text:
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Black Health
The Duchess & Thee Stallion: A Tale of Two Megs and their fight to be Believed
This is a story about two women sharing the same name, both beautiful, Black central figures in current conversations about self-advocacy and justice; and both fighting to be seen, heard, and believed in courts of law and public opinion.
I write this from the desk in my home office, fresh after viewing the final episodes of the new Netflix series Harry and Meghan, a deeply personal account of the royal couple's unprecedented journey, presented for the first time in their own words. Penning this post will undoubtedly delay my attendance at a family Christmas party, but that will have to wait, as there is a much more important conversation to be had with my community while sitting in these feelings at this exact time. As I type, bouldered emotions sit heavy in my spirit, like an elephant resting on a human chest, with my mind drawing the parallels between the two Megs - The Duchess and The Stallion - whose names have led countless headlines over the past few years. While I don't know either woman personally, I admire each for their transparency in the face of public atrocities committed against their character. I may not know them, but I know what it's like to be a Black woman in a world that seeks to write its own version of her story. And while my challenges pale in comparison to theirs playing out on social media and the public stage, it is through the common thread of their resilience that I have developed a deep sense of honor and respect. It's with all of this in mind, that I beg my community to remember that the rallying cry to protect Black women, also applies to these two.
Meghan Markle, Duchess of Sussex
Meghan Markle married Prince Harry of the United Kingdom on May 19, 2018, becoming the first woman of color in modern history to marry into the British Monarchy. Her marriage into House Windsor carried immense symbolism for the country and the family alike, leading many Black Brits and people across the diaspora to hope that this union signaled a turning point for the royals. Largely standing as a symbol of Old World opulence and tradition, the last hundred years has seen the monarchy often struggle to find its footing during challenges to modernize and keep pace with the ever-changing contemporary world. As Britain's longest reigning monarch, Queen Elizabeth II had steered her house through a number of scandals laced with numerous taboos of her day, including divorce and death. Her House was one of many in Old Europe known for its great lengths to preserve the "purity" (i.e., whiteness) of the bloodline, but finally, in 2018, the world saw a young, independent, successful, divorced, biracial, American woman marrying a British Prince, thus breaking many of the social barriers that had dictated, or dissolved, past royal pairings. Finally, here was a turning point and a symbol of change...or so we thought.
Almost immediately after their engagement and nuptials captivated millions around the globe, Meghan Markle, the new Duchess of Sussex found herself at the forefront of the British tabloids, with growing unpopularity. Despite the similarities drawn between her and the late Princess Diana, mother of Prince Harry and often called "The People's Princess" for her demeanor and humanitarian efforts, Meghan soon became the subject of a relentless slander campaign driven by the British media. Using every thinkable trope of racial bias against Black women, she was painted by tabloid writers as a bully to her in-laws, domineering spouse to her husband, and a disagreeable, uncivilized woman from the American ghetto who wasn't worthy of sharing the same space as royalty. As is custom, negativity sold papers and amassed millions of clicks online, making the falsehoods spread like wildfire. Meghan and Prince Harry discuss the full experience and aftermath of these attacks in their independently produced Netflix series, so I won't spoil the details here; but ultimately, the toll of racism and bullying from the media and within the royal family compromised Meghan's health, safety, and drove her toward the temptation to take her own life. Seeing the parallels between her and his mother's experiences, Prince Harry, the Duke of Sussex, took unprecedented measures to intervene for the protection of his wife and children.
Thanks to his advocacy, their collective sacrifice in taking a "step back" from their roles as senior royals, significant protective assistance from other Black stars like Tyler Perry, and now a legal win against The Daily Mail for invasion of privacy and copyright infringement, Meghan Markle leads a much more peaceful life. But it was hard fought through years of turmoil that almost broke her.
Megan Pete, known by her stage moniker "Megan Thee Stallion," has been a staple in headlines since she skyrocketed into mainstream hip-hop success in 2018. With songs like "Big Ole Freak," "Plan B," and "WAP" promoting female sexuality, sex-positivity, and mental health, the raptress has walked through the doors opened by her predecessors and continued thundering down virgin paths as she, and peers like Cardi B. and City Girls, continues to blaze a new trail. She has been both praised and condemned for her raunchy lyrics, even though sexually explicit lyrics are far from atypical in Hip-Hop. When a man chants detailed instructions for how a woman should "Slob on [his] Knob," or swipes a credit card down a woman's butt crack in a music video, there's no issue; but when Thee Stallion says that she "ain't lying about [her] nut just to make a n**** happy," or "if it don't hang, then he can't bang," while grinding in her own music video, it's a problem. The gender-based double standard around sexuality in Hip-Hop is as old as the genre itself, and unfortunately has influenced the undertone of discussions about Megan in the recent months leading to the trial of her alleged shooter, Tory Lanez.
To paraphrase the story, in August 2020, Megan alleged that Lanez shot her in the feet in after an altercation that began at a party they both attended in July. When the story originally broke about Lanez's arrest, it was reported that Megan had sustained injuries after stepping on glass. Later, however, she stated that she had undergone foot surgery after sustaining gunshot wounds to her feet and identified Lanez as the shooter. After two years of disputing claims between the two in music, on social media, and in interviews, Lanez trial finally began last week. Many people, especially Black women, have come to Megan's defense in the time since the allegations were initially made, but social media commentary has suggested that men have been more hesitant to believe her claims, with some outright dismissing them. Those who don't believe Megan often defend their doubt with arguments rooted in the criticism of the sex her music and persona, which are then used to discount her character. It's slut shaming in its ugliest form. Megan the Stallion has been open about her battles with mental health in the past, and when she took the stand to testify last week, she described how this situation has added to those challenges, admitting, "I don't feel like I want to be on this earth. I wish he would have just shot and killed me, if I knew I would have to go through this torture."
Here's the point that we should all take from the stories of Meghan Markle and Megan Thee Stallion:
We must stop condemning Black women who come forward with the truth of their traumatic experiences. It is unjust for us to continue gaslighting those who come forward with their traumas, while shielding the male forces hurting them, from accountability.
Some of the same social media intellectuals wondering why people like DJ tWitch silently succumb to their demons, turn around and condemn women like The Duchess and Thee Stallion for being honest about their struggles. Their trauma is discussed as though they somehow deserve it; for marrying a white royal, or for being just as explicit as the countless men screwing women in every position imaginable with their 16 bars of fame. If they aren't told that they deserved their pain, then these women are bombarded with reasons why they don't deserve our protection. Some said that Meghan Markle didn't deserve our defense because she wasn't Black enough, and Megan Thee Stallion doesn't deserve it because she twerks in leather costumes while rapping about "wet ass p***y."
No Black woman deserves the bad things that happen to her; not now, not ever. We can't exclaim that Black Lives Matter while invalidating Black women so terribly that they would rather be lifeless. All Black women are worthy of being seen, respected, protected, and believed.
#WeLoveYouMeghan #IStandWithMegan #BelieveBlackWomen #StayKultured
'22 Lessons
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Attorneys general from Connecticut, New Jersey and Delaware sent letters on October 10, 2008, to 11 companies (Avent America Inc., Disney First Years, Gerber, Handicraft Co., Playtex Products Inc., Evenflo Co., Abbott, Mead Johnson, PBM Products, Nature's One, and Wyeth) that make baby bottles and baby formula containers, asking that they no longer use the chemical bisphenol A (BPA) in their manufacturing because of the growing body of evidence that it is potentially harmful to infants.
Animal studies have linked BPA with breast, prostate and reproductive system abnormalities and some cancers. However, the Food & Drug Administration has tentatively concluded that BPA is safe for humans.


Planning authority performance data is a statistical collection undertaken to establish the number of planning applications determined by Scottish planning authorities, and their performance in processing them. The Scottish Government Communities Analysis Division collects quarterly data from all 32 Local Authorities and the two Planning Authorities (Cairngorms National Park and Loch Lomond & The Trossachs National Park) on the detail of planning decisions and timescales.
On 3rd August 2009 substantial changes to the statutory development management system, relating to the handling of planning applications, came into effect across the whole of Scotland. A few legacy cases that were validated pre 3rd August 2009 can badly skew results and therefore the analysis in Sections 2 to 6 covers applications validated in the current planning system post 3rd August 2009. For applications decided prior to quarter one 2017/18, where appropriate, figures are provided for both post August 2009 applications and all application including legacy cases. Analysis of the limited number of decided legacy cases is provided separately in Section 10.
Additional detailed excel tables of results as well as a copy of this summary, along with results for previous quarters and annual results for previous years are available in the Planning Authority Performance Statistics Section on the Planning Publications page of the Scottish Government's website at: http://www.gov.scot/Topics/Statistics/Browse/Planning/Publications.
For the purpose of planning applications, developments are put into one of three categories: local, major or national. The different types allow councils to treat developments in a way which is suited to their size, complexity and the issues they are likely to raise.
Local developments include applications for changes to individual houses and smaller developments for new housing as well as applications covering areas of development such as minerals, business & industry, waste management, electricity generation, freshwater fish farming, marine finfish farming, marine shellfish farming, telecommunications, Approval of Matters Specified in Conditions ( AMSCs  ) and other developments. Most applications for planning permission will be for local developments.
Major developments include applications for 50 or more homes, as well as certain waste, water, transport and energy-related developments, larger retail developments, and other types of major developments. Classification between local and major developments depends on the particular development type. For example, housing developments are classed as major when the application is for 50 or more dwellings or for a site that exceeds two hectares, whereas electricity generation is classed as major when the capacity of the generating station is or exceeds 20 megawatts. Typically there are only a small number of decisions made for major developments each quarter and therefore average decision times are likely to be volatile.
The sub category "other developments", used for both major and local developments, includes any developments not falling wholly within any of the specific categories of development for minerals, housing, business & industry, waste management, electricity generation, fish farming and AMSCs. It includes, but is not limited to, retail, leisure and entertainment, education, healthcare, community facilities, transport interchanges, water and drainage developments and projects. It can also include mixed use projects.
National developments are mainly large public works (for example, the regeneration of the former Ravenscraig steelworks and the redevelopment of the Dundee Waterfront) and are identified in the National Planning Framework ( http://www.gov.scot/Resource/0045/00453683.pdf).
National Developments are not included in the planning performance statistics analysed in this publication.
Details for the classification of all development types can be found here: http://www.gov.scot/Resource/Doc/278390/0083657.pdf.
The average decision time in weeks is calculated in days from the date of validation  to the date the decision is issued. The average weeks are then calculated by dividing the number of days by seven.
For further explanation of planning performance methodology please refer to Planning Performance Technical Notes.
In some cases it is appropriate to remove a length of time from the total decision time. We have called this process of removing a specific length of time "stopping the clock". Stopping the clock is used where there has been a lengthy delay caused by the applicant or external consultees, outwith the planning authority's control that has halted progress with an application. It is for individual authorities to decide what they consider 'lengthy' on a case by case basis. This should not be used for every application; it is about recording the data in a way which produces more accurate and relevant performance statistics that would otherwise have been skewed by extreme cases of delay outwith the planning authority's control.
Details of clock stops can be seen in Section 9.
The revisions policy for planning performance statistics has been developed in accordance with the UK Statistics Authority Code of practice for Official Statistics and further details are available at: http://www.gov.scot/Topics/Statistics/Browse/Planning/Revisions.
In this publication revisions have been made to the 2015/16 Quarter 4 All Applications figures detailed in Charts 6, 24, 26, 30 and 34 as a result of further review of the collected data.


