=\infty\right\}$.
The prior $\pi$ satisfies $\pi(I)<1$.
\end{itemize}

\begin{itemize}
\item[(S5)] There exists a sequence of sets $\mathcal G_n\rightarrow\Theta$ as $n\rightarrow\infty$
such that:
\begin{enumerate}
\item[(1)]
\begin{equation}
\pi\left(\mathcal G_n\right)\geq 1-\alpha\exp\left(-\beta n\right),~\mbox{for some}~\alpha>0,~\beta>2h(\Theta);
\label{eq:S5_1}
\end{equation}
\item[(2)]The convergence in (S3) is uniform in $\theta$ over $\mathcal G_n\setminus I$.
\item[(3)] $h\left(\mathcal G_n\right)\rightarrow h\left(\Theta\right)$, as $n\rightarrow\infty$.
\end{enumerate}
\end{itemize}
For each measurable $A\subseteq\Theta$, for every $\delta>0$, there exists a random natural number $\tau(A,\delta)$
such that
\begin{equation}
n^{-1}\log\int_{A}R_n(\theta)\pi(\theta)d\theta
\leq \delta+\underset{n\rightarrow\infty}{\lim\sup}~n^{-1}
\log\int_{A}R_n(\theta)\pi(\theta)d\theta,
\label{eq:limsup_2}
\end{equation}
for all $n>\tau(A,\delta)$, provided
$\underset{n\rightarrow\infty}{\lim\sup}~n^{-1}\log\pi\left(\mathbb I_A R_n\right)<\infty$.
Regarding this, the following assumption has been made by Shalizi:
\begin{itemize}
\item[(S6)] The sets $\mathcal G_n$ of (S5) can be chosen such that for every $\delta>0$, the inequality
$n>\tau(\mathcal G_n,\delta)$ holds almost surely for all sufficiently large $n$.
\end{itemize}
\begin{itemize}
\item[(S7)] The sets $\mathcal G_n$ of (S5) and (S6) can be chosen such that for any set $A$ with $\pi(A)>0$,
\begin{equation}
h\left(\mathcal G_n\cap A\right)\rightarrow h\left(A\right),
\label{eq:S7}
\end{equation}
as $n\rightarrow\infty$.
\end{itemize}

