According to leading Miami urologist Dr. George Suarez, the most recent innovation in the detection of prostate cancer is magnetic resonance imaging (MRI) of prostate and ultrasound biopsy, both which provide numerous benefits for both the patient and the urologist.
According to the National Cancer Institute estimates, in 2014 we'll see almost a quarter of a million new cases of prostate cancer, and almost 30,000 deaths from the disease.
Your Miami urologist advises that this new technology offers a more targeted biopsy, which refers to direct tissue sampling of suspect areas observed on the MRI. This is as opposed to the traditional method of random, systematic sampling that is basically performed sightlessly in various regions of the prostate.
We are utilizing MRI of prostate and fusing it with real-time ultrasound to locate prostate cancer with a higher degree of accuracy. This improves overall detection compared to standard biopsy and has the potential to give urologists and patients a more accurate image of their true disease burden by allowing improvements in staging.
Research of this new technique have indicated that it improves the overall cancer detection rate, elevates the high-risk cancer detection rate, and improves staging for patients who are considering active surveillance to monitor their low-risk prostate cancer for any changes.
According to your Miami urologist, the technique is expected to be especially successful in cases of men who have a history of negative biopsies and are still suspected of having prostate cancer because of a baffling elevated prostate-specific antigen (PSA) level.
A patient can undergo numerous biopsy sessions over a number of years, and have a consistently elevated PSA, yet each biopsy can be negative. But with MRI of prostate, we are able to target areas that we identified as areas of concern to reveal the condition that has been causing the elevated PSA that was hidden in all the previous biopsies.
This MRI of prostate is the first major advancement in prostate cancer detection in over 30 years, and it's a significant one.
If you are interested in more information regarding MRI of prostate contact the office of leading Miami urologist Dr. George Suarez to schedule a consultation.


The New Indoor 3W LED Wall Lamp is adoptable to any given conditions, like lighting up homes, offices. We can offer many different shapes and colors to meet any lighting requirements you may have. For usual house lighting, "cool" LED light bulb is excellent to be used in task lighting while "warm" LED light bulb is good to be used in accent illumination.


Researchers from the Miller School of Medicine have identified a gene associated with Alzheimer's disease in Caucasians. Published online October 27 in Neurobiology of Aging, their study demonstrated that a change in the C9ORF72 gene, called a repeat expansion, could be a new cause for late-onset Alzheimer's disease, mostly in Caucasians.
The research was led by Margaret A. Pericak-Vance, Ph.D., Director of the Hussman Institute for Human Genomics, and Stephan Züchner M.D., Interim Chair of the Dr. John T. Macdonald Foundation Department of Human Genetics. Dr. Pericak-Vance's post-doctoral fellow, Martin A. Kohli, Ph.D., was the first author on the paper, Repeat expansions in the C9ORF72 gene contribute to Alzheimer's disease in Caucasians.
"We are excited about this research finding because it solidifies that rare genetic variants can contribute to risk in late onset Alzheimer's disease," Pericak-Vance said.
Previously, researchers identified a stretch in DNA, in this case an intronic hexanucleotide repeat expansion in the C9ORF72 gene, which accounted for a significant portion of families affected by neurodegenerative disorders, frontotemporal dementia and amyotrophic lateral sclerosis (ALS). Healthy individuals appear to have 0 to about 20 repetitions of this stretch of DNA, whereas at-risk patients have more than 30 expansions, up to a thousand. The pathogenic range of expansions interferes with expression and thus proper functioning of the C9ORF72 gene.
Since causes for neurodegenerative disorders have been found to overlap, researchers in this study hypothesized that DNA stretches in the C9ORF72 gene could also cause Alzheimer's disease in some individuals. They examined more than 1,100 patients with Alzheimer's disease and found that approximately one percent of the cases also had large expansions in the C9ORF72 gene, whereas none of the controls had large expansions. In contrast, no large expansions were detected in patients of African-American ethnicity. Therefore, repeat expansions in the C9ORF72 gene appear to be a new cause for Alzheimer's disease, but only in Caucasians.
Moreover, these ethnic differences in the prevalence of C9ORF72 expansion carriers may provide further insight into the underlying molecular mechanisms of this genetic cause of neurodegenerative disease in both Caucasians and African-Americans.
Other UM researchers included Krista John-Williams, M.S., research associate; Patrice Whitehead, B.S., director of research laboratory; Kara Hamilton, M.P.H., project manager of research support; Gary Beecham, Ph.D., assistant professor of human genetics; Eden R. Martin, Ph.D., professor of human genetics; John Gilbert, Ph.D., professor of human genetics; Michael Benatar, M.D., Ph.D., associate professor of neurology; Clinton Wright, M.D., associate professor of neurology; Deborah Mash, Ph.D., professor of neurology; Regina M. Carney, M.D., assistant professor of clinical psychiatry and behavioral sciences; and Elizabeth Crocco, M.D., assistant professor of clinical psychiatry and behavioral sciences.
They were joined by Harry E. Gwirtzman, M.D. (Vanderbilt), Rosalyn Lang, Ph.D. (North Carolina A&T), Gary W. Small, M.D. (UCLA), Goldie Byrd, Ph.D. (North Carolina A&T), and Jonathan L. Haines, Ph.D. (Vanderbilt).
Margaret A. Pericak-Vance, Ph.D., and Stephan Züchner, M.D.


Masseduction is the fifth studio album by American musician St. Vincent. It was released on October 13, 2017, through Loma Vista Recordings. The album peaked at number 10 on the Billboard 200, becoming her first top ten album in the United States.


You can contact us vis telephone, email, online chat or skype. If you wish to receive a quote please fill in our medical form enlosing your most recent photos. The photos should be taken from a medium range distance from front and side views.
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With the competition season over Cheryl Maas and Sarka Pancochova head to Sarka's home base of Colorado. Little did they know it would dump with epic snow conditions on arrival! Roll out the sleds, snowshoes and powder boards for one last powder blast up Vail Pass before some slushy park laps in Breckenridge to round out the winter.