You've come to the right place for answers to your most commonly asked questions about the Space in Brighton, MI. We want to ensure you can come to the Space as you are, get exactly what you need, and leave feeling better! No question is too silly for our team, so if you can't find the information you're looking for below please contact us.
We're located at the Green Oak Village Place in Brighton, MI. Our space is located directly next to Fraser Bicycle and near Buffalo Wild Wings. We're close to US-23, I-96, and other great amenities for your convenience!
Parking is available at the front and back of the Space, but you are free to park at any location at Green Oak Village.
You can experience infrared heat in our yoga room, during our heated classes, and in our sauna. There are a few reasons why infrared heating is better than traditional forced air heat. Infrared heating can increase metabolism and collagen production. It can also help support the natural detoxification process in your body, help the body age slower on a cellular level, and help soothe sore muscles. Lastly, infrared heating can encourage healthy cognitive function, cardiovascular health, and immune function.
Each class we offer is unique in style, and in experience level. On our classes page, you'll find a full description of what you can expect in class and a difficulty level rating. the Calm, the Dance, the Fusion, and the Contrast are ideal for all levels and are moderate in difficulty, while the Release, the Strength, and the Flow are great for those with experience and are challenging. Our team will provide those in the class a 'template'. Each member in class will be free to move their body as they please and 'freelance' to their needs within the workout. Lots of hands-on instruction is available if you need it. If you're curious about what the instructor will be like, please see our Staff page for more information on our team!
the Flow, the Fusion, and the Release classes are heated via our infrared ceiling as well as forced air. You'll discover this way of heating won't dry the air or reduce oxygen in the Space, provides the benefits of natural sunlight without the dangers of UV rays, heats directly, and promotes healthy blood flow. All other classes are non-heated and are done in room temperature – keep in mind, you will be warm from the movement of your body!
We have a purified water station that you can use at anytime! We recommend you bring a water bottle to class that you can fill up as needed.
HOW DO I RESERVE A SPOT OR CANCEL A RESERVATION? DO I NEED TO?
All reservations and cancellations can be done through MindBody. We ask each tribe member and visitor to sign-up, reserve, and pay for each class online before arriving - it allows for a hassle-free experience for us all! If you're having trouble canceling online, please give us a call in advance at (810) 522-5367. If you do not show for class without notice or cancellation, it will still count towards your package and you will still be charged.
Yes, however, we have a limited amount of mats that can be borrowed or rented. We recommend you bring your own mat for comfort! We also offer mat cleaning spray so you can keep your mat in good shape.
IS THERE A PLACE TO CHANGE OR SHOWER?
We have bathrooms and showers where you can change or get freshened up!
I'M NEW TO YOGA. WHAT CAN I EXPECT?
The yoga culture can seem completely different at each and every studio you visit. At the Space, we believe movement is personal. Our instructors offer guidance and lots of hands-on assistance if you need it, but they will also teach you to listen to the teacher within and move the way that feels best to your body. The team at the Space has a true passion for providing something deeper than just a workout.
WHAT SHOULD I WEAR? WHAT SHOULD I DO IF I FORGET MY CLOTHES?
We recommend wearing clothing that is light, airy, and allows for a lot of movement. We also recommend wearing what's comfortable to you! If you happen to forget your clothes for class, we have a retail section filled with exclusive brands and clothing for you to shop!
We understand the intimidating and confusing journey of transformation, the everyday struggles of finding your truest self, and the constant need to "fit in". We've created a studio fitness environment founded on acceptance and release for yogis and non-yogis alike. We can't wait to share it you!
Whether your body is asking for strength, release, or calm, we've got something for everyone at the Space. Our selection of classes involves Vinyasa and Hatha yoga styles, high intensity interval training (HIIT), strength, cardio, and dance. We invite you to move your body with freedom, push fiercely towards your goals, and to achieve just what you came in for while being 100% accepted as you are.


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REPEATABLE_READ, DIRTY_READ, and READ_COMMITTED are mutually exclusive (i.e. can't be used together). EXCLUSIVE_READ_LOCK can be joined with any of them.
Transactions using the REPEATABLE_READ isolation level wait until the object that is write-locked by other transactions are unlocked before they acquire their own locks. This prevents them from reading "dirty" data. In addition, because other transactions cannot update or delete an object that is locked by a transaction using the REPEATABLE_READ isolation level, non-repeatable read situations are avoided.
If the read operation is under a transaction, there is no need to "enlist" the space object in the transaction (unless its already enlisted).
Refer to the Locking and Blocking page for general XAP locking and blocking rules.
Setting MATCH_BY_ID changes the matching algorithm, such that once a SpaceID is set the other fields' values are ignored. The matching is only done according to the SpaceID value.


text:
\section{Introduction}

The successes of modern deep learning have highlighted that good performance on tasks such as image classification \citep{krizhevsky2012imagenet} requires deep models with lower layers that have the flexibility to learn good representations.
Up until very recently, this was only possible in feature-based methods such as neural networks (NNs).
Kernel methods did not have this flexibility because the kernel could be modified only using a few kernel hyperparameters.
However, with the advent of deep kernel processes \citep[DKPs;][]{aitchison2020deep}, we now have deep kernel methods that offer neural-network like flexibility in the kernel / top-layer representation.
DKPs introduce this flexibility by taking the kernel from the previous layer, then sampling from a Wishart or inverse Wishart centered on that kernel, followed by a nonlinear transformation.
The sampling and nonlinear transformation steps are repeated multiple times to form a deep architecture.
Remarkably, deep Gaussian processes \citep[DGPs;][]{damianou2013deep,salimbeni2017doubly}, standard Bayesian NNs, infinite-width Bayesian NNs  \citep[neural network Gaussian processes or NNGPs;][]{lee2017deep,matthews2018gaussian,novak2018bayesian,garriga2018deep} and infinite NNs with finite width bottlenecks \citep{aitchison2019bigger} can be written as DKPs \citep{aitchison2020deep}.
In e.g.\ DGPs the random variables used in variational inference are ultimately features, even though a kernel is computed as a function of the features.
By contrast, in a DKP, there are no features \textit{at all}: the only random variables are the positive semi-definite kernel matrices themselves, which are sampled directly from distributions over positive semi-definite matrices such as the (inverse) Wishart.

\citet{aitchison2020deep} argued that DKPs should have considerable advantages over related feature-based models, because feature-based models have pervasive symmetries in the true posterior, which are difficult to capture in standard variational approximate posteriors.
For instance, in a neural network, it is possible to permute rows and columns of weight matrices, such that the activations at a given layer are permuted, but the network's overall input-output function remains the same \citep{mackay1992practical,sussmann1992uniqueness,bishop1995neural}.
These permutations result in network weights with exactly the same probability density under the true posterior, but with very different probability densities under standard variational approximate posteriors, which are generally unimodal.
However, these issues do not arise with DKPs, because all permutations of the hidden units correspond to the same kernel (see Appendix D in \citet{aitchison2020deep} for more details).

Deep Wishart processes (DWPs) are the most important DKP because their prior is equivalent to the DGP prior.
While \citet{aitchison2020deep} showed this equivalence, they were not able to do inference in DWPs because they were not able to find a sufficiently flexible distribution over positive semi-definite matrices to form the basis of an approximate posterior.
Instead, they were forced to work with a different DKP: the deep \emph{inverse} Wishart processes (DIWPs), which was easier because the inverse Wishart itself forms a suitable approximate posterior.
We show how to create a sufficiently flexible approximate posterior for DWPs, thereby enabling us to compare directly to the equivalent DGPs.
In particular, our contributions are:

\begin{itemize}
  \item We develop a new family of flexible distributions over positive semi-definite matrices by generalising the Bartlett decomposition (Sec.~\ref{sec:gen_wishart}).
  \item We use this distribution to develop an effective approximate posterior for the deep Wishart process which incorporates dependency across layers (Sec.~\ref{sec:approx_post}).
  \item We develop a doubly stochastic inducing-point inference scheme for the DWP. While the derivation mostly follows that for deep inverse Wishart processes \citep{aitchison2020deep}, we need to give a novel scheme for sampling the test/training points conditioned on the inducing points, as this is very different in the DWP compared to the previous DIWP (Sec.~\ref{sec:dsvi}).
  \item We empirically compare DGP and DWP inference with the \textit{exact same prior}. This was not possible in \citet{aitchison2020deep} as they only derived an inference scheme for deep inverse Wishart processes, whose prior is not equivalent to a DGP prior. As expected, DWPs show strong benefits over DGPs.
\end{itemize}

\section{Background}

\subsection{Wishart distribution}

The Wishart, $\mathcal{W}\b{\S, \nu}$, is a distribution over positive semi-definite $P \times P$ matrices, $\mathbf{G}$, with positive definite scale parameter $\S\in\mathbb{R}^{P\times P}$ and a positive, integer-valued degrees-of-freedom parameter, $\nu$.
The Wishart distribution is defined by taking $\nu$ vectors $\mathbf{n}_\lambda \in \mathbf{R}^P$ sampled from a zero-mean Gaussian.
These vectors can be generated from standard Gaussian vectors, $\boldsymbol{\xi}_\lambda$, by transforming them with the Cholesky, $\L$ of the scale parameter, $\S = \L \L^T$,
\begin{align}
  \L \boldsymbol{\xi}_\lambda &= \mathbf{n}_\lambda \sim \mathcal{N}\b{{\bf {0}}, \S} &\text{where } \boldsymbol{\xi}_\lambda &\sim \mathcal{N}\b{{\bf {0}}, \mathbf{I}}
\end{align}
Both $\mathbf{n}_\lambda$ and $\boldsymbol{\xi}_\lambda$ can be stacked to form $P \times \nu$ matrices, $\mathbf{N}$ and $\mathbf{\Xi}$,
\begin{align}
  \mathbf{N} &= \b{\mathbf{n}_1 \quad \mathbf{n}_2 \quad \dotsm \quad \mathbf{n}_\nu} &
  \mathbf{\Xi} &= \b{\boldsymbol{\xi}_1 \quad \boldsymbol{\xi}_2 \quad \dotsm \quad \boldsymbol{\xi}_\nu}.
\end{align}
Wishart samples are defined by taking the sum of the outer products of the $\mathbf{n}_\lambda$'s, which can be written as a matrix-multiplication,
\begin{align}
  \label{eq:def:wishart}
  \sum_{\lambda=1}^\nu \mathbf{n}_\lambda \mathbf{n}_\lambda^T = \mathbf{N} \mathbf{N}^T = \L \mathbf{\Xi} \mathbf{\Xi}^T \L = \L \mathbf{Z} \L^T = \mathbf{G} &\sim \mathcal{W}\b{\S, \nu}
  \intertext{where $\mathbf{Z}=\mathbf{\Xi} \mathbf{\Xi}^T$ is a sample from a standard Wishart (i.e.\ one with an identity scale parameter,)}
  \label{eq:def:standard_wishart}
  \sum_{\lambda=1}^\nu \boldsymbol{\xi}_\lambda \boldsymbol{\xi}_\lambda^T = \mathbf{\Xi} \mathbf{\Xi}^T = \mathbf{Z} &\sim \mathcal{W}\b{\mathbf{I}, \nu}.
\end{align}
Note that therefore the Wishart has mean,
\begin{align}
  \label{eq:mean_wishart}
  \mathbb{E}\bcket{[}{]}{\mathbf{G}} &= \nu \mathbb{E}\bcket{[}{]}{\mathbf{n}_\lambda \mathbf{n}_\lambda^T} = \nu \S
\end{align}

\subsection{Bartlett decomposition}
However, sampling $\mathbf{\Xi}$ can be computationally expensive for very large values of $\nu$.
Instead, it is possible to sample a Wishart by writing down the distribution over the Cholesky of $\mathbf{Z}$, denoted $\mathbf{A}$ \citep{bartlett1933on}.
Taking $\mathbf{Z} = \mathbf{A} \mathbf{A}^T$, the distribution over $\mathbf{A}$ is,
\begin{subequations}
  \label{eq:PA}
  \begin{align}

    \label{eq:PA_ondiag}
    \P{A_{jj}^2} &= \text{Gamma}\b{A_{jj}^2; \alpha{=}\tfrac{\nu-j+1}{2}, \beta{=}\tfrac{1}{2}},\\
    \label{eq:PA_offdiag}
    \P{A_{j > k}} &= \mathcal{N}\b{A_{jk}; 0, 1}.
  \end{align}
\end{subequations}
i.e.\ the square of the on-diagonal elements are Gamma distributed and the off-diagonal elements are IID standard Gaussian.