\end{appendix}

redpajama_set_name: RedPajamaArXiv


text:
Speeding Ticket & Traffic Violations
Residential & Commercial Real Estate
Robert J. Brown
Ellen B. Fichtelman
Richard E. Jonas
Rebecca J. Pomeroy
Johnathan L. Rhyne Jr.
Martha C. Rhyne
Rebecca J. Yoder
Lincolnton: 704.666.1210
Denver: 704.483.2016
Serving Our Community's Legal Needs Since 1912
Harvey A. Jonas Jr. Memorial
Harvey was born May 12, 1919 in Lincolnton, he was the son of the late Harvey A. Jonas Sr. and Maude Crowell Jonas, and grew up on the family homesite that is now the location of the Charles Jonas Library on West Main Street. He earned his Eagle Scout award while attending Lincolnton High school and graduated from UNC-Chapel Hill in 1940. He earned his law degree at UNC in 1942 before entering World War II as a U.S. Navy Officer.
Mr. Jonas was a charter member of the Lincolnton post of the Veterans of Foreign Wars and was a former commander of the local American Legion Post. A former Master of Lincoln Lodge 137, AF&AM, and a Shriner, he also served as president of the Lincolnton Chamber of Commerce and the Lincolnton Kiwanis Club. A lifelong member and leader of Lincoln's First United Methodist Church, he formerly served as Gastonia district Lay Leader and served for several years as a trustee of the Rock Springs Campground.
He was chairman of the board of First Federal Savings Bank for many years and served on the board for more than 60 years. He also was a former board member of Gordon Crowell Memorial Hospital; Cochrane Furniture Co.; Cronland Warp Roll Co.; and Lincoln Frozen Foods. He served on the local board of First Citizens Bank & Trust Co. and was a charter member and president of the Lincoln County Horseman's Association.
Harvey retired as senior partner from the Jonas Law Firm in 1980 but remained "of counsel" and continued to take care of the legal needs of his longtime clients and friends. He was a former member of the N.C. Bar Counsel and was one of the first lawyers selected for the N.C. Bar Association's General Practice Hall of Fame. He was admitted to practice law before the U.S. Supreme Court in 1965 and was recognized by the NC State Bar for practicing law for 50 years and continued to practice for a total of 76 years. It was his desire to die a practicing member of the NC State Bar and NC Bar Association. He was a devoted family man who loved his country, the Republican Party and his law firm where he practice for a number of years with his father, the late Harvey A. Jonas Sr., his cousin, former U.S. Congressman Charles R. Jonas, Charles' son, Richard E. Jonas and for 21 years with his granddaughter, Rebecca J. Pomeroy.
Practice Areas Menu
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text:
Sun kills off much delayed UltraSPARC IIIi+ chip
Way past its prime
Ashlee Vance
Sun Microsystems has turned to half-truths where its UltraSPARC IIIi+ processor is concerned. The company has cancelled the product, once slated for 2005, in favor of concentrating on the UltraSPARC T1 - aka Niagara - processor line.
The cancellation business is news to us. Fadhi Azhari, a processor marketing manager at Sun, told us earlier this month that the UltraSPARC IIIi+ had seen "no delays." We thought that a stretch at the time, since Sun had originally planned to ship the chip in the later half of 2005. Now, however, we realize how much of a stretch the "no delay" claim really was.
"We canceled it last fiscal year to focus on the ramp (up) of UltraSparc IV+, Niagara and Niagara 2," Sun's server chief John Fowler told CNET today. "The big push in the volume area is Niagara 2."
Apparently you have to ask the right people to get a straight answer.
Last month, we exposed Sun's decision to do little more than perform basic upgrades with the upcoming Sun Fire V215, V245 and V445 servers. Sun will continue to rely on the UltraSPARC IIIi processor in these boxes and add support for PCI Express.
Delays and cancellations are nothing new for Sun's UltraSPARC group. The chip unit, however, has received a boost from the hot-selling UltraSPARC T1 chip and looks to gain ground on the competition when the Rock processor line ships in 2008. ®


text:
Authoring framework for choice fiction. An extraction from **[Muturangi](http://maetl.net/talks/narrative-anxiety)**.

## Disclaimer

This is a very basic, very sketchy draft extraction of some of the components I wrote to support turning my novel manuscript into an interactive choice-driven experience. It doesn't yet support the full range of features of the original code and won't be considered production ready until **Muturangi** is in production.

## License

Copyright (c) 2015-2016 [Editorial Technology](http://editorial.technology). See the `LICENSE` file at the root of this software distribution for more information.

redpajama_set_name: RedPajamaGithub


Bring a dish to share and come enjoy a Christmas Dinner with Neighbors with the Harford Road Missional Community! Please do not bring any dishes containing pistachios or cashews, due to a severe allergy in the home. For more details, email megan@frontporchbaltimore.org.


text:
Pereza ist eine spanische Rock-Band aus Madrid.

Geschichte
Im Jahr 2001 schlossen sich die drei Musiker Leiva, Rubén und Tuli, die bereits in anderen Bands Erfahrung im Musikgeschäft gesammelt hatten, zusammen. Schlagzeuger Tuli hat die Band mittlerweile verlassen, so dass Gitarrist Rubén und Bassist Leiva nur noch ein Duo bilden.

2006 veröffentlichten sie in Zusammenarbeit mit anderen spanischen Musikern wie Amaral und Iván Ferreiro mit Los Amigos de los Animales  eine CD mit Neuaufnahmen ihrer bisherigen Hits.