Non-executive Directors: Independent of management, our Non-executive Directors bring diverse skills and experience, vital to constructive challenge and debate. Exclusively, they form the Audit, Nominations and Remuneration Committees, and have an important role in developing proposals on strategy.
Career and experience: Paul is a Chartered Engineer with a doctorate in mechanical engineering, Fellow of the Royal Academy of Engineering, the Institution of Engineering and Technology, the Institution of Mechanical Engineers, the Energy Institute and City of Guilds. He was awarded honorary degrees from Aston University and Cranfield University. Paul was an Executive Director of Clayhithe plc, before going on to join E.ON UK plc where he was Chief Executive and later Chairman. Paul also held previous appointments as Non-executive Chairman of AEA Technology Group plc and Chairman of EngineeringUK.
Skills and competencies: Paul has a lifelong passion for engineering and innovation, and has spent his career in the energy, governmental and regulatory sectors. He brings the skills of an experienced Chairman and Chief Executive to his role as a Non-executive Director. He adds a valuable perspective to debates on UK regulatory and strategic issues. His deep understanding and specific experience in safety and risk management is crucial to his role as Chairman of the Safety, Environment and Health Committee.
External appointments: Chairman of Costain Group plc, the UK National Air Traffic Services and a member of the Prime Minister's Council for Science and Technology.
Committees: Chairman of the Safety, Environment and Health Committee, member of the Audit and Nominations Committees.


text:
Q: O que significa #document? Boa noite pessoal, estou analisando um código fonte para fazer um parse e me deparei com um trecho do código em que aparece um #document porém nunca vi isso na vida, dei um Google aqui mas não encontrei nada referente. Alguém sabe o que é isso?

A: Este #document é um "elemento virtual", ou seja, é uma forma específica do seu browser te indicar que é parte de um DOM (incorporado) diferente. Não é possível utilizá-lo ou consultá-lo, pois o "elemento virtual" não existe como parte do DOM do código.
Em resumo geral o #document não significa realmente nada!

redpajama_set_name: RedPajamaStackExchange


Groundwork Creswell, Ashfield and Mansfield has been working closely with Local Authorities for over 20 years. A partnership with parish, city and borough councils means we can deliver a range of projects to improve communities and bring significant change to local areas.
The Landscape and Design team can help communities from the initial project proposal. If you have an idea, we can help make it happen. We are able to offer a complete management package tailored to your individual project; from initial consultation and raising funds to project completion.
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Groundwork has 20 years of experience gained in landscape design and design-build projects. These range from small-scale community gardens and playgrounds up to substantial derelict land remediation and townscape improvement projects. We work on a huge range of projects including urban renaissance, living spaces, working spaces, learning environments and projects that promote biodiversity and protect our heritage. Our landscape architects have a wealth of experience and range of expert knowledge that can apply to each projects aims and objectives.
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text:
Q: How to avoid empty rows in SSIS Excel Destination? Does anyone have a way to avoid empty rows when using SSIS to export to Excel. Here's a simple example of one data flow task:
OLE DB Source:

Data Conversion (to handle the annoying UNICODE / NON-UNICODE deal):

The end result is either of the two below depending on value of "FirstRowHasColumnName" in the Excel Connection Manager. Note, the blank rows.

A: I tested this in SSIS from a 2008 R2 install and Excel 2010.
Using the Excel 97-2003 output format in the connection manager, I was able to reproduce the case where there was the header name in the first row, then a blank, then the data. But changing the "First row has column names" had a different effect for me: I still got the header row, but no blank.
Using the Excel 2007 format, I wasn't able to suppress the header row either, though in both cases, it didn't output a blank row.
You didn't say if you want the header row or not, or which output format you're using; based on my testing, you're using 97-2003, so all you might have to do is switch to the 2007 format.
Attaching a Data Viewer just before the Excel Destination shows only data rows, so the extra blank is being added by the Destination component. So if the Destination isn't configurable to get rid of the blank, you're out of luck.
Suffice to say, though, since the behaviour of this component seems very buggy and the whole setup process to even get this to work at all was such a pain, I would strongly consider switching to a more stable component like the Flat File Destination to output CSV, which you can open directly in Excel, too.

A: I ran into similar issue recently, what worked for me was to trick the Data Flow Task by adding a File System Task to copy an empty file with same table structure. Here is my 3 step solution:

*

*I created the Data flow task with the excel destination and parameterized connection string.

*I created a folder which I named "Default folder" and copied the empty excel destination created in step 1.

*I added a File System Task to copy the default file from [Default folder] into my designated destination folder with an overwrite if the file exists.

A: I ran into the same issue (SSIS 2012 and Excel 2007 format)
I had my original header line, a bunch of empty lines and then the exported SSIS data.
I resolved the issue by selecting the whole spreadsheet except for the title row and by deleting the selection (right-clik and choose delete). This must have removed any left-behind formatting of blank cells.
Then I put my selection to cell A1, saved the empty xlsx file and now SSIS starts writing the output on line 2.

A: I had same issues with the blank rows in excel being inserted.  I ended up creating a brand new excel file with the same headers.  Somehow the old file had some formatting in place that was not visible to me that I could fix in order to remove the blank space.


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text:
"Pedant Needs More Paunch": Reviving Robertson Davies's Annotations from his Performance Copy of The Taming of the Shrew
Cathleen Mckague+−
Director Robertson Davies's 1949 production of The Taming of the Shrew with the Peterborough Little Theatre won the Louis Jouvet Trophy for best direction at the Dominion Drama Festival in Toronto, garnering critical praise as the DDF's first full-length Shakespearean play. Davies's performance copy for this production has recently been discovered in his personal book collection, donated to Queen's University's W. D. Jordan Special Collections and Music Library. Never previously examined or transcribed, this text incorporates director's notes and critiques, music cues, blocking, omissions, and alterations, offering invaluable insights into Davies's 1949 Elizabethan interpretation of this play. Reviving Davies's annotations reveals new information about this specific production, one that challenged mid-twentieth century Canadian staging conventions. PLT's Shrew emerges as a lighthearted and detail-oriented comic romp invested in historicity, its Commedia dell'Arte flavour likely imported from Tyrone Guthrie's 1939 Old Vic production in which Davies acted. The mentorship model that Davies implemented with PLT he replicates in his first novel, Tempest-Tost (1951), where he draws directly upon his Shrew notes and directorial experience as well. Revising the "maturation myth" narrative of Canadian theatre, McKague's study of Davies's play situates nonprofessional theatre as intrinsic to the fabric and history of Canadian culture.
Mckague, C. (2019). "Pedant Needs More Paunch": Reviving Robertson Davies's Annotations from his Performance Copy of The Taming of the Shrew. Theatre Research in Canada / Recherches théâtrales Au Canada, 39(2). Retrieved from https://journals.lib.unb.ca/index.php/TRIC/article/view/28775