\subsection{Deep Gaussian proceses (DGPs)}
In a DGP, we progressively sample features, $\mathbf{F}_\ell$, from a Gaussian process, conditioned on features from the previous layer,
\begin{subequations}
\label{eq:deepgp}
\begin{align}
  \label{eq:deepgp:top:F}
  \P{\mathbf{F}_\ell| \mathbf{F}_{\ell-1}} &= {\textstyle \prod}_{\lambda=1}^{\nu_\ell} \mathcal{N}\b{\mathbf{f}_\lambda^\ell; {\bf {0}}, \mathbf{K}^\ell_\text{features}\b{\mathbf{F}_{\ell-1}}} & \text{with } \mathbf{F}_0 &= \mathbf{X},\\
  \P{\mathbf{Y}| \mathbf{F}_{L+1}} &= {\textstyle \prod}_{\lambda=1}^{\nu_{L+1}} \mathcal{N}\b{\mathbf{y}_\lambda; \mathbf{f}^{L+1}_\lambda, \sigma^2 \mathbf{I}}
\end{align}
\end{subequations}
where $\mathbf{X} \in \mathbb{R}^{P\times \nu_0}$ is the input and $\mathbf{F}_\ell \in \mathbb{R}^{P \times \nu_\ell}$ are the features.
We use $P$ for the number of input points and $\nu_\ell$ for the width of layer $\ell$; thus $\nu_0$ is the number of inputs and $\nu_{L+1}$ is the number of outputs.
In addition, the features and targets can be written as a stack of vectors, $\mathbf{f}_\lambda^\ell \in \mathbb{R}^P$ and $\mathbf{y}_\lambda \in \mathbb{R}^P$, i.e.\
\begin{align}
\mathbf{F}_\ell &= (\mathbf{f}_1^\ell \quad \mathbf{f}_2^\ell \quad \dotsm  \quad \mathbf{f}_{\nu_\ell}^\ell) &
\mathbf{Y} &= (\mathbf{y}_1 \quad \mathbf{y}_2 \quad \dotsm  \quad \mathbf{y}_{\nu_{L+1}}).
\end{align}
The function $\mathbf{K}^\ell_\text{features}\b{\mathbf{F}_{\ell-1}}$ takes the features at the previous layer and returns the corresponding $P\times P$ kernel matrix.
We mainly consider isotropic kernels such as the squared exponential, which can be written as a function of $R_{ij}^{\ell-1}$, the distance between input features $i$ and $j$,
\begin{align}

  K^\ell_{\text{features}; ij} &= k_\ell(R^{\ell-1}_{ij}),\\
  \label{eq:def:R}
  R_{ij}^{\ell-1} &= \tfrac{1}{N_\ell} {\textstyle \sum}_{\lambda=1}^{N_\ell} \b{F^{\ell-1}_{i\lambda} - F^{\ell-1}_{j\lambda}}^2.
\end{align}

\subsection{Deriving equivalent deep Wishart processes}
\label{sec:dwp_deriv}
Following \citet{aitchison2020deep}, we show how the DGP model of Eq.~\eqref{eq:deepgp} can be expressed as a deep Wishart process. We first consider the $P\times P$ Gram matrices defined as
\begin{align}
  \mathbf{G}_\ell &= \tfrac{1}{\nu_\ell} \mathbf{F}_\ell \mathbf{F}_\ell^T = \tfrac{1}{\nu_\ell} \sum_{\lambda=1}^{\nu_\ell} \mathbf{f}^\ell_{\lambda} (\mathbf{f}^\ell_{\lambda})^T,
\end{align}
where $\mathbf{f}^\ell_\lambda$ are IID and multivariate-Gaussian distributed conditioned on the features at the previous layer (Eq.~\ref{eq:deepgp:top:F}).
Thus, $\mathbf{G}_\ell$ follows the definition of the Wishart (Eq.~\ref{eq:def:wishart}), and we can sample $\mathbf{G}_\ell$ directly,
\begin{align}
  \Pc{\mathbf{G}_\ell}{\mathbf{F}_{\ell-1}} &= \mathcal{W}\b{\mathbf{G}_\ell; \tfrac{1}{\nu_\ell} \mathbf{K}^\ell_\text{features}(\mathbf{F}_{\ell-1}), \nu_\ell}.
\end{align}
To work entirely with Gram matrices rather than features, we need to be able to compute the kernel, $\mathbf{K}^\ell_\text{features}(\mathbf{F}_{\ell-1})$ as a function of the Gram matrix at the previous layer, $\mathbf{G}_{\ell-1}$.
Remarkably, this is possible for a large family of practically relevant kernels \citep{aitchison2020deep}.
In particular, note that it is possible to recover the distance from the Gram matrix,
\begin{align}
  R_{ij}^\ell &= \tfrac{1}{N_\ell} {\textstyle \sum}_{\lambda=1}^{N_\ell} \b{\b{F^\ell_{i \lambda}}^2 - 2F^\ell_{i \lambda}F^\ell_{j \lambda} + \b{F^\ell_{j \lambda}}^2} = G^\ell_{ii} - 2 G^\ell_{ij} + G^\ell_{jj}.
\end{align}
Thus, at least for isotropic kernels, which depend only on the distance, it is possible to obtain $\mathbf{K}_\ell(\cdot)$, which takes the Gram matrix from the previous layer and returns the same kernel matrix as that returned by applying $\mathbf{K}_\text{features}$ to the features from the previous layer:
\begin{align}
  \mathbf{K}^\ell_\text{features}(\mathbf{F}_{\ell-1}) = \mathbf{K}_\ell(\mathbf{G}_{\ell-1}) = \mathbf{K}_\ell(\tfrac{1}{\nu_\ell} \mathbf{F}_{\ell-1} \mathbf{F}_{\ell-1}^T).

\end{align}
By using the equivalent kernel written as a function of the Gram matrix at the previous layer, we can entirely eliminate intermediate layer features, resulting in a deep Wishart process,
\begin{subequations}
\label{eq:def:deepiw}
\begin{align}

  \Pc{\mathbf{G}_\ell}{\mathbf{G}_{\ell-1}} &= \mathcal{W}\b{\mathbf{G}_\ell; \tfrac{1}{\nu_\ell} \mathbf{K}_\ell(\mathbf{G}_{\ell-1}), \nu_\ell} & \text{with } \mathbf{G}_0 &= \tfrac{1}{\nu_0} \mathbf{X} \mathbf{X}^T,\\
  \Pc{\mathbf{F}_{L+1}}{\mathbf{G}_L} &= {\textstyle \prod}_{\lambda=1}^{\nu_{L+1}} \mathcal{N}\b{\mathbf{f}_\lambda^{L+1}; {\bf {0}}, \mathbf{K}_{L+1}\b{\mathbf{G}_L}},\\
  \Pc{\mathbf{Y}}{\mathbf{F}_{L+1}} &= {\textstyle \prod}_{\lambda=1}^{\nu_{L+1}} \mathcal{N}\b{\mathbf{y}_\lambda; {\bf {0}}, \sigma^2 \mathbf{I}}.
\end{align}
\end{subequations}

\subsection{DWP formulation captures true-posterior symmetries while DGP does not}
\label{sec:symmetry}
We now have two equivalent generative models: one phrased in terms of features, $\mathbf{F}_\ell$ and another phrased in terms of Gram matrices, $\mathbf{G}_\ell$.
Is there any reason to prefer one over the other?
It turns out that there is.
In particular, consider a transformation of the features, $\mathbf{F}_\ell' = \mathbf{F}_\ell \mathbf{U}$ where $\mathbf{U}$ is a unitary matrix, such that $\mathbf{U} \mathbf{U}^T = \mathbf{I}$.
Remarkably, the true posterior is symmetric under these transformations, in the sense that all unitary transformations of the underlying features have the exact same true-posterior probability density \citep[see][Appendix D.2]{aitchison2020deep},
\begin{align}
  \P{\mathbf{F}'_1,\dotsc,\mathbf{F}'_L, \mathbf{F}_{L+1}| \mathbf{X}, \mathbf{Y}} &= \P{\mathbf{F}_1,\dotsc,\mathbf{F}_L, \mathbf{F}_{L+1}| \mathbf{X}, \mathbf{Y}}.
\end{align}
It would be desirable for variational approximate posteriors to capture these true posterior symmetries.
However, the usual family of Gaussian approximate posteriors over features fails to capture these symmetries because they use non-zero means.
Worryingly, the failure to capture these symmetries can bias variational inference to focus on low-mass areas of the true posterior \citep{moore2016symmetrized, pourzanjani2017improving}.

In contrast, the deep Wishart process sidesteps this issue by phrasing posteriors entirely in terms of Gram matrices, $\mathbf{G}_\ell = \tfrac{1}{\nu_\ell} \mathbf{F}_\ell \mathbf{F}_\ell^T$.
Critically, the Gram matrix is invariant to unitary transformations of the features,
\begin{align}
  \label{eq:gram-rotation}
  \mathbf{G}_\ell &= \tfrac{1}{N_\ell} \mathbf{F}_\ell \mathbf{F}_\ell^T = \tfrac{1}{N_\ell} \mathbf{F}_\ell \mathbf{U}_\ell \mathbf{U}_\ell^T \mathbf{F}_\ell^T = \tfrac{1}{N_\ell} \mathbf{F}^\prime_\ell \mathbf{F}_\ell^{\prime T}.
\end{align}
As such, DWP approximate posteriors written in terms of $\mathbf{G}_\ell$ implicitly respect this unitary symmetry over the features.

\section{Methods}
As detailed in \citet{aitchison2020deep}, the key difficulty in obtaining a variational inference scheme for DWPs is the difficulty of providing a sufficiently flexible approximate posterior.
In particular, as we are working with a probabilistic process, the number of input points, $P$, can be arbitrarily large, and thus there is always the possibility that $\nu < P$ and hence that our sampled Gram matrices are low-rank.
We therefore need to form flexible variational approximate posteriors over rank $\nu$ Gram matrices.
An obvious first choice is the Wishart distribution itself with degrees of freedom $\nu$, so as to match the rank of matrices sampled from the prior.
However, for fixed degrees of freedom the Wishart variance,
\begin{align}
  \label{eq:var_wishart}
  \mathrm{Var} \bcket{[}{]}{G_{ij}} &= \nu \b{\Sigma_{ij}^2 + \Sigma_{ii} \Sigma_{jj}}
\end{align}
cannot be specified independently of the mean (Eq.~\ref{eq:mean_wishart}), which is essential for a variational approximate posterior that can flexibly capture potentially narrow true posteriors.
An alternative approach would be to work with a non-central Wishart, which is defined by taking $\mathbf{\Xi}$, which is IID standard Gaussian in the case of the Wishart, to have non-zero mean.
However, the non-central Wishart has a probability density function that is too difficult to evaluate in the inner loop of a deep learning algorithm \citep{koev2006efficient}.
Instead, we develop a new Generalised Singular Wishart distribution, based on the Bartlett decomposition, which modifies the Wishart to give independent control over the mean and variance of sampled matrices.