Diskografie

Studioalben

Weitere Studioalben
 Pereza (2001)
 Algo para cantar (2003)

Singles

Weblinks
 Offizielle Webpräsenz (spanisch)

Quellen

Rockband
Spanische Band
Band (Madrid)

redpajama_set_name: RedPajamaWikipedia


text:
The world's best independent fantasy films and series on a platform owned by and for creators.
Browse Forums Search HOME RENEW CREATE WIKI DISCORD LOGIN WITH PATREON Start Free Trial Sign in
Abracadavers
Chris' mom died in a freak hair salon chair accident. Ever since then, he's been attached to the chair responsible for her death. Gabriel, his closest friend, believes this is more than an obsession and, along with their group of friends, kidnaps him on a trip. On the road-trip everyone discovers, as Chris had previously begun to suspect, that the salon chair gives those close to it superpowers. Chris, on the other hand, comes to understand he's in fact struggling with Obsessive Compulsive Disorder (OCD) and learns, along with his friends, that dealing with super-human abilities is sometimes easier to grapple with than what's going on in your mind. The group trips, stumbles, but ultimately works and grows together to understand their new-found abilities and, more importantly, themselves.
Abracadavers Episode 1: A Kidnapping
Chris's obsession with the hair chair that killed his mother has gone too far and his friends are ready to do something about it.
Abracadavers Episode 2: Putt-Putt of Peculiar Fantasy
Maria is glowing and now something is up with Paul. Things are going off the rails. Chris opens up.
Abracadavers Episode 3: Blow Up
Maria reaches a breaking point. Gabriel reaches a breaking point. Someone is following. Paul gets carried away and the group splits.
Abracadavers Episode 4: Destination
The first step.
Zombie Orpheus Entertainment Arrowstorm Entertainment The Forge Studios Forums Help Terms Privacy Cookies Sign in


is a global marketplace connecting art, design, jewelry dealers to potential buyers online. As such, the company has a complex digital ecosystem that requires an advanced analytical capability. In order to create a data-driven business, the company worked with Cardinal Path, a Google Analytics Certified Partner and Premium Authorized Reseller, in order to collect, process, store, and visualize all their data.
The companies started by discussing the most important key performance indicators (KPIs), which would be used to measure both failures and successes in their efforts. This understanding led to more customized and effective data collection, including features such as Custom Dimensions, User ID and Enhanced Ecommerce.
Using Google Analytics Premium to power their decisions, 1stdibs saw a 47% lift in transactions on paid media campaigns and a 10% gain in overall return on ad spend (ROAS). Plus, newly optimized email strategies led to a 34% increase in email click-through rates.
To learn more, read the full case study.
1stdibs.com is a global marketplace connecting art, design, jewelry dealers to potential buyers online. As such, the company has a complex digital ecosystem that requires an advanced analytical capability. In order to create a data-driven business, the company worked with Cardinal Path, a Google Analytics Certified Partner and Premium Authorized Reseller, in order to collect, process, store, and visualize all their data.


I'm Kelly Peacock, a 22 year-old Brooklyn-based poet and Thought Catalog writing intern.
In May 2018, I graduated Fairleigh Dickinson University with my Bachelor of Arts Degree in Creative Writing and with 3 (!!) poetry awards under my belt.
In the past, I've held the position as a Social Media Intern for The Typewriter Project through The Poetry Society of New York and the NYC Poetry Festival on Governor's Island and in Brooklyn, as well as an Editorial Intern at a literary magazine known as The Literary Review.
As a contributing writer for online publications known as Aura 32, Lucy Out Loud, She Did What She Wanted, and CREER Magazine, I conducted interviews with musicians, YouTubers, photographers, and social media influencers; reviewed albums, artists and concerts of different genres; and wrote advice pieces on self-love, relationships and the law of attraction.
Currently in the works is my poetry collection—so keep your eyes peeled for poems, updates + more! If you're interested in purchasing one of my poems or getting access to poems before anyone else, go here!