redpajama_set_name: RedPajamaCommonCrawl


text:
BrianHicks / elm-string-graphemes
BrianHicks/elm-string-graphemes
GitHub makes it easy to scale back on context switching. Read rendered documentation, see the history of any file, and collaborate with contributors on projects across GitHub.
elm-string-graphemes/README.md
BrianHicks add "why a drop-in replacement" question 46805a5 Jul 5, 2019
1 contributor
Users who have contributed to this file
146 lines (99 sloc) 7.7 KB
Graphemes
Do string operations based on graphemes instead of codepoints or bytes. Compare:
import String.Graphemes
String.toList "🦸🏽♂" --> [ '🦸', '🏽', '\u{200D}', '♂', '\u{FE0F}' ]
String.Graphemes.toList "🦸🏽♂" --> [ "🦸🏽♂" ]
What's going on here? Graphemes? What're those?
Unicode defines a system for encoding text as symbols called codepoints. For example, a is codepoint 97, usually written in hex as 0061. There are a wide range of possible codepoints (from 0000 to 10FFFF), although not all of those numbers are assigned a symbol.
A codepoint takes between one to four bytes in memory, both to maintain compatibility with older documents and to save space. For example, a can be encoded in one byte but 🦸 takes four. If they didn't vary in length, you would have to pad out a with 3 bytes worth of zeros just to support both in the same string!
Some codepoints stand on their own (like a), but others need to be used in combinations. We call these combinations "graphemes", and they map to what you'd intuitively think of as a single character in a writing system. So, for example, you can put a and ¨ together to get ä. You can do this for all kinds of diacritics: ´, ˜, ˆ, et cetera.
If you get really wild here, you end up with z̴̙͒ả̴̫̼̫̀̅ĺ̴̔̿͜g̷̨͇͉̊͐̚o̶̳̣̯͌̓!
But this system isn't just for diacritics. Tons of writing systems use graphemes: Hangul, Devanagari, Thai, and Tamil among others! An easy one to talk about for English speakers is emoji: 🦸🏽♂ is composed of 🦸 + 🏽 + zero-width joiner (200D) + ♂ + variation selector 16 (FE0F). You tend to think of 🦸🏽♂ as a single character—a very definite expression which can't really be broken up into constituent parts. That means it's a grapheme! But if you used 🦸 by itself it's a grapheme too. The point is not "what codepoints are there?", it's "what is the smallest useful unit when writing?"
The above means that when we ask questions like "how long is this string?" or "what is the first character here?" we sometimes mix three levels:
the byte level. Operations like String.length and String.left operate here. You should operate here if you're working with ASCII or ISO 8859-1 data, but working at the byte level with unicode text is a good way to break your document.
the codepoint level. Here, our base superhero emoji is only one character, but our skin tone and gender (🦸🏽♂) take more like we discussed. This combination happens to be 17 bytes but only 5 codepoints. Operations like String.foldl operate here (so you can measure length with operations like String.foldl (\_ len -> len + 1) 0.) You should operate here if you're implementing higher-level operations on the codepoints, like grapheme segmentation (hi!) or normalization.
the grapheme level. Despite being 5 codepoints, 🦸🏽♂ is only one grapheme. Operations like String.Graphemes.toList operate here. You should operate here if you're modifying unicode text in ways meaningful to a user.
To underscore, if you're modifying text that the user has entered, work at the grapheme level. This reduces the possibility of errors and increases the possibility that your program will "do the right thing."
Still not convinced? Here are some practical reasons you should work at the grapheme level in the browser:
If you operate at the byte level, you will split multi-byte characters into invalid unicode sequences. If you do the wrong thing with these sequences, you'll crash your user's browser. In fact, that's what started me writing this library! Everyone does it occasionally, but there are better ways.
If you operate at the character level, you will split skin tones and genders off of people emoji, split flags into country codes, and move diacritic marks around. Your user entered this text precisely in these cases, don't lose their meaning!
Think you don't need those pesky diacritic marks? Think again! They're crucial to understanding in a lot of languages! For example, in Spanish, papa (potato) is different than papá (father.) Don't make your users call their dad a potato!
What spec does this package implement?
The Grapheme Cluster Boundaries section of UAX #29.
Does this package correctly reverse strings with diacritics?
Yes! It reverses the order of the graphemes, not the codepoints. This means that it does not move diacritics around and emoji are perfectly safe.
-- äo without normalization
String.Graphemes.reverse "a\u{0308}o" --> "oa\u{0308}"
-- compare with String
String.reverse "a\u{0308}o" --> "o\u{0308}a"
Does this package do normalization?
No, and it probably never will. It's a whole 'nother spec in the Unicode standard which doesn't really fit in this package.
That said, it looks like you could implement it in a similar way as the internal String.Graphemes.Parser, so give it a go in a new package of your own!
(n.b. normalization in this case means turning "a\u{0308}" into "ä", usually for the purposes of improving equality checks.)
Does this package segment words or sentences?
No, and it probably never will. Segmenting words and sentences is locale- and implementation-dependent, so it's really hard to address them in a general way. Rather than introducing confusion ("it should segment here… why doesn't it?") we only segment graphemes.
That said, word and sentence segmentation rely on grapheme segmentation, so you're on the right track by asking this! UAX #29 has guidance here.
Why not "fix" elm/core's String instead of writing a new package?
The String module solves a different—but overlapping—set of problems. For example, you do not always want to work with graphemes: sometimes you need to be able to decompose into codepoints or operate at the byte level. As usual, it's all tradeoffs.
That said, if it eventually becomes obvious that merging into core would be a good thing we may do that. In that case, we would probably just keep equivalents of String.Graphemes.uncons and String.Graphemes.foldl. Everything else is implemented in terms of those two operations.
Why a drop-in replacement? / Why does the code refer to String functions so much?
Unless you've worked with unicode strings a lot, it can be tricky to know which level (bytes, codepoints, or graphemes) you're operating at with any given time. So instead of giving you the functions you might need, and leaving you to implement the rest on your own, we provide all of them and only change the ones where you'd run into trouble.
But not all of the functions in String need to be modified. In those cases, we just pass through to the String functions!
This way, you don't have to worry about it. You could potentially do import String.Graphemes as String in a module, fix the type errors, and all of a sudden all your string operations work with graphemes.
This code in this project is licensed under the BSD 3-Clause license, located at LICENSE in the source.
The documentation strings in String.Graphemes are derived from those in elm/core's String, © 2019 Evan Czaplicki, and licensed under the BSD 3-Clause license.
The grapheme break property data used here are © 2019 Unicode®, Inc., and licensed under their terms of use.