\subsection{Singular Bartlett decomposition}
To define the Generalised Singular Wishart distribution, we first need to generalise the Wishart construction to potentially singular matrices (i.e.\ those for which $\nu < P$).
Remembering that $\mathbf{Z} = \mathbf{A} \mathbf{A}^T$, in the singular case $\mathbf{A}$ is given by
\begin{align}
  \label{eq:A}
  \mathbf{A} &= \begin{pmatrix}
    A_{11}    & \hdots & 0          \\
    \vdots    & \ddots & \vdots     \\
    A_{\nu 1} & \hdots & A_{\nu \nu} \\
    \vdots    & \vdots & \vdots     \\
    A_{P 1}   & \hdots & A_{P \nu}
  \end{pmatrix},\\
  \P{A_{jj}^2} &= \text{Gamma}\b{A_{jj}^2; \tfrac{\nu-j+1}{2}, \tfrac{1}{2}}\\
  \P{A_{i > j}} &= \mathcal{N}\b{A_{ij}; 0, 1}.
\end{align}
The full probability density is
\begin{align}
  \P{\mathbf{A}} &= \prod_{j=1}^{\min(p, \nu)} 2 A_{jj} \text{Gamma}\b{A_{jj}^2; \tfrac{\nu-j+1}{2}, \tfrac{1}{2}} \prod_{i=j+1}^p \mathcal{N}\b{A_{ij}; 0, 1}.
\end{align}
where the $2 A_{jj}$ accounts for the Jacobian for the transformation from $A_{jj}^2$ to $A_{jj}$.
Now we transform from $\mathbf{A}$ to $\mathbf{G}$ using the Jacobians in Appendix~\ref{app:J_CT} and Appendix~\ref{app:J_LLT} (i.e. by first transforming $\mathbf{A} \rightarrow \L \mathbf{A}$ then transforming $\L \mathbf{A} \rightarrow \L \mathbf{A} \mathbf{A}^T \L^T$.)
\begin{align}

  \P{\mathbf{G}} &= \b{\prod_{j=1}^p \frac{1}{L_{jj}^{\min(j,\nu)}}} \prod_{j=1}^{\min(p, \nu)} \frac{\text{Gamma}\b{A_{jj}^2; \tfrac{\nu-j+1}{2}, \tfrac{1}{2}}}{A_{jj}^{p-j} L_{jj}^{p-j+1}} \prod_{i=j+1}^p \mathcal{N}\b{A_{ij}; 0, 1}.

\end{align}
In Appendix~\ref{app:singular_proof} we prove that this corresponds to the known full rank and singular Wishart distribution.

\subsection{Generalised singular Wishart distributions}
\label{sec:gen_wishart}
Our goal is to develop a generalisation of the Wishart distribution, $\mathcal{W}\b{\S, \nu, \boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\mu}, \boldsymbol{\sigma}}$ based on the Bartlett decomposition, which will turn out to have additional parameters, $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ for the on-diagonal elements of $\mathbf{A}$ and $\boldsymbol{\mu}$ and $\boldsymbol{\sigma}$ for the off-diagonal elements.
As we will be using this distribution for the approximate posterior, we write densities under this distribution as $\Q{\cdot}$,
\begin{align}

  \Q{\mathbf{G}} &= \mathcal{W}\b{\mathbf{G}; \S, \nu, \boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\mu}, \boldsymbol{\sigma}}.
\end{align}
To specify this distribution, we generalise the Bartlett decomposition using
\begin{subequations}
\begin{align}
  \Q{A_{jj}^2} &= \text{Gamma}\b{A_{jj}^2; \alpha_j, \beta_j} && \text{for } j \leq \nu\\
  \Q{A_{i > j}} &= \mathcal{N}\b{A_{ij}; \mu_{ij}, \sigma_{ij}^2}.
\end{align}
\end{subequations}
Thus, the full probability density for $\mathbf{A}$ is,
\begin{align}
  \Q{\mathbf{A}} &= \prod_{j=1}^\nu 2 A_{jj} \text{Gamma}\b{A_{jj}^2; \alpha_j, \beta_j} \prod_{i=j+1}^p \mathcal{N}\b{A_{ij}; \mu_{ij}, \sigma_{ij}^2},
\end{align}
applying the same transformations and Jacobians as in the previous section, this implies a distribution over $\mathbf{G}$ of,
\begin{align}
  \Q{\mathbf{G}} &= \b{\prod_{j=1}^p \frac{1}{L_{jj}^{\min(j,\nu)}}} \prod_{j=1}^{\min(p, \nu)} \frac{1}{A_{jj}^{p-j} L_{jj}^{p-j+1}} \text{Gamma}\b{A_{jj}^2; \alpha_j, \beta_j} \prod_{i=j+1}^p \mathcal{N}\b{A_{ij}; \mu_{ij}, \sigma_{ij}^2}.

\end{align}

\subsection{Full approximate posterior distribution}
\label{sec:approx_post}
Inspired by the across-layer dependencies in \citet{aitchison2020deep}, we use an Generalised Wishart approximate posterior for $\mathbf{G}_\ell$ with dependencies across layers,
\begin{subequations}
\label{eq:def:ap}
\begin{align}

  \Qc{\mathbf{G}_\ell}{\mathbf{G}_{\ell-1}} &= \mathcal{W}\b{\mathbf{G}_\ell; (1-p_\ell) \tfrac{1}{\nu_\ell} \mathbf{K}_\ell(\mathbf{G}_{\ell-1}) + p_\ell \mathbf{V}_\ell \mathbf{V}_\ell^T,\, \nu_\ell,\, \boldsymbol{\alpha}_\ell,\, \boldsymbol{\beta}_\ell,\, \boldsymbol{\mu}_\ell,\, \boldsymbol{\sigma}_\ell},
\end{align}
\end{subequations}
where the approximate posterior parameters are $\sl{\mathbf{V}_\ell, \boldsymbol{\alpha}_\ell, \boldsymbol{\beta}_\ell, \boldsymbol{\mu}_\ell, \boldsymbol{\sigma}_\ell, p_\ell}$, where $0 < p_\ell < 1$ is a scalar, and $\mathbf{V}_\ell\in \mathbb{R}^{p \times p}$.
Note that the exact form for the across layer dependencies in our approximate posterior is inspired by the approximate posterior in \citep{aitchison2020deep}.

\subsection{Doubly stochastic inducing-point variational inference in deep inverse Wishart processes}
\label{sec:dsvi}
For efficient inference in high-dimensional problems, we take inspiration from the DGP literature \citep{salimbeni2017doubly} by considering doubly-stochastic inducing-point deep Wishart processes.
We begin by decomposing all variables into inducing and training (or test) points $\mathbf{X}_\text{i}\in\mathbb{R}^{P_\text{i}\times N_0}$ and $\mathbf{X}_\text{t}\in\mathbb{R}^{P_\text{t}\times N_0}$ where $P_\text{i}$ is the number of inducing points, and $P_\text{t}$ is the number of testing/training points,
\begin{align}
  \mathbf{X} &= \begin{pmatrix}
    \mathbf{X}_\text{i}\\
    \mathbf{X}_\text{t}
  \end{pmatrix}, &
  \mathbf{F}_{L+1} &= \begin{pmatrix}
    \mathbf{F}^{L+1}_\text{i}\\
    \mathbf{F}^{L+1}_\text{t}
  \end{pmatrix}, &
  \mathbf{G}_\ell &=\begin{pmatrix}
    \mathbf{G}_\text{ii}^\ell & \mathbf{G}_\text{it}^\ell\\
    \mathbf{G}_\text{ti}^\ell & \mathbf{G}_\text{tt}^\ell
  \end{pmatrix}, &

\end{align}
where e.g.\ $\mathbf{G}_\text{ii}^\ell$ is $P_\text{i} \times P_\text{i}$ and $\mathbf{G}_\text{it}^\ell$ is $P_\text{i} \times P_\text{t}$.
The full ELBO including latent variables for all the inducing and training points is
\begin{align}
  \label{eq:def:elbo}
  \mathcal{L} &= \mathbb{E}\bcket{[}{]}{\log \P{\mathbf{Y}| \mathbf{F}_{L+1}} + \log \frac{\P{\slt{\mathbf{G}_\ell},\mathbf{F}_{L+1}| \mathbf{X}}}{\Q{ \slt{\mathbf{G}_\ell},\mathbf{F}_{L+1}| \mathbf{X}}}},
\end{align}
where the expectation is taken over $\Q{\slt{\mathbf{G}_\ell},\mathbf{F}_{L+1}| \mathbf{X}}$.
The prior is given by combining all terms in Eq.~\eqref{eq:def:deepiw} for both inducing and test/train inputs,
\begin{align}
  \P{\slt{\mathbf{G}_\ell},\mathbf{F}_{L+1}| \mathbf{X}} = \bcket{[}{]}{{\textstyle \prod}_{\ell=1}^{L} \P{\mathbf{G}_\ell| \mathbf{G}_{\ell-1}}} \P{\mathbf{F}_{L+1}| \mathbf{G}_L},
\end{align}
where the $\mathbf{X}$-dependence enters on the right because $\mathbf{G}_0 = \tfrac{1}{\nu_0} \mathbf{X} \mathbf{X}^T$.
Taking inspiration from \citet{salimbeni2017doubly}, the full approximate posterior is the product of an approximate posterior over inducing points and the conditional prior for train/test points,
\begin{multline}
  \label{eq:Q_inducing_fac}
  \Q{\slt{\mathbf{G}_\ell},\mathbf{F}_{L+1}| \mathbf{X}} =\\
  \Q{\slt{\mathbf{G}^\ell_\text{ii}},\mathbf{F}_\text{i}^{L+1}| \mathbf{X}_\text{i}} \P{\slt{\mathbf{G}^\ell_\text{it}},\slt{\mathbf{G}^\ell_\text{tt}},\mathbf{F}_\text{t}^{L+1}| \slt{\mathbf{G}^\ell_\text{ii}},\mathbf{F}_\text{i}^{L+1}, \mathbf{X}}.
\end{multline}
And the prior can be written in the same form,
\begin{multline}
  \label{eq:P_inducing_fac}
  \P{\slt{\mathbf{G}_\ell},\mathbf{F}_{L+1}| \mathbf{X}} =\\
  \P{\slt{\mathbf{G}^\ell_\text{ii}},\mathbf{F}_\text{i}^{L+1}| \mathbf{X}_\text{i}} \P{\slt{\mathbf{G}^\ell_\text{it}},\slt{\mathbf{G}^\ell_\text{tt}},\mathbf{F}_\text{t}^{L+1}| \slt{\mathbf{G}^\ell_\text{ii}},\mathbf{F}_\text{i}^{L+1}, \mathbf{X}}.
\end{multline}
We discuss the second terms (the conditional prior) in Eq.~\eqref{eq:condP}.
The first terms (the prior and approximate posteriors over inducing points), are given by combining terms in Eq.~\eqref{eq:def:deepiw} and Eq.~\eqref{eq:def:ap},
\begin{align}
  \P{\slt{\mathbf{G}^\ell_\text{ii}},\mathbf{F}_\text{i}^{L+1}| \mathbf{X}_\text{i}} &= \bcket{[}{]}{{\textstyle \prod}_{\ell=1}^{L} \P{\mathbf{G}^\ell_\text{ii}| \mathbf{G}^{\ell-1}_\text{ii}}} \P{\mathbf{F}_\text{i}^{L+1}| \mathbf{G}_\text{ii}^L},\\
  \label{eq:Q_inducing}
  \Q{\slt{\mathbf{G}^\ell_\text{ii}},\mathbf{F}_\text{i}^{L+1}| \mathbf{X}_\text{i}} &= \bcket{[}{]}{{\textstyle \prod}_{\ell=2}^{L} \Q{\mathbf{G}^\ell_\text{ii}| \mathbf{G}^{\ell-1}_\text{ii}}} \Q{\mathbf{F}_\text{i}^{L+1}| \mathbf{G}_\text{ii}^L}.
\end{align}
Substituting Eqs.~(\ref{eq:Q_inducing_fac}--\ref{eq:Q_inducing}) into the ELBO (Eq.~\ref{eq:def:elbo}), the conditional prior cancels and we obtain,
\begin{align}