23:0Verses 1-12 are about Balaam's first prophecy.
1 Balaam said, 'Build 7 altars here. Kill 7 bulls and 7 male sheep.' 2 Balak did this. Balaam and Balak gave a gift of a bull and a male sheep on each altar.
3 Then Balaam said to Balak, 'Stand next to your burnt-offering, and I will go. Perhaps the LORD will come to meet me. And whatever he shows to me I will tell to you.' So Balaam went to the top of a hill where there were no trees.
5 God told Balaam what he should say to Balak.
6 So Balaam returned to Balak, and Balak was still standing by his burnt-offering. And all the leaders of Moab's people were with him.
8 But I cannot curse these people because the LORD does not curse them. I cannot speak bad words to these people because the LORD does not speak bad words to them.
9 I look at them when I am on the hills. I see people. They are separate from us. They know that God blesses them.
23:12Verses 13-24 are about Balaam's second prophecy.
14 They went to the field of Zophim on the top of the mountain called Pisgah. He built 7 altars. He killed 7 bulls and 7 male sheep. And he offered a bull and a male sheep on each altar.
16 The LORD told Balaam what he should say to Balak.
18 Balaam spoke his message, 'Balak, son of Zippor listen to me.
19 God is not a man. He does not speak words that are not true. If he promises something, it happens.
20 He told me that I must bless the Israelites. He has blessed them, so I must do it too.
27 Balak said to Balaam, 'Let me take you to a different place. Perhaps God will let you curse the Israelites from this place.' 28 Balak took Balaam to the top of the mountain called Peor. From there they could look over the desert. 29 Balaam said, 'Build 7 altars here. Kill 7 bulls and 7 male sheep.' 30 Balak did as Balaam asked him to. He offered a bull and a male sheep on each altar.


text:
package ace

import (
	"bytes"
	"io"
	"strings"
)

// htmlComment represents an HTML comment.
type htmlComment struct {
	elementBase
}

// WriteTo writes data to w.
func (e *htmlComment) WriteTo(w io.Writer) (int64, error) {
	var bf bytes.Buffer

	// Write an open tag.
	bf.WriteString(e.opts.DelimLeft)
	bf.WriteString(preDefinedFuncNameHTML)
	bf.WriteString(space)
	bf.WriteString(doubleQuote)
	bf.WriteString(lt)
	bf.WriteString(exclamation)
	bf.WriteString(doubleQuote)
	bf.WriteString(e.opts.DelimRight)
	bf.WriteString(hyphen)
	bf.WriteString(hyphen)

	// Write the HTML comment
	if len(e.ln.tokens) > 1 {
		bf.WriteString(space)
		bf.WriteString(strings.Join(e.ln.tokens[1:], space))
	}

	// Write the children's HTML.
	if len(e.children) > 0 {
		bf.WriteString(lf)

		if i, err := e.writeChildren(&bf); err != nil {
			return i, err
		}
	} else {
		bf.WriteString(space)

	}

	// Write a close tag.
	bf.WriteString(hyphen)
	bf.WriteString(hyphen)
	bf.WriteString(gt)

	// Write the buffer.
	i, err := w.Write(bf.Bytes())

	return int64(i), err
}

// ContainPlainText returns true.
func (e *htmlComment) ContainPlainText() bool {
	return true
}

// newHTMLComment creates and returns an HTML comment.
func newHTMLComment(ln *line, rslt *result, src *source, parent element, opts *Options) *htmlComment {
	return &htmlComment{
		elementBase: newElementBase(ln, rslt, src, parent, opts),
	}
}


"Demand appears to be cooling. During the third quarter, office demand registered 22.6 million sq. ft. down 16% from the 26.8 million sq. ft. of net absorption posted in the second quarter.
"The report found that the value of commercial property declined 1.2% in September from the previous month. Particularly hard hit were apartments in the West and office property in most states other than California.
"'This is the first full month of data since the acceleration of the liquidity crisis, and we saw a downturn in prices as the economy slowed, the availability of capital decreased and the cost of capital increased,' said Sally Gordon, a senior vice president at Moody's Investors Service.
Over the next year, 'we'll see a net decline in values,' Gordon said.
But prices are coming off a high point that will provide a cushion if the downturn continues.