text:
KSC, Inc.
Collin College Technical Campus
With the projected growth in Collin County, eyes are on the future. In recent years, countless hours have been spent preparing to take on the higher education needs of North Texas. The Collin County Community College Technical Campus is a recently completed project by McCarthy Building Companies, Inc. and Perkins + Will that will do just that. The four-building, three-story campus is a prime example of state-of-the-art college architecture, located on 31.8 acres in Allen, Texas. The facility encompasses 340,000 square feet, and can serve over 7,000 students in the automotive, construction, health care, information technology and manufacturing fields.
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There was a strong desire to present the feel of a true college environment, one that is rarely felt among community college campuses. With careful planning and every detail examined, the ground work for that desire was laid. An underground parking garage, sidewalk connections to the mixed-use developments located along the Highway 121 Corridor, a complete academic center and cantilevers on both ends of the building that provide a shaded place to study or relax all came together to bring this vision to life.
Over 3,400 Aluminum Composite Material Alpolic 4MM FR core panels were manufactured and installed, using the KSC 2100 ventilated rain-screen mounting system. ACM was selected because its clean aesthetic appearance and versatility are a perfect fit to meet the requirements of a functional campus. The result: a modernistic facade system sculpted to endure the test of time.
The overall project design implements a low-site impact methodology and is considered to be one of the more striking campuses in the area.
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Collin County Community College
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ACM Systems
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Uwe Zahn, aka Arovane, has been one of the most influential characters in the electronic music scene over the last few years. His classic 'Tides' album was not only the first full-length release on the City Centre Offices label (and a huge influence on the label's roster) but is also often cited as one of the key electronic music releases of the last decade. After 4 long years, arovane finally returns."Lilies" is a follow-up to "Tides" in every sense, exploring a trip to japan and drawing on shimmering textures and the sort of melodies that you might need some time to recover from. There's a hugely evocative sense to these tracks, emotionally driven, free of complexity or conceit, piano melodies providing the central focus for a twilight cascade of light that seems perfect for the Tokyo skyline - just as the sun sets. It's an album that radiates warmth and vulnerability, fusing the technological might at the heart of each track (and at the heart of the city) with an age-old understanding that certain echoes of sound, small melodic changes and cushioned lullabies can imprint sounds on your mind like childhood memories - remembered forever. Like a dreamlike score, or maybe even an alternate soundtrack to "Lost in Translation" - the sort of music that intertwines with images and stays in your mind indefinately. After coming back from tokyo and completing the production of "lilies", Arovane disassembled his studio in his big flat in an old building in berlin's prenzlauerberg district and stored it away in boxes. He needs a break from making music now. Who knows how long for. Gorgeous.


text:
\section{Introduction}
This paper considers the use of Bayesian regularized  linear regression models for the purpose of estimating a treatment effect from observational data. Treatment effects ---  the amount some response variable would change if the value of the treatment variable were changed by a given amount --- can only be properly estimated from observational data by taking into account all of the various explanatory factors that may otherwise account for the observed correlation between the treatment and response variables. In the case of a linear regression model (assuming it to be correct) this ``adjustment for confounding'' means that the model includes a sufficient set of control variables as regressors in addition to the treatment variable.

Practical implementation of regression modeling for estimating treatment effects from observational data is complicated by two related issues. First, the minimal set of sufficient control variables is almost never known and second, the set of candidate control variables is often quite large relative to the available sample size. This consideration suggests that statistical regularization has a role to play in reliable treatment effect estimation. It may therefore come as a surprise that naive deployment of Bayesian shrinkage priors in the context of treatment effect estimation can yield exceptionally poor estimators. Exploring this phenomenon and providing a straightforward solution is the main contribution of this paper. We show that regularization can indeed provide statistical improvements over maximum likelihood estimation, but that it must be imposed carefully, in a sense we will make precise.

\subsection{Previous literature}
Treatment effect estimation is an important topic with a long and varied literature; a comprehensive review is beyond the scope of this paper. For review articles from an expressly Bayesian perspective, see \cite{li2014bayesian} or \cite{heckman2014treatment}. This paper focuses more narrowly on the impact of regularization or ``shrinkage" priors on the estimation of treatment effects from observational studies. Our use of regularization priors in this context addresses a practical data analysis problem that has been recognized since at least \cite{leamer1983let}: regression analyses including very many potential control variables often produce unsatisfyingly imprecise effect estimates. \cite{leamer1983let} admonishes those who react to this dilemma by hand-selecting a small subset of the potential controls and proceeding with analysis as if the others were irrelevant. See also \cite{leamer1978specification} for an early Bayesian treatment of this problem.

More specifically, this paper represents a contribution to the small but growing literature on Bayesian approaches to treatment effect estimation via linear regression with many potential controls. Specifically, we propose a conceptual and computational refinement of ideas first explored in \cite{wang2012bayesian}, where Bayesian adjustment for confounding is addressed via hierarchical priors. Our proposed method can be seen as an alternative to \cite{wang2012bayesian}, with certain conceptual and computational advantages, namely ease of prior specification and posterior sampling. Other papers elaborating upon this approach include \cite{wang2015accounting}, \cite{lefebvre2014effect} and \cite{talbot2015bayesian}; see also \cite{jacobi2016bayesian}. \cite{zigler2014uncertainty} and \cite{an2010bayesian} focus on Bayesian propensity score models (for use with binary treatment variables). \cite{wilson2014confounder} takes a decision theoretic approach to variable selection of controls.  Again, each of these previous approaches cast the problem as one of selecting appropriate controls; posterior treatment effect estimates are obtained via model averaging.  Here, we argue that if the goal is estimation of a certain regression parameter (corresponding to the treatment effect, provided the model is correctly specified), then questions about which specific variables are necessary controls is a means to an end rather than an end in itself. Other recent papers looking at regularized regression for treatment effect estimation include \cite{ertefaie2015variable} and \cite{ghosh2015penalized}, but even here the focus is on variable selection via the use of $1$-norm penalties on the regression coefficients.

Finally, treatment effect estimation is clearly a sub-topic within the broader field of causal inference. Here, we do not emphasize this connection, focusing instead on the specifics of the important special case that is linear regression. For excellent book length treatments on causal inference, we recommend  \cite{imbens2015causal} and \cite{morgan2014counterfactuals}. Like \cite{wang2012bayesian}, our work has forebears in earlier work based on joint modeling of treatment and response variables as functions of control variables, notably \cite{rosenbaum1983central} and \cite{robins1992estimating}, as well as \cite{mccandless2009bayesian}.