  \mathcal{L} &= \mathbb{E}\bcket{[}{]}{\log \P{\mathbf{Y}| \mathbf{F}^{L+1}_\text{t}} + \log \frac{\bcket{[}{]}{{\textstyle \prod}_{\ell=1}^{L} \Q{\mathbf{G}^\ell_\text{ii}| \mathbf{G}^{\ell-1}_\text{ii}}} \Q{\mathbf{F}_\text{i}^{L+1}| \mathbf{G}_\text{ii}^L}}{\bcket{[}{]}{{\textstyle \prod}_{\ell=1}^{L} \P{\mathbf{G}^\ell_\text{ii}| \mathbf{G}^{\ell-1}_\text{ii}}} \P{\mathbf{F}_\text{i}^{L+1}| \mathbf{G}_\text{ii}^L}}}.
\end{align}
The first term is a summation across test/train datapoints, and the second term depends only on the inducing points, so as in \citet{salimbeni2017doubly} we can compute unbiased estimates of the expectation by taking only a minibatch of datapoints.
We also never need to compute the density of the conditional prior in Eq.~\eqref{eq:P_inducing_fac}, we only need to be able to sample from it,
\begin{multline}
  \label{eq:condP}
  \P{\slt{\mathbf{G}_\text{ti}^\ell, \mathbf{G}_\text{tt}^\ell},\mathbf{F}_\text{t}^{L+1}|  \slt{\mathbf{G}^\ell_\text{ii}},\mathbf{F}_\text{i}^{L+1}, \mathbf{X}} =\\
  \P{\mathbf{F}^{L+1}_\text{t}| \mathbf{F}^{L+1}_\text{i}, \mathbf{G}_{L}} {\textstyle \prod}_{\ell=1}^{L} \P{\mathbf{G}^\ell_\text{ti}, \mathbf{G}^\ell_\text{tt}| \mathbf{G}^\ell_\text{ii}, \mathbf{G}_{\ell-1}}.
\end{multline}
The first distribution, $\P{\mathbf{F}^{L+1}_\text{t}| \mathbf{F}^{L+1}_\text{i}, \mathbf{G}_{L}}$, is a multivariate Gaussian, and can be evaluated using methods from the GP literature \citep{williams2006gaussian,salimbeni2017doubly}.
Specifically, we use the global inducing point scheme from \citet{ober2020global}.
The second distribution is more difficult to sample from.
To address this issue, we go back to features,
\begin{align}
  \label{eq:FFT}

  \mathbf{F}_\ell \mathbf{F}_\ell^T = \nu \mathbf{G}_\ell &\sim \mathcal{W}\b{\mathbf{K}_\ell\b{\mathbf{G}_{\ell-1}}, \nu},
\end{align}
with
\begin{align}
  \mathbf{F}_\ell &= \begin{pmatrix}
    \mathbf{F}^\ell_\text{i} \\ \mathbf{F}^\ell_\text{t}
  \end{pmatrix} &
   \mathbf{K}_\ell\b{\mathbf{G}_{\ell-1}} &= \begin{pmatrix}
    \mathbf{K}_\text{ii} & \mathbf{K}_\text{ti}^T\\
    \mathbf{K}_\text{ti} & \mathbf{K}_\text{tt}
  \end{pmatrix}
\end{align}
where $\mathbf{F}_\ell \in \mathbb{R}^{(P_\text{i}+P_\text{t})\times \nu_\ell}$, $\mathbf{F}_\text{i}\in\mathbb{R}^{P_\text{i}\times \nu_\ell}$ and $\mathbf{F}_\text{t}\in\mathbb{R}^{P_\text{t}\times \nu_\ell}$.
Our goal is to sample $\mathbf{G}_\text{it}^\ell$ and $\mathbf{G}_\text{tt}^\ell$ given $\mathbf{G}_\text{ii}^\ell$.
Our approach is to note that, $\mathbf{F}_\text{t}$ conditioned on $\mathbf{F}_\text{i}$ is given by a matrix normal, \citep[][page 310]{eaton2007wishart},
\begin{align}
  \label{eq:Ft|Fi}
  \Pc{\mathbf{F}^\ell_\text{t}}{\mathbf{F}^\ell_\text{i}} &= \mathcal{MN}\b{\mathbf{K}_\text{ti}^T \mathbf{K}_\text{ii}^{-1} \mathbf{F}_\text{i}, \mathbf{K}_{\text{tt}\cdot \text{i}}, \mathbf{I}},
\end{align}
where
\begin{align}
  \mathbf{K}_{\text{tt}\cdot \text{i}} &= \mathbf{K}_\text{tt} - \mathbf{K}^T_\text{it} \mathbf{K}_\text{ii}^{-1} \mathbf{K}_\text{it}.
\end{align}
Note that we sample each test/train point one-at-a-time/independently, in which case, $P_\text{t}=1$ and $\S_{22 \cdot 1}$ is scalar.

Then $\mathbf{G}_\ell$, which includes $\mathbf{G}^\ell_\text{it}$ and $\mathbf{G}^\ell_\text{tt}$ is given by,
\begin{align}
  \mathbf{G}_\ell &=\begin{pmatrix}
    \mathbf{G}_\text{ii}^\ell & \mathbf{G}_\text{it}^\ell\\
    \mathbf{G}_\text{ti}^\ell & \mathbf{G}_\text{tt}^\ell
  \end{pmatrix} =
  \tfrac{1}{\nu}
  \begin{pmatrix}
    \mathbf{F}_\text{i}^\ell \b{\mathbf{F}_\text{i}^\ell}^T & \mathbf{F}_\text{i}^\ell\b{\mathbf{F}_\text{t}^\ell}^T \\
    \mathbf{F}_\text{t}^\ell \b{\mathbf{F}_\text{i}^\ell}^T & \mathbf{F}_\text{t}^\ell\b{\mathbf{F}_\text{t}^\ell}^T
  \end{pmatrix}

  = \tfrac{1}{\nu}\mathbf{F}_\ell \mathbf{F}_\ell^T
\end{align}
For $\mathbf{F}_\text{i}$, we can use any value as long as $\mathbf{G}^\ell_\text{ii} = \mathbf{F}_\text{i}^\ell \b{\mathbf{F}_\text{i}^\ell}^T$, as the resulting distribution over $\mathbf{G}_\ell$ arising from Eq.~\eqref{eq:FFT} does not depend on the specific choice of $\mathbf{F}_\text{i}$ (App.~\ref{sec:app:F}).
Remembering that to sample $\mathbf{G}_\text{ii}$, we explicitly sample its potentially low-rank Cholesky, $\L_\ell \mathbf{A}_\ell$, we can directly use
\begin{align}
  \mathbf{F}_\text{i}^\ell &= \L_\ell \mathbf{A}_\ell
\end{align}
However, this only works if $\nu \leq P_\text{i}$, in which case, $\L_\ell \mathbf{A}_\ell \in \mathbb{R}^{P_\text{i} \times \nu}$.
In the unusual case where we have fewer inducing points than degrees of freedom, $P_\text{i}<\nu$, then $\L_\ell \mathbf{A}_\ell \in \mathbb{R}^{P_\text{i} \times P_\text{i}}$, so we need to pad to achieve the required size of $P_\text{i} \times \nu_\ell$,
\begin{align}
  \mathbf{F}_\text{i}^\ell &= \begin{pmatrix}
    \L_\ell \mathbf{A}_\ell & {\bf {0}}
  \end{pmatrix}.
\end{align}

Finally, note that we can optimise all the variational parameters using standard reparameterised variational inference \citep{kingma2013auto,rezende2014stochastic}.
For an algorithm, see Alg.~\ref{alg}.

\begin{algorithm}[tb]
\caption{Computing predictions/ELBO for one batch}
\label{alg}
\begin{algorithmic}
  \STATE {\bfseries P parameters:} $\sl{\nu_\ell}$.
  \STATE {\bfseries Q parameters:} $\sl{\mathbf{V}_\ell, p_\ell, \boldsymbol{\alpha}_\ell, \boldsymbol{\beta_\ell}, \boldsymbol{\mu}_\ell, \boldsymbol{\sigma}_\ell}, \mathbf{X}_\text{i}$.
  \STATE {\bfseries Inputs:} $\mathbf{X}_\text{t}$; {\bfseries Targets:} $\mathbf{Y}$
  \STATE \textcolor{gray}{combine inducing and test/train inputs}
  \STATE $\mathbf{X} = \begin{pmatrix} \mathbf{X}_\text{i} & \mathbf{X}_\text{t} \end{pmatrix}$
  \STATE \textcolor{gray}{sample first Gram matrix and update ELBO}
  \STATE $\mathbf{G}_0 = \tfrac{1}{\nu_0} \mathbf{X} \mathbf{X}^T$
  \FOR{$\ell$ {\bfseries in} $\{1,\dotsc,L\}$}
    \STATE \textcolor{gray}{sample inducing Gram matrix and its Cholesky, $\L_\ell \mathbf{A}_\ell$ and update ELBO}
    \STATE $\L_\ell \mathbf{A}_\ell \mathbf{A}_\ell^T \L_\ell^T = \mathbf{G}_\text{ii}^\ell \sim \Q{\mathbf{G}_\text{ii}^\ell| \mathbf{G}_\text{ii}^{\ell-1}}$
    \STATE $\mathcal{L} \leftarrow \mathcal{L} + \log \P{\mathbf{G}_\text{ii}^\ell| \mathbf{G}_\text{ii}^{\ell-1}} - \log \Q{\mathbf{G}_\text{ii}^\ell| \mathbf{G}_\text{ii}^{\ell-1}}$
    \STATE \textcolor{gray}{sample full Gram matrix from conditional prior}
    \STATE $\S = \tfrac{1}{\nu_\ell} \mathbf{K}_\ell(\mathbf{G}_{\ell-1})$
    \STATE $\S_{\text{tt}\cdot \text{i}} = \S_\text{tt} - \S^T_\text{it} \S_\text{ii}^{-1} \S_\text{it}$
    \STATE $\mathbf{\hat{F}}_\text{i}^\ell = \L_\ell \mathbf{A}_\ell$
    \STATE $\mathbf{\hat{F}}^\ell_\text{t} \sim \mathcal{MN}\b{\S_\text{ti}^T \S_\text{ii}^{-1} \mathbf{\hat{F}}_\text{i}, \S_{\text{tt}\cdot \text{i}}, \mathbf{I}}$

    \STATE $\mathbf{G}_\ell =
    \begin{pmatrix}
      \mathbf{G}_\text{ii}^\ell & \mathbf{\hat{F}}^\ell_\text{i} (\mathbf{\hat{F}}^\ell_\text{t})^T\\
      \mathbf{\hat{F}}^\ell_\text{t} (\mathbf{\hat{F}}^\ell_\text{i})^T & \mathbf{\hat{F}}^\ell_\text{t} (\mathbf{\hat{F}}^\ell_\text{t})^T
    \end{pmatrix}$

  \ENDFOR
  \STATE \textcolor{gray}{sample GP inducing outputs and update ELBO}
  \STATE $\mathbf{F}_\text{i}^{L+1} \sim \Q{\mathbf{F}_\text{i}^{L+1} | \mathbf{G}^L_\text{ii}}$
  \STATE $\mathcal{L} \leftarrow \mathcal{L} + \log \P{\mathbf{F}_\text{i}^{L+1}| \mathbf{G}_\text{ii}^L} - \log \Q{\mathbf{F}_\text{i}^{L+1}| \mathbf{G}_\text{ii}^L}$
  \STATE \textcolor{gray}{sample GP predictions conditioned on inducing points}
  \STATE $\mathbf{F}_\text{t}^{L+1} \sim \Q{\mathbf{F}_\text{t}^{L+1} | \mathbf{G}^L, \mathbf{F}_\text{i}^{L+1}}$
  \STATE \textcolor{gray}{add likelihood to ELBO}
  \STATE $\mathcal{L} \leftarrow \mathcal{L} + \log \P{\mathbf{Y}| \mathbf{F}_\text{t}^{L+1}}$
\end{algorithmic}
\end{algorithm}

\subsection{Computational complexity}
Recalling that $\nu_\ell$ is the width of the $\ell$th layer, $P_i$ is the number of inducing points, and $P_t$ is the number of train or test points, the computational complexity of one DWP layer is given by $O(P_i^3 + P_t P_i^2)$. This is a decrease of a factor of $\nu_{\ell + 1}$ over the complexity for standard DGP inference, such as doubly stochastic variational inference \citep{salimbeni2017doubly}, which has complexity $O(\nu_{\ell+1}(P_i^3 + P_t P_i^2))$. The difference arises from the fact that in a DGP, $\nu_{\ell+1}$ Gaussian processes are sampled in each layer, whereas for a DWP we sample a single Gram matrix.