text:
How to Feed a Fat Feline
Last January, Ingrid Duthie's favorite pants became very uncomfortable. Holiday parties led to extra calories, along with less time at the gym, for the 40-year-old Detroit native. She wasn't surprised she'd put on a few additional pounds, but she was taken aback when she noticed some extra girth on her 6-year-old cat, Felix, too. "I was probably more aware of Felix's belly since I was thinking a lot about my own, but I'd always thought of him as skinny, since he was very thin as a kitten," she remembers.
At Felix's annual checkup the following month, his veterinarian confirmed that Duthie's small-boned feline had added two pounds in the last year -- increasing his weight by more than 20 percent. He prescribed the following treatment: fewer calories, more calorie-burning. Below, veterinarian Trisha Joyce, DVM, of New York City Veterinary Specialists, weighs in on how cats acquire tubby tummies and how you can help to reverse the damage.
Why Cats Get Fat
According to Dr. Joyce, obesity in domestic felines has reached epidemic proportions in the U.S. This is accompanied by health problems all too familiar to Americans: diabetes, arthritis and even premature death. But unlike dogs and humans, underlying health issues usually do not cause obesity in cats, according to Dr. Joyce. "In dogs we'll see hypothyroidism, but it almost never happens in cats," she says, explaining that this condition slows down an individual's metabolism. Hyperthyroidism, which causes just the opposite to occur, is far more common among felines.
So what does make Fluffy fat? Lifestyle. "Overweight cats are that way because of overfeeding and lack of exercise," she says. "Some cats are good at regulating their food intake, but others are not. If too much food is made available to them, they will eat it." Indoor cats generally don't have much stimulation in their environment -- they aren't stalking, chasing, jumping. Indoor cats are safe from trauma, which is obviously critical to their well-being, but they also sit a lot.
How Cats Get Thin
"Weight loss diets have the most successful outcomes," says Dr. Joyce. This includes both a commercial weight loss formula and portion control. "I recommend wet food, but only because it seems easier for owners to regulate how much they are feeding. Also, it's eaten in one shot, so if there are two cats in the household, there is less chance for the fatter cat to eat the skinnier cat's portion." Dr. Joyce also recommends strictly limiting treats and eliminating table scraps altogether. A grain-free food containing the amino acid L-carnitine may also be beneficial, as it helps to burn fat while maintaining lean muscle mass.
Though a gym membership may not be in your kitty's future, exercise should be on the menu. But be prepared. Getting Fluffy moving may take a dedicated and creative owner. "See what your cat responds to," Dr. Joyce says. "Some enjoy chasing a laser light, which you can operate from the couch. You can put a cat on a harness and an extendable leash and let it run around an enclosed yard with you. Make your cat work for its food -- take the bowl and ask it to follow you around the house to get it." If adopting a kitten is an option, a younger companion's eagerness to play and chase may also get your adult cat off the couch.
Enlisting Professional Help
Unlike dogs and humans, cats cannot tolerate severe calorie restriction. Consult your veterinarian about acceptable portion sizes for weight loss in order to avoid fatty liver syndrome, a serious condition affecting the liver that results when cats do not consume enough calories. Adds Dr. Joyce, "Fatty liver syndrome is also something to be aware of if you introduce a new cat into the home. The first cat may go on a hunger strike, to act out, which can seriously endanger its liver functioning." She suggests monitoring the first cat's food intake closely while the felines become acquainted.
by Darcy Lockman
cat food cat nutrition activities and play


text:
Pale Male with eyass, Red Tailed Hawk of Fifth Ave, NY, NY.
Pale Male moved to New York as a juvenile hawk in the early 1990s. He has been widely successful as a predator, siring many offspring and maintaining a hunting grounds in Central Park. He lives at 927 Fifth Ave, with his 8th mate, Octavia.
Reports are that Pale Male idolizes Horatio Alger and is a strong supporter of Reaganomics.
Juvenile Brown Pelican at the Salton Sea in Southern California
This pelican stands on a beach made of fish bones, enjoying one of the few wet places in the Mojave Desert. The Salton Sea is an important stop for birds of many species as they migrate for winter.
Several hundred years ago, a giant ancient sea called Lake Cahuilla existed in the area of the Salton Sink in Southern California. Before European explorers reached Lake Cahuilla, it evaporated due to natural changes in the weather. In the early 1900s, dam engineers miscalculated and water diverted from the Colorado River intended for Los Angeles residents and surrounding farmland flowed into the Salton Sink, filling the ancient sink hole where Lake Cahuilla had been. During the 1950s, the Salton Sea became a popular tourist spot for weekend vacationers. Fish species were introduced, and children came to play at the crowded Bombay Beach.
However, the Salton Sea does not have sufficient water to sustain itself. Water is becoming saltier and more polluted. Many of the introduced fish species have died off, and their bones fill the shorelines, appearing like sand from a distance.
Yellow Sweet Clover, Melilotus officinalis, a common invasive species in North America. A native species of Eurasia