\subsection{Outline}
The paper is structured as follows. In section \ref{reparametrization}, we describe how naive regularization can corrupt treatment effect estimation and present a reparametrized linear model that avoids this pitfall. Section \ref{simulations} presents extensive simulation studies demonstrating the performance of the new model relative to standard alternatives. Section \ref{levitt} reanalyses the data of \cite{donohue2001impact}, which considers the impact of abortion laws on crime rates, following the similar recent (frequentist) analysis of \cite{belloni2014inference}.

\section{Regularized linear regression for treatment effect estimation}\label{reparametrization}
In this paper, we focus on linear regression models
\begin{equation}\label{responseeqn}
	Y_i = \alpha Z_i + \mbox{X}_i \beta + \nu_i,
\end{equation} where $\mbox{X}_{i}$ is a row vector of control variables, $\beta$ is a column vector of the control effects, $Z_{i}$ is a continuous scalar treatment variable and $\alpha$ is a scalar regression coefficient. When these variables are meant to be interpreted as random variables, they will be denoted in capital letters; when they are to be interpreted as observed quantities they will either be lower case, to indicate a scalar quantity, roman font, to indicate a vector, or bold, to indicate a matrix.  We assume the errors, $\nu_{i}$, are normally distributed with zero mean and unknown variance. Under these assumptions, the ordinary least squares estimator gives unbiased estimates with valid coverage.

Our goal is to accurately estimate the treatment effect, and this is done by including the proper controls in the equation. Specifically, ``proper" in this context means that:
\begin{equation}\label{covcond}
	\mbox{cov}(Z_{i}, \nu_{i} \vert \mbox{X}_{i}) = 0.
\end{equation}This {\em exogeneity} condition guarantees that estimates of $\alpha$ will have the desired counterfactual interpretation as ``the amount $Y$ {\em would change} if $Z$ {\em were changed} by one unit'': $\alpha = \mbox{E}(Y \mid Z = z+1, \mbox{X}) - \mbox{E}(Y \mid Z = z, X)$. For a detailed discussion of why (\ref{covcond}) licenses a causal interpretation, see e.g. \cite{imbens2015causal} section 12.2.4 or \cite{morgan2014counterfactuals} section 6.2.

It will be assumed throughout that this model is correctly specified so that attention may be focused narrowly on the impact that regularization has on posterior inferences regarding parameter $\alpha$. To emphasize, a thorough regression analysis for causal inference should including a sensitivity analysis to gauge robustness of one's inferences to various modeling assumptions. In this paper we intentionally set these practically important concerns aside for conceptual clarity: the phenomenon of ``regularization-induced confounding'' is an independent issue that arises even if the model and exogeneity assumptions are all satisfied. For a complete introduction to the host of additional issues surrounding causal inference, see again \cite{imbens2015causal}.

The most common approach to estimating the parameters of linear regression models is via ordinary least squares (OLS), which in the present model is equivalent to maximum likelihood estimation. Estimating the model via OLS guarantees that (\ref{covcond}) is satisfied by construction: it is well-known, and easy to show, that the residual vector produced by OLS has zero correlation with the observed treatment vector $Z$. However, (\ref{covcond}) will not in general be satisfied by the residuals corresponding to a shrinkage estimator of $\beta$. Accordingly, in finite samples we have two competing criteria -- the shrinkage prior over $\beta$ and the sampling distribution for $Y_{1:n}$ -- which combine to form our eventual estimate. What can happen in this setting is that posterior inferences can be affected by the prior in such a way that (\ref{covcond}) is violated in-sample, making the causal interpretation of the $\alpha$ estimate suspect.

Intuitively, the prior ``prefers'' to have ``small'' elements of $\beta$; in the case of strong confounding, a very similar in-sample fit can be achieved by over-stating the magnitude of the treatment effect parameter $\alpha$ (which is one-dimensional) while simultaneously attenuating the control variable coefficients. To observe this phenomenon formally, we can examine the bias of the posterior mean of $\alpha$ in the case of a standard normal (ridge) prior over $\beta$. In this case, considering $\mathrm{z}$ and $\mathbf{X}$ fixed, the bias of $\hat{\alpha}_{rr}$ ($rr$ for ``ridge regression'') under an independent non-informative prior, may be expressed as:
\begin{equation}\label{bias}
\mbox{bias}(\hat{\alpha}_{rr}) = -\left((\mathrm{z}^t\mathrm{z})^{-1}\mathrm{z}^t\mathbf{X}\right)(\mathbf{I}_p + \mathbf{X}^t(\mathbf{X} - \hat{\mathbf{X}}_Z))^{-1}\beta.
\end{equation}
The first term is a $p$-vector of regression coefficients corresponding to univariate regressions of each $X_j$ on $Z$; $\hat{\mathbf{X}}_Z$ is the $n$-by-$p$ matrix of fitted values from these $p$ regressions. Note that the bias is not a function of the true value of $\alpha$, but is a function of every element of the true (unknown) $\beta$ vector, with weights proportional to how well $X_j$ is predicted by $Z$. To put this formula into the context of treatment effect estimation, it says roughly that the stronger the confounding is, the worse the bias on the treatment effect parameter $\alpha$ will be.

\subsection{A reparametrized model for regularized treatment effect regressions}
Consider the two equation model:
\begin{equation} \label{themodel0}
	\begin{split}
	\text{Selection Eq.:} &\hspace{5mm}	Z = \mbox{X}\gamma + \epsilon, \hspace{15.5mm} \epsilon \sim N(0,\sigma_{\epsilon}^{2}),
		\\
		\text{Response Eq.:} &\hspace{5mm}	Y = \alpha Z + \mbox{X}\beta + \nu, \hspace{5mm} \nu \sim N(0,\sigma_{\nu}^{2}).
	\end{split}
\end{equation}
Without loss of generality, assume that our variables are zero centered (in practice, one may include an intercept term).

The designation ``selection" refers to the impact that the control variables have on the level of treatment, $Z$, received. Prototypically, certain individuals are ``selected" to receive treatment. The ``response" equation describes the impact of the treatment and controls, $\mbox{X}$, on the conditional expectation of the response (outcome) variable, $Y$. Prototypically, $Y$ records some diagnostic measure on individuals. Because $\mbox{X}$ appears in both equations, the selection equation reflects the confounding influence of the controls, and the residual variance of this equation, $\sigma_{\epsilon}$, gauges the extent of the confounding.