\section{Results}

The DWP prior is equivalent to a DGP prior (Sec.~\ref{sec:dwp_deriv}) \citep{aitchison2020deep}; the only difference is that in a DGP, we use features as the latent variables, whereas in the DWP we use Gram matrices.
Using Gram matrices as in the DWP should be beneficial as the true posteriors are expected to be simpler than in the DGP (Sec.~\ref{sec:symmetry}).

We trained a DWP and a DGP with the exact same generative model with a squared exponential kernel.
We trained both models for 20000 gradient steps using the Adam optimizer \cite{kingma2014adam}; we detail the exact experimental setup in Appendix~\ref{app:exp-details}.
We report ELBOs and test log likelihoods for depth 5 in Table~\ref{tab:uci_comb}; we report other depths in Appendix~\ref{sec:tables}.
We found strong benefits for the DWP, which are especially evident if we look at the ELBOs and smaller datasets (boston, concrete, energy, wine and yacht).
On larger datasets, the benefits become smaller, as accurate uncertainty modelling is less relevant.
Note that we compared against the recently introduced DGP method based on global inducing points \citep{ober2020global}.
Global inducing point methods were particularly important in our setting because we use a standard feedforward architecture without skip connections to ensure equivalence between the DGP and DWP.
Standard DSVI has considerable difficulties with optimizing the approximate posterior in such models; to get optimization to work effectively \citet{salimbeni2017doubly} were forced to modify the prior to introduce skip connections.

\begin{table}[t]
  \caption{
    ELBOs and log-likelihoods for UCI datasets from \citep{Gal2015DropoutB} for a five-layer network.
    See Appendix~\ref{sec:tables} for other depths.
    Significantly better results are highlighted.
  }
  \label{tab:uci_comb}
  \centering
  \begin{tabular}{lrcc}
    \toprule
& dataset & DWP & DGP\\
\midrule
& boston & \textbf{-0.38 $\pm$ 0.01} & -0.46 $\pm$ 0.01 \\
& concrete & \textbf{-0.49 $\pm$ 0.00} & -0.55 $\pm$ 0.00 \\
& energy & \textbf{1.41 $\pm$ 0.00} & 1.37 $\pm$ 0.00 \\
& kin8nm & \textbf{-0.14 $\pm$ 0.00} & -0.18 $\pm$ 0.00 \\
ELBO & naval & 3.62 $\pm$ 0.07 & \textbf{3.74 $\pm$ 0.08} \\
& power & \textbf{0.03 $\pm$ 0.00} & 0.01 $\pm$ 0.00 \\
& protein & \textbf{-1.01 $\pm$ 0.00} & -1.02 $\pm$ 0.00 \\
& wine & -1.19 $\pm$ 0.00 & -1.19 $\pm$ 0.00 \\
& yacht & \textbf{1.63 $\pm$ 0.01} & 1.30 $\pm$ 0.02 \\
\midrule
& boston & \textbf{-2.39 $\pm$ 0.04} & -2.48 $\pm$ 0.04 \\
& concrete & \textbf{-3.13 $\pm$ 0.01} & -3.18 $\pm$ 0.01 \\
& energy & -0.70 $\pm$ 0.03 & -0.73 $\pm$ 0.03 \\
& kin8nm & \textbf{1.40 $\pm$ 0.01} & 1.38 $\pm$ 0.01 \\
LL & naval & 8.20 $\pm$ 0.04 & 8.15 $\pm$ 0.07 \\
& power & \textbf{-2.77 $\pm$ 0.01} & -2.79 $\pm$ 0.01 \\
& protein & -2.73 $\pm$ 0.00 & -2.74 $\pm$ 0.01 \\
& wine & -0.96 $\pm$ 0.01 & -0.96 $\pm$ 0.01 \\
& yacht & \textbf{-0.46 $\pm$ 0.12} & -0.77 $\pm$ 0.03 \\
\bottomrule
  \end{tabular}
\end{table}

\section{Related Work}

The DWP prior was introduced by \citet{aitchison2020deep}.
However, they were not able to do variational inference with the DWP because they did not have a sufficiently flexible approximate posterior over positive semi-definite matrices.
Instead, they were forced to work with a deep \textit{inverse} Wishart process, which is easier because the inverse Wishart itself is a suitable approximate posterior.
Here, we give a flexible generalised Wishart distribution over positive semi-definite matrices which is suitable for use as a variational approximate posterior in the DWP.
As the deep Wishart process prior is equivalent to a DGP prior, we were able to directly compare DGP and DWP inference in models with the exact same prior.
Such a comparison with equivalent priors was not possible in \citet{aitchison2020deep}, because their deep \textit{inverse} Wishart process priors are not equivalent to DGP priors.

There is an alternative line of work using \textit{generalised} Wishart processes \citep[][as opposed to our \textit{deep} Wishart processes]{wilson2010generalised}.
Note that the ``generalised Wishart process'' terminology does not seem to have spread as widely as it should, but it is very useful in our context.
A generalised Wishart process specifies a distribution over an infinite number of finite-dimensional Wishart-distributed matrices.
These matrices might represent e.g.\ the noise covariance in a dynamical system, in which case there might be an infinite number of such matrices, one for each time or location in the state-space \citep{wilson2010generalised,heaukulani2019scalable,jorgensen2020stochastic}.
In contrast, the Wishart process \citep{dawid1981some,bru1991wishart} describes finite dimensional marginals of a single, potentially infinite dimensional matrix.
In our context, we stack (non-generalised) Wishart processes to form a deep Wishart process.
Importantly, these generalised Wishart priors do not have the flexibility to capture a DGP prior because the underlying features at all locations are jointly multivariate Gaussian \citep[Sec.~4 in][]{wilson2010generalised} and therefore lack the required nonlinearities between layers.
Further, not only do the underlying stochastic processes (deep vs generalised Wishart process) differ, inference is also radically different.
In particular, work on the generalised Wishart performs inference on the underlying multivariate Gaussian feature vectors (Eq.~\ref{eq:def:wishart} e.g.\ Eq.\ 15-18 in \citealt{wilson2010generalised}, Eq.~12 in \citealt{heaukulani2019scalable} or Eq.~24 in \citealt{jorgensen2020stochastic}).
Unfortunately, variational approximate posteriors defined over multivariate Gaussian feature vectors fail to capture symmetries in the true posterior (Sec.~\ref{sec:symmetry}).
In contrast, we define approximate posteriors directly over the symmetric positive semi-definite Gram matrices themselves, which required us to develop new, more flexible distributions over these matrices.

\section{Conclusions}

We introduced a flexible distribution over positive semi-definite matrices which formed the basis of a variational approximate posterior for the deep Wishart process.
We adapted the doubly stochastic variational inference scheme from \citet{aitchison2020deep} to the deep Wishart process.
Thus, we were able to directly compare the performance for inference in a DWP vs.\ DGP with exactly the same prior.
This isolates the effects on performance of the prior and the inference procedure.
We found considerable benefits, both in terms of predictive performance and the ELBO from doing inference in the DWP rather than the DGP.

redpajama_set_name: RedPajamaArXiv


text:
[![Build Status](https://travis-ci.org/dpolansky/ytc.png?branch=master)](https://travis-ci.org/dpolansky/ytc)

ytc is a utility for fetching a YouTube video's comments. Loading all comments for a video manually is tedious and makes it difficult to, for example, see all the comments made by the uploader.

## Installation
```
go get -u github.com/dpolansky/ytc
```

## Usage
```
Usage: ytc [options...] <videoID>

The GOOGLE_API_KEY environment variable must be set, which can be found from https://console.developers.google.com. The 'Youtube Data API' must be enabled.

  -n int
        The number of top-level comments to fetch (replies are always included), -1 (all) by default. (default -1)
  -o string
        The order of the comments, either 'relevance' or 'time' (default "relevance")
```

## Color key
- **Blue**: top-level comments (the start of a comment-thread)
- **Green**: replies to a top-level comment (all replies to a comment are listed underneath their parent)
- **Yellow**: number of likes

![Sample Output](https://raw.githubusercontent.com/dpolansky/ytc/master/sample.png)

redpajama_set_name: RedPajamaGithub


text:
SELECT COUNT(*) FROM sys.databases
WHERE state <> 0


Nice... you're doing good work! I must have told you that the marathon I ran 11 years ago in Alaska was through Team In Training for the LLS, right? Our little team of three raised about $15,000. I believe you even contributed to it!


Exporting is to send goods or services to another country for sale. Exporting is an important aspect of the apparel industry.
In the United States, we important more clothing than we export. However, exporting is still a critical component of the fashion and textile industry.
Manufacturing garments and accessories in developing countries can offer fashion-related companies opportunities to cut fixed costs through reductions in the price of production labor. Overseas employers pay lower wages based on local standards of living and many other factors. Lower labor costs are a key contributor to clothing prices. Countries with lower labor costs often have a higher volume of exports.
Export.gov: helping U.S. Companies export. 95% of the world's consumers live in other countries and the benefits of exporting are enormous. Learn how to export in their Export Basics section or register to access market research and trade leads. Trade Specialists are also available in your area to help plan your international business. Call 1-800- USA-TRAD(E) to find a trade specialist and get answers to your export questions.
The Trade Compliance Center, the TCC, in the U.S. Department of Commerce's International Trade Administration, is the U.S. Government's focal point for monitoring foreign compliance with trade agreements to see that U.S. firms and workers get the maximum benefits from these agreements. The TCC is your one-stop shop for getting U.S. government assistance in resolving the trade barriers or unfair situations you encounter in foreign markets.
The Bureau of Export Administration (BXA) plays an important role in export controls, guarding the national security, foreign policy and non-proliferation goals of the United States, and balancing them against the need of U.S. firms to be viable and competitive in the world marketplace. The BXA oversees the development, licensing and enforcement of export control laws. Their recently overhauled site now includes access to the Export Administration Regulations (by subscription), the List of Denied Parties, Fact Sheets on obtaining Export Licenses, descriptions of their programs, and office location and contact information.
As the DOC's international trade assistance program, the ITA is dedicated to helping U.S. businesses compete in the global marketplace. The site has a number of assistance related workspaces and information resources, and includes the NAFTA Home Page, the Trade Information Centers, and the very helpful Export Assistance Center. The U.S. & Foreign Commercial Service also prints (along with DOS) the extremely helpful Country Commercial Guides which include worldwide contacts, marketing info, best prospects and more. Their site has also recently added a Trade Compliance Center which helps U.S. exporters receive the fullest benefits from the more than 200 trade agreements the United States has concluded. Also, the OTEA has extensive trade statistics.
Learn about government resources that are helpful to the fashion industry.