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Want to find out HOW and WHERE you can catch Lake Erie's giant rainbow trout (steelhead) during the WINTER AND SPRING in local tributary streams and rivers? Interested in learning about steelhead biology? Want to learn more about spinning and fly fishing techniques for steelhead? Want to know if steelhead are safe to consumer.
Learn all of this and more during Ohio Sea Grant's Steelhead Trout Fishing Seminar to be held in Bay Village at the Lake Erie Nature & Science Center, 28728 Wolf Road. Scheduled for Wednesday, February 23, 2011 from 7 P.M. – 9:30 P.M., the seminar is co- sponsored by the Ohio State University's Sea Grant College Program and The Lake Erie Nature & Science Center.
Pre-registration and payment of $5.00 per person (to help support the Nature and Science Center) will be necessary to guarantee seating.
Seating will be limited, and registration at the door will be accepted only if seating is available. Call the Nature and Science Center at 440/871-2900 for more information and details regarding registration and pre-payment for this seminar.
The seminar will feature two locally renowned steelhead fishing experts; Craig Lewis, professional steelhead guide and owner of Erie Outfitters Tackle Shop in Sheffield Lake, and North Olmstead's Jeff Liskay, professional steelhead guide and regionally known steelhead angling seminar speaker.
Jeff Liskay will reveal his secrets regarding where, when and how to catch steelhead in local Ohio Lake Erie tributary streams using fly fishing equipment.
Dave Kelch, Sea Grant Extension Specialist with the Ohio State University's Ohio Sea Grant College Program, will cover basic steelhead biology and provide and update on the state's successful steelhead stocking program.
The boys from Erie Outfitters after another great day of ice fishing!


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Briefly noted
The (Anglican) Church of England's main legislative body said February 10 in London that it recognizes and affirms the desire of the breakaway Anglican Church of North America to remain in the Anglican fold. But the General Synod simultaneously said that it was not ready yet to be in full communion with the conservative group. American Anglican Council President David Ander son, a former Episcopal priest, told Episcopal News Service he was pleased with the outcome, despite the lack of full acceptance. A representative of the Episcopal Church's presiding bishop, Katharine Jefferts Schori, said that the U.S. church does not interfere in the decision-making of other Anglican provinces but that Episcopal leaders continue to object to the fact that some foreign provinces have "actively interfered in the affairs of the Episcopal Church."
The general secretary of the World Council of Churches, Olav Fykse Tveit, has written to the finance ministers of the Group of Seven leading industrialized nations to thank them for pledging to write off Haiti's outstanding debts to G7 countries. At the same time, Tveit, a Norwegian Lutheran, warned that the approval by the International Monetary Fund of more loans to Haiti after its earthquake in January would cause extra problems for the stricken country. "The decision of the IMF approving more loans to Haiti after the earthquake will only add to Haiti's burden, nearly doubling the country's debt to that institution, as there is no clear willingness or definitive moves yet to cancel the country's current debt," Tveit wrote.
The top executive of the National Council of Churches is challenging investment giant Goldman Sachs to use half of its $20 billion bonus pool to help rebuild Haiti after its devastating earthquake. Haiti's entire gross domestic product (the basic measure of a country's overall economic output) is $8.5 billion, which is less than half of Goldman Sachs's bonus pool. The government's bailout in 2009 left Wall Street's biggest name with a much-discussed amount of money. Last September, Goldman Sachs CEO Lloyd Blankfein said, "Compensation continues to generate controversy and anger. In many respects, much of it is understandable and appropriate." In light of the dire need in Haiti, Michael Kinnamon, the NCC general secretary, and George Hunsinger, professor of systematic theology at Princeton Theological Seminary, urged Goldman Sachs to donate half of its bonuses to Haitian relief.
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Hopes rise in Haiti for new Episcopal cathedral
by Gary G. Yerkey
Forgive Haitian debt, religious leaders urge: Grants, not loans, for rebuilding
Global faith-based aid alliance says a billion hungry wrong
by Stephen Brown
ELCA shifts $500,000 for cholera fight in Haiti


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