These equations correspond to the factorization of the joint distribution $$f(Y, Z \mid X, \gamma, \beta, \sigma_{\epsilon}, \sigma_{\nu}) = f(Y \mid X, Z, \beta, \sigma_{\epsilon}, X)f(Z \mid X, \gamma, \sigma_{\nu},).$$ This factorization implies a complete separation of the parameter sets; specifically, independent priors on the regression parameters $\pi(\beta, \gamma, \alpha) = \pi(\beta)\pi(\gamma)\pi(\alpha)$ imply that only the response equation is used in estimating $\beta$ and $\alpha$.

It is possible, as investigated in \cite{wang2012bayesian}, to incorporate information concerning $\gamma$ into the inference for $\beta$ via a joint prior $\pi(\beta \mid \gamma)\pi(\gamma)$ which would then be updated by the treatment data as $\pi(\gamma \mid Z)$ whereupon it can be incorporated with the response likelihood via the integrated prior $\pi(\beta \mid Z) = \int_{\gamma} \pi(\beta \mid \gamma)\pi(\gamma \mid Z)$. Our approach will be more direct, placing widely-used independent priors in a transformed parameter space.

Specifically, we introduce the following transformation:
\begin{equation}\label{transformation}
\begin{pmatrix}
\alpha\\
\beta +  \alpha \gamma\\
\gamma
\end{pmatrix} \rightarrow \begin{pmatrix}
\alpha \\
\beta_d\\
\beta_c
\end{pmatrix},
\end{equation}
which yields the model
\begin{equation} \label{themodel}
	\begin{split}
	\text{Selection Eq.:} &\hspace{5mm}	Z = \mbox{X}\beta_c + \epsilon, \hspace{31mm} \epsilon \sim N(0,\sigma_{\epsilon}^{2}),
		\\
		\text{Response Eq.:} &\hspace{5mm}	Y = \alpha(Z - \mbox{X}\beta_c) + \mbox{X}\beta_d + \nu, \hspace{5mm} \nu \sim N(0,\sigma_{\nu}^{2}).
	\end{split}
\end{equation}

Our approach will be to place  independent regularization priors over $\beta_c$ and $\beta_d$ and to update our prior using the likelihood of both of the above equations.

This parametrization tidily separates the distinct roles that covariates can play in a regression analysis of causal effects. Specifically, in previous literature, a ``prognostic'' or ``predictive'' variable refers to variables $X_j$ with $\beta_{c,j} = 0$, $\beta_{d,j} \neq 0$ and ``confounder'' refers to variables $X_j$ with $\beta_{c,j} \neq 0$, $\beta_{d,j} \neq 0$. Here we refer to a confounder as any variable with $\beta_{c,j} \neq 0$, with the understanding that this is a {\em necessary} but not sufficient condition to be a confounding variable in the usual sense. Likewise, the term ``direct effect'' has other meanings in some related literature; here we will use it simply to refer to variables with $\beta_{d, j} \neq 0$. Moreover, our parametrization makes transparent how the linear regression (the response equation) ``controls for" confounding: the parameter $\alpha$ gives the rate of change in the response as a function of changes in treatment level due to ``random fluctuation'' ($\epsilon = Z  - \mbox{X}\beta_c$). Intuitively, with $\beta_c$ and $\mbox{X}$ in hand, we have access to a randomized experiment from which to infer $\alpha$. Crucially, the $Z$ likelihood enforces this interpretation of $\beta_c$ and hence $\alpha$.

Note also that this transformation leaves the likelihood unchanged. In particular, if one fits the selection equation via OLS and then substitutes the associated residuals into then response equation and then fits OLS, the resulting estimate of $\alpha$ will be exactly as if one used the original parameterization and fit the model via a single application of OLS. However, in terms of imposing regularization, the two parametrizations are quite different --- under our transformation the selection equation likelihood plays a role in dictating the degree of posterior shrinkage, because $\beta_c$ appears in both likelihoods.

Finally, note that this parametrization greatly mitigates the bias of $\hat{\alpha}$: given $\beta_c$, the expression for the bias under a flat prior for $\alpha$ and a standard normal prior for $\beta_d$ is
\begin{equation}\label{bias2}
\mbox{bias}(\hat{\alpha}) = -\left((\mathrm{r}^t\mathrm{r})^{-1}\mathrm{r}^t\mathbf{X}\right)(\mathbf{I}_p + \mathbf{X}^t(\mathbf{X} - \hat{\mathbf{X}}_R))^{-1}\beta_d.
\end{equation}
where $\mathrm{r} = \mathrm{z}-\mathbf{X}\beta_c$. By construction, $(\mathrm{R}^t\mathrm{R})^{-1}\mathrm{R}^t\mathbf{X}$ will be close to the zero vector, because $R_i = Z_i - X_i\beta_c$ is independent of $X_i$. Of course, $\beta_c$ (and hence $R$) is not known, but the new model is conditionally approximately unbiased for $\alpha$ and the $Z$ likelihood provides information on $\beta_c$. In fact, expressions (\ref{bias}) and (\ref{bias2}) indicate that the naive model will have higher bias the stronger the confounding (as measured by small $\sigma_{\epsilon}$), which is exactly when the new parametrization has more information about $
\beta_c$ and so will be closer to unbiased. This observation is borne out in the simulation studies below.

\section{Simulation studies}\label{simulations}
This section reports simulation studies which demonstrate the success of the reparametrized model in avoiding the mis-identified shrinkage of naive regularization. The four methods being compared are ordinary least squares (OLS) applied to the response equation, ``naive regularization" which applies a shrinkage prior over $\beta$ and uses only the response equation likelihood, the new approach, which places independent shrinkage priors over $\beta_c$ and $\beta_d$ and uses both the response and treatment likelihoods, and ``oracle OLS" which performs OLS using only the variables with non-zero coefficients. Note that oracle OLS is not possible to implement in applied problems. Non-informative priors over the remaining parameters are the same for both Bayesian approaches: $\alpha \propto 1$, $\sigma_{\epsilon} \propto 1/\sigma_{\epsilon}$,  $\sigma_{\nu} \propto 1/\sigma_{\nu}$.