However you obviously have to eat something. 2. Brushing Your Teeth After Every Meal As an example: Beside these yoga has many more benefits for surfers because it dominates over the surfing workouts within the following approach to supply physique a controlled Parasympathetic Nervous System with controlled sub cortical regions of brain.
One of the best thing about my divorce was not simply being aside from my husband, however not having to be tortured by her. My husband launched his new girlfriend to Cynthia, and she called her a "whore". His new girlfriend was stronger than me. So how does chiropractic help folks with low again pain? To effectively answer this query, one should first identify causes of low again pain. There are various factors that may start low again pain signs and symptoms.
Thought difficult or cognitive restructuring, the process of correcting unfavourable pondering patterns and replacing them with extra optimistic thoughts, involves three steps. 4. Knows how you can monitor development Bamboo Palm—This largely green plant can survive in corners where there's not much light. And that is great as a result of it is excellent at absorbing formaldehyde which may be off-gassing from furnishings or carpet.
2. Lack of physical motion vii) It helps in speedy healing of burns and wounds. 2) Sondergaard D, Meyer O, Wurtzen G. Magnesium stearate given perorally to rats. A brief term study. Toxicology 198 17(1) 51-fifty five. Tel: 020 8952 2800E-mail: [email protected]: * Flexibility and joint vary of motion Pole dancing is the brand new exciting technique to preserve fit. Lap dancing exercise health poles are fast being sold to homes, dance clubs and health centres across the UK.
A moist or dirty nappy that has been left touching the skin for too long Arthritis: Adding using treadmills to your exercise program has advantages. A treadmill will provide you with an superior cardiovascular workout for your coronary heart, your lungs, and your circulation. There may be less pain and it?s easier while you exercise when there is low influence on your joints.


text:
It's The Economy
It's been three decades since a presidential race was won based largely on the slogan, "It's the economy, stupid." And once again, a troubled economy has become a major driver in U.S. politics. Americans are struggling with a pandemic shutdown hangover, mind-boggling inflation, shocking gas and grocery prices, and shortages of both workers and products. Today in our cover story, straight talk from two experts with opposing views on a lot of topics, but agreement that we're not close to out of the woods, yet.
Back in 1992, there was a recession, and Bill Clinton was running to unseat President George H.W. Bush. Clinton adviser James Carville made the economy the linchpinof the campaign, coining the phrase: "The economy, stupid," which he later explained in an interview.
James Carville: (C-SPAN interview, July 13, 2012) In '92, we thought the economy should be the focus of everything we did went. What I was trying to get the campaign more focused, get the people in the campaign more focused. And it was like: 'It's the economy, stupid.'
Clinton won.
Today, nearly thirty years later, the economy is again shaping up to be a political game-changer. Polls show the economy was the top issue on voter's minds in the recent upset in the Virginia governor's race.
In a state that has far more registered Democrats. The Republican, Glenn Youngkin, somehow beat the Democrat, former governor Terry McAuliffe,
But to whose advantage will the economy prove to be in the 2022 Congressional elections? And, most importantly, what's the outlook for your personal finances?
Wendy Edelberg: It's a mixed bag, there's a lot to be very excited about. The economy is growing quickly, but it's coming out of a very deep hole and the recovery from the pandemic-induced recession has been nothing short of chaotic.
Tyler Goodspeed: We seem to be plateauing at levels that are still well below where we were pre-pandemic, at the same time the inflation is at some of the highest levels that we've observed in recent decades.
We get dueling assessments from Tyler Goodspeed, who headed up the White House Council of Economic Advisers under President Trump, and Wendy Edelberg, once the top economist at the Congressional Budget Office.
Sharyl: Tyler, when I say "supply chain," which is something I think most Americans haven't spent much time at all thinking about until fairly recently, what thoughts come to your mind?
Goodspeed: I think the pressures are going to persist for the next few quarters and perhaps a bit beyond. Some of the policy choices that have been made that I think have exacerbated some of the labor shortages and increased demand at the same time that supply remained somewhat constrained.
Sharyl: Specifically to supply chain, what do you see in the next four to six months?
Edelberg: If we're just talking about the container ships, we need resolution all the way across that system. This is a near term problem. This is a three month problem, a six month problem.
Sharyl: Tyler, do you think the supply chain issues overall, were largely just entirely unavoidable or do you see that different policies would have or could have mitigated some of this?
Goodspeed: I think that the continuation of enhanced federal unemployment insurance benefits had probably modest effect, but nonetheless, a net negative effect on the return to work of a lot of American workers. In July, we had the rollout of the new Child Tax Credit, the design of which, at the margin, discourages the return to work. Policy has certainly exacerbated some of these supply side problems that were there already present.
Edelberg: I think we actually now have a whole lot of evidence in hand, that the expiration of the Unemployment Insurance benefits did very little in aggregate to spur more people to come back into the labor market. But one reason why we know that it's not US fiscal policy, that has driven these supply constraints, is that they're happening globally. We're seeing supply constraints in Asia, we're seeing supply constraints in the UK, we're seeing supply constraints in Australia, we're seeing inflationary pressure across the globe.
Sharyl: We had become what most people consider energy independent, I think starting around 2017 or 2018 or so, no longer are we, and we are now asking or talking with Russia, Saudi Arabia and OPEC nations about their output to try to help ease pressure on our gas prices. What's going on with energy? And what can we expect in the near future?
Edelberg: So I would still characterize us as largely energy independent. We have actually an enormous capacity to ramp up our own oil and natural gas production in this country. And I think that if we see oil prices and natural gas prices remain elevated, for any longer than what we've already seen, I would fully expect our production to start to respond and start to respond pretty quickly.
Sharyl: Tyler?
Goodspeed: I agree with Wendy that historically, the US domestic energy production has been pretty nimble. I think that has been throttled to a certain extent in recent months, because unfriendly policy is staring them in the face.
Sharyl: What's that?
Goodspeed: Well, that's first of all, competition down the road from heavily subsidized non fossil fuel based sources of energy. So in the near term, I think this could be an adverse energy supply shock, not like the 1970s but probably more than we've seen in a few decades.
Sharyl: Tyler, looking ahead, going into Christmas, there may be a significant number of vaccine mandate related retirements and firings. What do you see as the impact of that on our nation and the cost of that?
Goodspeed: We're actually still 5 million jobs shorter where we were in February 2020. We're also 8 million jobs short of where we ought to be or would have been if we have continued at the Pre-pandemic pace of jobs growth. So, if I were in policy now, I would be focusing on measures to incentivize return to employment as quickly as possible so that we can keep that labor market recovery on track.
Edelberg: People are scared of the pandemic, and the more we can get people vaccinated, the less scared people will be. So I think worrying about how mandates, vaccine mandates, will constrain labor supply, misses something that I think is quite significant on the other side of the ledger.
Sharyl: The economy as a political issue in the midterm elections, what do you say about that?
Edelberg: The economy is always a political issue. So there's nothing particularly surprising there.
Sharyl: Does that help or hurt one side or the other, do you think? Where we are now if it stays about the same?
Edelberg: Oh, geez, I'm going to ask Tyler to answer that first.
Goodspeed: I think it's a negative for the party that occupies the White House and has majorities or effective majorities in both houses of Congress.
Edelberg: Alright, let me take a shot at this. Acknowledging that there's a huge amount of uncertainty, it is quite possible that a year from now, we will have largely recovered from the pandemic, knock on something. And we will have logged a bunch of quarters of very strong growth, we may well be looking forward to a slowing, as the economy has to have a glide path back toward what is a reasonable state of growth.
Sharyl: What is something significant economically that you think was done right under the Trump administration, something very important that comes to mind?
Goodspeed: I think the 2017 tax law was absolutely pivotal in transitioning us from what had been a historically slow economic recovery with historically slow real income growth to a state of affairs, that on the eve of the pandemic, in 2019 alone, we observed real income gains for the typical American household that were more in one year than in the entire 16 years through 2016.
Edelberg: I do think one of the most extraordinary and positive things that happened in that four year span is that policymakers generally allowed the economy and pushed the economy to grow quite fast, in a way that actually worried me somewhat at the time. but turned out that we were able to grow that quickly, without spurring really unwanted inflation. All of those great benefits of strong growth, I would pin far more on monetary policy, and what Congress was doing with defense and non-defense spending than anything having to do with the tax act.
Sharyl: Is there something very important and critical you can point to, that has been done under President Biden that's been positive for the economy?
Edelberg: Whoa, wow. So I think much of what was in the American Rescue Plan was very, very positive for the economy. I think it provided a significant amount of fiscal support to ensure that we would have a strong recovery.
Goodspeed: One of the good things about the relief package was the funding and support it provided for rolling out vaccine distribution and for helping schools to safely reopen.
Sharyl (on-camera): When looking at the Covid time period, a Gallup poll says we recently reached a new high in terms of people mentioning an economic issue as the top problem facing the U.S.: 26%.

redpajama_set_name: RedPajamaCommonCrawl


Warning: The following post features ferocious meat lovers. If you are tender, juicy, or have a tendency to hop, they might eat you. Read at your own risk.
The Great White Shark is Sovereign of the Sea!
The Timber Wolf is Emperor of the Forest!
PRINCE OF THE PICTURE BOOK!!
It wasn't enough that back in 2005, Aaron spiced up my ho-hum existence with Chicks and Salsa. No. He got me to wiggle my wattle and actually tolerate football with Buffalo Wings in 2007. Did he stop there? Not a chance.
Last year he terrified me with a bunch of Creepy Carrots, but I've since forgiven him because at least now I know the chopped salad I've been smelling under my pillow is real.


When transitioning to a DevOps model, organizations must remember that people are essential to a successful switchover. It's people who must learn new workflows, collaboration techniques, and tools during the move. This process will cause at least some disruption over a period as long as two years. Needless to say, they will need patience and ample support to get through such a substantial change.
Fortunately, organizations can help their people acclimate to the new way of doing things. They can start by identifying key stakeholders and pitching their plan for the transition. By listening to that feedback, enterprises can make sure everyone is on the same page and shape their plan accordingly to avoid common obstacles during the transition process.
Here are five problems in particular to avoid when changing over to DevOps.
Organizations need a firm idea of why they're implementing DevOps and what they hope it will mean for the company. In other words, they need a goal in mind against which they can measure the progress of and propose changes to the transition. Companies can achieve this end by asking development, operations and QA to fill out a questionnaire in which they're given an opportunity to define DevOps, express what privileges they're willing to lose and/or take on as well as propose boundaries that they feel should separate DevOps systems from the rest of the organization.


Here are the dates – we hope to see you at one of these. Come and see us between 5.30pm to 7.30pm.
Complete the form below, confirming the branch you would like to attend, and we will confirm this with you shortly.


text:
Eminem & Dr. Dre to Contribute New Music For 'Bodied' Soundtrack
blame it on JES7 September 10, 2017
Last month, we reported that Eminem would be reuniting with esteemed film and video director Joseph Kahn for a new satire film exploring the underbelly of battle rapping.
The film, directed and co-written by Kahn and Alex Larsen (aka real life battle rapper Kid Twist) and produced by Eminem, was recently premiered at the Toronto International Film Festival, with a cast which includes Dumbfoundead, Charlamagne Tha God, Calum Worthy, Rory Uphold, Anthony Michael Hall and battle rap vets Dizaster, Hollow Da Don and Loaded Lux.
Now, according to The Hollywood Reporter, the film's soundtrack will feature a "number of songs" from Eminem, in addition to enlisting "Dr. Dre for a couple of songs." And with Eminem and longtime business partner Paul Rosenberg serving as the film's music supervisors, it's safe to say we're in for a real treat.
Eminem & Dr. Dre to Contribute New Music For 'Bodied' Soundtrack was last modified: March 31st, 2018 by JES7
Tags:dr. dre, eminem
Quan – "This Fire" (Video)
Therman Munsin – "I Ain't Wit The Evil Empire" f. Big Twins
The first single of 2022 from the songstress.