In this paper, the shrinkage prior we employ is
\begin{equation}\label{horseshoe}
\begin{split}
\pi(\beta_j) &\propto \frac{1}{v}\log{\left(1 + \frac{4}{(\beta_j/v)^2}\right)},\\
\pi(v) &\sim \mbox{C}^{+}(0,1),
\end{split}
\end{equation}
where $v$ is a global scale parameter common across all elements $j = 1, \dots p$, and $\mbox{C}^{+}(0,1)$ denotes a folded standard Cauchy distribution. This prior is a close proxy of the horseshoe prior of \cite{carvalho2010horseshoe}. Such priors have proven empirically to be a fine default choice for regression coefficients:  they lack hyperparameters, forcefully separate strong from weak predictors, and exhibit robust predictive performance. This modified representation permits the model to be fit using an elliptical slice sampler of \cite{hhl}; as reported there, when $p = 1000$ this sampler can produce 10,000 posterior samples in less than a minute (for any sample size strictly larger than $p$). We defer the computational details of our approach to the appendix.  We stress, however, that the key patterns revealed in our simulation study are a byproduct primarily of our reparameterization, and can be expected to arise under any similar regularization prior. Although not reported here, simulation studies were also conducted under ridge priors (with empirical Bayes selection of the shrinkage parameter) and the basic conclusions do not change under these variations. We also include one simulation study using point-mass model selection priors (using within-model $g$-priors) for applications where $p > n$.

\subsection{\cite{wang2012bayesian} simulations}

In this section, we consider two simulations from the analysis of \cite{wang2012bayesian}.  In the first simulation, the true model for the data is: $Y_{i} = \alpha Z_{i} + \beta_{1}X_{1i} + \beta_{2}X_{2i} + \epsilon_{i}$, where $i = 1, ..., 1000$ and $\epsilon_{i} \sim N(0,1)$.  The vector of treatment and covariates is distributed as $(Z_{i},X_{1i},X_{2i}) \sim N(0,\Sigma)$ where $\Sigma_{kk} = 1$ for $k=1,2,3$, $\Sigma_{12} = \Sigma_{21} = \rho$, and $\Sigma_{13}=\Sigma_{31}=\Sigma_{23}=\Sigma_{32}=0$.  The potential confounders are $(X_{1}, X_{2})$ with 49 additional independent random variables drawn from a standard normal. We set the parameters as $\rho=0.7$ and $\alpha=\beta_{1}=\beta_{2}=0.1$ and generate 1000 data sets for analysis.

The results from the first simulation are displayed in table \ref{domsim1}.  We show average bias, interval length, and mean squared error across all generated data sets as well as the probability of covering the true treatment effect (coverage).

In the second simulation, a larger set of potential confounders is considered, and they are correlated with both the treatment and response variables.  The true model is: $Y_{i} = \alpha Z_{i} + \beta_{1}X_{1i} + \cdots + \beta_{14}X_{14i} + \epsilon_{i}$, where $i = 1, ..., 1000$ and $\epsilon_{i} \sim N(0,1)$.  The vector of treatment and covariates is distributed as $(Z_{i},X_{1i}, ...,X_{7i}) \sim N(0,\Sigma)$.  The covariance matrix $\Sigma$ is designed so that weak and strong correlations among the treatment and confounders exist.  We set $\Sigma_{kl} = 1$ if $k=l$ and $\Sigma_{kl}=\rho^{k+l-2}$ if $k \neq l$ and $k,l \in \{1,...,8\}$.  The remaining covariates are $(X_{8},..., X_{14})$ are drawn from a standard normal. The entire set of potential confounders is $\mbox{X}_{1},...X_{14}$ with 43 additional random variables drawn from a standard normal.  Similar to the first simulation, we set the parameters as $\rho=0.7$ and $\alpha=\beta_{1}=\cdots= \beta_{14}=0.1$ and again consider 1000 replications of this data set.

In both simulations, naive regularization performs poorly in coverage and is severely biased.  The new approach successfully reduces bias and has comparable performance to OLS in coverage, statistical power as measured by interval length, and mean squared error. The similar performance of our method and OLS in this case is due to the relatively large sample size for the given signal-to-noise level (\cite{wang2012bayesian} report nearly identical results as well). In the following section, we construct a simulation that shows when the data generating process has certain realistic properties, the new approach can outperform OLS in interval length and mean squared error (while naive regularization continues to underperform).

\begin{table}[H]
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
&Bias & Coverage&I.L.& MSE\\
\hline
New Approach  &  0.0024  &  0.959  &  0.1754  &  0.002 \\
OLS  &  0.0014  &  0.96  &  0.1786  &  0.002 \\
Naive Regularization  &  0.0479  &  0.35  &  0.0774  &  0.0053 \\
Oracle OLS  &  0.0015  &  0.958  &  0.1738  &  0.0019 \\
\hline
\end{tabular}
\end{center}
\caption{\cite{wang2012bayesian}: Simulation Study 1.}
\label{domsim1}
\end{table}

\begin{table}[H]
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
&Bias & Coverage&I.L.& MSE\\
\hline
New Approach  &  0.0034  &  0.955  &  0.201  &  0.0027 \\
OLS  &  -0.002  &  0.956  &  0.2022  &  0.0026 \\
Naive Regularization  &  0.0822  &  0.597  &  0.1889  &  0.0097 \\
Oracle OLS  &  -1e-04  &  0.94  &  0.1985  &  0.0028 \\
\hline
\end{tabular}
\end{center}
\caption{\cite{wang2012bayesian}: Simulation Study 2.}
\label{domsim2}
\end{table}

\subsection{Further simulations: shrinkage estimation in the presence of confounding}

In this section, we show results from a simulation designed to capture a variety of scenarios a data analyst may face. We consider changing the relative strengths of the confounding and direct effects as well as the number of such variables.  Specifically, we use the two equation model (\ref{themodel}) to generate our data. We set the marginal variance of the treatment and response variables to one, $\mbox{var}(Z)= \mbox{var}(Y)=1$, and we center and scale the control variables $\mbox{X}$ to have mean zero and unit variance.