The image sees.The image feels.The image acts. (Bennett, CB, 195)The image gives.The image is given.The image proliferates.The image betrays.The image for gives.The image is for giving.The image is for exposition.The image is for beauty.The image is from the good.The image is mother, and is father, is both mother and father, and neither mother nor father; for it is the child. The image is the parent, and the children, both parent and children, and neither parent nor children.


text: Q: how does android gesture builder work? What is the technology behind android's GestureBuilder tool, that allows you to draw a varieties of gestures and later able to recognize user gestures as one of the gestures you've drawn? Some links and pointers would be greatly appreciated!!

redpajama_set_name: RedPajamaStackExchange


Video footage recorded by Humberside Police's helicopter reveals the moment a "violent" attacker hurled tiles from a rooftop in a stand-off with officers.
Clint Clay, 24, , of Frodingham Road, Scunthorpe, assaulted Jason Potts with a machete and a hammer before taking to the roof of a house on Alexandra Road in Scunthorpe on 6 July last year.
He denied assault and three other charges but admitted affray, and was sentenced at Grimsby Crown Court on Monday.


Samajwadi Party (SP) chief Akhilesh Yadav will file his nomination papers from Azamgarh parliamentary seat on April 18 and later address a public meeting in Baitholi village along Azamgarh-Mau highway. Local SP leaders have intensified the preparations for the nomination day with the confirmation of Akhilesh's schedule.
They are buoyed by the fact that four-time MP Ramakant Yadav quit BJP and joined the Congress party on Friday. BJP has fielded Bhojpuri singer and actor Dinesh Lal Yadav Nirahua against Akhilesh from the constituency, currently represented by the latter's father and SP patriarch Mulayam Singh Yadav.
He said that the party along with alliance partner Bahujan Samaj Party (BSP) has completed all poll preparations in all assembly areas of Azamgarh parliamentary constituency.


text:
Home » Popular » +2468 [Image] Motivate yourself
+2468 [Image] Motivate yourself
B-L-O-C-K-S
+2468 – [Image] Motivate yourself
+1825 Renovated flat with detailed stucco...
+1917 chest workout
ThinkRichGrowRich March 30, 2022 at 1:37 am
Unfortunately this is true. The cavalry isn't coming.
ACuteMonkeysUncle March 30, 2022 at 1:37 am
My dog growing up loved holding the leash in her mouth. I think it's because she (rightly) saw the leash as a form of power, and when she lad the leash, she had the power. So, while it's kind of weird, it kind of makes sense too.
imeatingpizzaritenow March 30, 2022 at 1:37 am
It's ok to ask for help too 💛
Harvey_Wongstein March 30, 2022 at 1:37 am
So true as a lonely single man in his late 20s
Burnermcfakename March 30, 2022 at 1:37 am
Can I be a dog
furiousfran March 30, 2022 at 1:37 am
That dog is still going to need someone to open doors for him. Can't do everything all yourself all the time.
Deegedeege March 30, 2022 at 1:37 am
Or, this breed of dog simply loves to carry something in it's mouth. It's mouth is its hands.
Epicwindow March 30, 2022 at 1:37 am
the most reliable one,is yourself,but don't forget to rely on others too
HiramNinja March 30, 2022 at 1:37 am
…every. damned. day.
My ammends aren't coming, because the people they'd be coming from have either died or don't think they did anything wrong. So, I tell the Universe to apply those ammends to my outstanding karmic debt, and march along, knowing that, perhaps I will skip the dung beetle stage of reincarnation and move up to something better, like a tree or a nice flower.
bigmikemcbeth756 March 30, 2022 at 1:37 am
iormangund March 30, 2022 at 1:37 am
Anyone else lost their leash?
LukeNew March 30, 2022 at 1:37 am
This is me trying to learn calculus after being out of education for 15 or so years, while my gf struggles with both her dad and aunt dying of covid and cancer. Trying to be strong but it's hard. She's a mess. Don't know what to do. Pushing forward but I'm so tired because of sleep apnoea(which I'm hoping will be fixed soon)
GiantRetortoise March 30, 2022 at 1:37 am
This is borderline toxic. Everyone needs help sometimes, and you should not be afraid or ashamed to ask for it.
feraicrag March 30, 2022 at 1:37 am
Adulthood sucks, this post helped me smile.
Sometimes shitty weeks last a couple of months, nobody notice your efforts but what else you could do?.
6CummyBunny9 March 30, 2022 at 1:37 am
No dude this doesn't motivate fucking anything holy shit
PhantomCAT_MEOW March 30, 2022 at 1:37 am
***fuck.***


Four years in the making (since the previous version).
One hundred eighty Proposals for Change (PFCs), of which half were ranked four or five in a scale of one to five (ascending order).
Six new protocols (for a total of 33 individual Chief Complaints in the system).
The Fire Priority Dispatch System™ (FPDS®) has come a long way since version 1.0 released in August 2000. And as the numbers show, it's the people in emergency dispatch who continue to generate the advances.
Two new protocols cover situations never before addressed in FPDS: outdoor misadventures.
The protocol is specific to emergencies involving injury or death or the inability to self-rescue because of conditions, such as climbers trapped in a crevasse, single or multiple skiers trapped in an avalanche, or spelunkers lost in a cave.
Nearly every backcountry search begins with a 911 call, either from the missing person or other people, someone familiar with the group's or individual's intentions, family members, or someone reporting a vehicle parked for a long time at the trailhead.
The new protocols include questioning about a person's outdoor/backcountry experience. After all, as most seasoned outdoor enthusiasts will admit, it's not only the novice who gets lost. Most people who spend a lot of time in the wilderness have confused their whereabouts at some point. Collecting information early on when people go missing and transmitting information as it changes are critical steps in getting them to safety. Expediency limits the distance the person can go, and decreases the likelihood that weather and other factors will obliterate the clues.
A third new protocol focuses on bad weather and natural/manmade disaster situations. Protocol 83: Weather/Disaster Situations prioritizes helping individuals trapped by a hurricane, tornado, earthquake, flooding, or other weather-related hazards. PDIs give emergency dispatchers the information to relay when the fire department is in disaster mode ("I cannot tell you when the fire department will get there.") and when EMS is overwhelmed and cannot immediately respond to callers ("If it's safe to do so, try to find transportation to the hospital/doctor. Call us back if you find transportation.").
In addition to structural fires and collapse, electrical hazards, and explosive gases and liquids during large scale emergencies, fire departments can encounter lapses in command and control, operating procedures, and shortages of equipment.
The new protocols are not cookie-cutter response approaches, and they also consider the potential risk to the public and safety of responders.
Three protocols new to version 7.0 are derived from existing protocols to heighten the urgency of these situations as shown in their own unique set of Key Questions, PAIs, PDIs, and Determinant Codes.
Similar to the reasoning behind outdoor misadventure and weather/disaster protocols, these protocols address the singular critical nature of unpredictable events.
The protocol provides a level of response based on the size of the tank—LARGE and SMALL—with size determinations for gas, chemical, and fuel storage units left to local definition. Suffixes further delineate what the fire threatens (such people, animals, buildings) and the number of people injured.
Protocol 81 does not replace Protocol A: Water Rescue. The Chief Complaint includes questions for sinking vehicles and vehicles in floodwater and instructs the emergency dispatcher to suspend questioning when necessary to give PDIs. It also provides reminders to determine a specific and clear meeting point for emergency crews.
Revised Protocol 72: Water/Ice/Mud Rescue eliminates Key Questions, Determinant Codes, and Additional Information sections involving a vehicle in floodwater or sinking vehicle and adds a CHARLIE-level Determinant Code for an individual stranded in a building/structure due to flood. All vehicle in floodwater and sinking vehicle questions and instructions are now part of the new Protocol 81.
Protocol 82 addresses one of the most pressing environmental problems: vegetation, brush, wildland, and grass fires. Key Questions include asking the caller the size of the area burning (and if the caller struggles to determine the size of the fire, the emergency dispatcher asks them to relate the size to a familiar area, such as a football field). The fire's size, the type (wildland, brush/grass), and structures involved help determine levels of response. Emergency dispatchers coordinate with EMS when people are in immediate danger and/or injured.
One hundred years of fire suppression leading to denser vegetation/forests and fires that burn with greater intensity.
A less aggressive strategy on large fires for safety reasons.
Fires allowed to burn naturally without full suppression for environmental concerns.
Changes in the initial attack, responding with less equipment and personnel.
A notable change involves PDIs in sheltering situations during fires that cause a direct threat to life and health. Callers are advised to get out, when possible, rather than shelter in place. The instruction applies to more than one overall situation, considering variables such as the type of building, type of fire (wild fire, residential, outside tank fire, high rise), and the people involved (such as someone with a disability).
Telling people to get out without waiting for response can save lives, Thompson said.
Deciding whether to shelter in place or evacuate to safety (i.e., get away from a threat or hazard) is among the most important decisions that must be made during an emergency. In any emergency, local authorities may or may not be able to provide information immediately to assess the situation or may lack the resources to safely rescue victims.
FPDS v7.0 affects nearly every protocol, with the addition of Chief Complaint Selection Rules, Definitions, and Determinant Suffixes, along with changes in Case Entry, Key Questions, Determinant Descriptors, PDIs, and a new ECHO-level Determinant Codes (Threatened/TRAPPED by WILDLAND fire, 82-E-1). DLS instructions were expanded with detailed instructions for Building Evacuation and Health/Life Safety, Backcountry Hazards, USAR/Vehicle and Other Hazards, and Wildland Fire Hazards.
A new purple font in Key Questions identifies jurisdictionally approved questions. The local Fire Administration/Agency determines whether these questions should be asked. For example, Protocol 55: Electrical Hazard, Key Question 8 "Which electrical/power company provides service to this location?" is presented in purple font.
1"Disaster Prevention." United Nations. http://research.un.org/en/disaster/un-resources (accessed Aug. 13, 2018).
2"Weather-related disasters are increasing." The Economist. 2017; Aug. 29. https://www.economist.com/graphic-detail/2017/08/29/weather-related-disasters-are-increasing (accessed Aug. 13, 2018).
3"Steady Increase in Climate Related Natural Disasters." AccuWeather. 2019; Nov. 15. https://www.accuweather.com/en/weather-blogs/climatechange/steady-increase-in-climate-rel/19974069 (accessed Aug. 13, 2018).
4Shelley C. "Storage Tank Fires: Is Your Department Prepared?" Fire Engineering University (PenWell). https://www.fireengineering.com/content/dam/fe/online-articles/documents/FEU/Shelley-Nov08.pdf (accessed Aug. 13, 2018).
5Pierre-Louis K. "This is how much of the world is currently on fire." Popular Science. 2017; Aug. 4. https://www.popsci.com/global-wildfire-maps (accessed Aug. 14, 2018).
6Gabbert B. "Visualizing California fires over the last 18 years." Wildfire Today. 2018; Jan. 10. https://wildfiretoday.com/2018/01/24/visualizing-california-fires-over-the-last-18-years/ (accessed Aug. 14, 2018).
Audrey Fraizer is Managing Editor of the Journal, and is poster child for an editorial personality. She has a focused streak difficult to distract, calls library research a hobby, and believes she fools her co-workers into thinking she's listening when she's actually not.