To ensure we consider a range of data compositions, we parametrize our simulations using an ANOVA style decomposition. Defining the $\ell$-2 norms (squared Euclidean distance) of the confounding and direct effects as $\rho^{2}=\| \beta_c \|_{2}^{2}$ and $\phi^{2}=\| \beta_d \|_{2}^{2}$, we may decompose the marginal variances as
\begin{equation}
	\begin{split}\label{vardec}
		\mbox{var}(Z) &= \rho^{2} + \sigma_{\epsilon}^{2}
		\\
		\mbox{var}(Y) &= \alpha^{2}(1-\rho^{2}) + \phi^{2} + \sigma_{\nu}^{2},\\
		& = \kappa^2 + \phi^2 + \sigma_{\nu}^2,
	\end{split}
\end{equation}
because the control variables are standardized. Fixing the marginal variances to one implies $\sigma_{\epsilon}^{2} = 1 - \rho^{2}$ and $\sigma_{\nu}^{2} = 1 - \alpha^{2}(1-\rho^{2}) - \phi^{2}$. This decomposition admits the following interpretation: $\rho^{2}$ is the percentage of the treatment's variance due to confounding (strength of the confounding effect), $\phi^{2}$ is the percentage of the response variance due to the direct impact of the control variables on the response (strength of the direct effect), and $\kappa^2 := \alpha^{2}(1-\rho^{2})$ is the percentage of the response variance due to quasi-experimental variation of the treatment variable.

Next, observe that as the confounding becomes stronger ($\rho^2$ getting larger), the independent variation from which we infer the treatment effect ($Z - \mbox{X}\beta_c$) becomes smaller ($1 - \rho^2$). This means that for a fixed level of treatment effect, $\alpha$, and a fixed marginal variance, stronger confounding makes treatment effect inference harder in that the residual variance becomes correspondingly larger:  $1 - \alpha^2(1 - \rho^2) - \phi^2$. This makes it more difficult to get a clear picture of whether or not the confounding {\em per se} is making the problem difficult, or if problems with strong confounding just happen to be more difficult in this artificial way. To avoid this problem, we fix $\kappa^2 := \alpha^{2}(1-\rho^{2})$ to a constant, and allow $\alpha$ to vary as $\rho^2$ is varied. In this way we can examine the impact of confounding for a fixed difficulty of inference (as measured by the residual variance, which is held fixed at $1 - \kappa^2 - \phi^2$).

In our simulations, we fix a decomposition of the response variance given in \ref{vardec} and vary the strength of the confounding effect, $\rho^{2}$.  This amounts to specifying values for $\kappa^{2}$, $\phi^{2}$, and $\sigma_{\nu}^{2}$ that sum to one, and simulating data sets for several values  of $\rho^{2}$ between $0$ and $1$.  Again, because $\kappa^{2} = \alpha^{2}(1-\rho^{2})$ is fixed, as $\rho^{2}$ varies, $\alpha$ will vary as well.

Next, the components of $\beta_c$ and $\beta_d$ must be specified.  The nonzero entries of each identify which $\mbox{X}_{i}$'s are confounders, direct effects, and both, as previously defined. We define the first $k$ elements of $\mbox{X}$ to be confounders, the next $k$ to be both confounders \textit{and} direct effects, and the final $k$ elements to be direct effects.  We achieve this in our simulation by setting $\beta_{c}^{1:2k}$ to ones and $\beta_{d}^{(k+1):3k} \sim \mbox{N}(0,1)$.  These vectors are then rescaled to have magnitudes $\rho^{2}$ and $\phi^{2}$, respectively.  This sets the overall $\beta$ vector ($\beta = \beta_d - \alpha \beta_c$) to have $3k$ nonzero entries. (Note that under continuous priors for $\beta_c$ and $\beta_d$, every variable is a confounder and no variables are strictly prognostic.)

Let $n$ be the number of observations and $p$ be the number of columns of $\mbox{X}$.  In our simulation, we set $n=100,50$ and $p=30$.  Additionally, we consider the following response variance decompositions: $\{\kappa^{2}=0.05, \phi^{2}=0.7, \sigma_{\nu}^{2}=0.25\}$, $\{\kappa^{2}=0.05, \phi^{2}=0.05, \sigma_{\nu}^{2}=0.9\}$ and vary $\rho^{2} \in \{0.1,0.3,0.5,0.7,0.9\}$.

Tables \ref{simtab1} and \ref{simtab2} show results for the variance decomposition $\{\kappa^{2}=0.05, \phi^{2}=0.7, \sigma_{\nu}^{2}=0.25\}$ and $n=100$ and $50$, respectively.  In this scenario, the direct effect drives 70\% of variance in the response while the treatment effect drives 5\%.  Tables displaying the numbers used to generate these plots are shown in the appendix.  Similar to the \cite{wang2012bayesian} example, we compare the new, OLS, and naive regularization approaches in the presence of weak to strong confounding ($\rho^{2} \in \{0.1,0.3,0.5,0.7,0.9\}$). Again, the oracle OLS result is given for comparison. The four metrics we evaluate are bias, mean squared error (MSE), interval length (I.L.), and coverage.  First, note the poor performance of the naive approach.  As confounding strength increases, bias grows and coverage decays exponentially for both sample sizes.  In addition, MSE explodes for increasing confounding strength.  Nevertheless, the naive approach does produce a small interval length resulting from the regularization prior.

As table \ref{simtab1} demonstrates, the new approach and OLS are comparable when the data size is large relative to the number of potential confounders with MSE and I.L. gains using the new approach when confounding strength is large ($\rho^{2}>0.9$).  When the data size is smaller (table \ref{simtab2}), the gains of using the new approach over OLS are seen across the board.   The new approach outperforms OLS in both interval length and MSE for confounding levels varying from weak to strong. This is the benefit of ``betting on sparsity" when the data generating process is in fact sparse.

Tables \ref{simtab3} and \ref{simtab4} show results for a different response variance configuration: $\{\kappa^{2}=0.05, \phi^{2}=0.05, \sigma_{\nu}^{2}=0.9\}$.  In this scenario, the treatment and direct effects contribute 5\% each to the response variance and the remaining 90\% is residual noise.  This is a problem that, using any method for estimation, is inferentially difficult because of the low signal-to-noise ratio of the response.  In both the large data set ($n=100$, table \ref{simtab3}) and small data set ($n=50$, table \ref{simtab4}) relative to the number of potential controls, we again see underperformance of the naive approach.

In contrast to the previous example with a strong direct effect, the weak direct effect contributes to good performance of the new approach relative to OLS for both $n=100$ and $n=50$. Again, we see that the new approach has increased power through smaller interval lengths and lower mean squared error, especially for data sets with strong confounding. And again, we see the benefit of ``betting on sparsity" when the data generating process is in fact sparse.

\begin{table}[H]
\begin{center}
\begin{tabular}{|l|l|c|c|c|c|}
\hline
$\rho^{2}$ & & Bias & Coverage & I.L. & MSE \\
\hline
0.1  &  New Approach  &  -0.0032  &  0.943  &  0.2357  &  0.0037 \\
 &  OLS  &  -0.0016  &  0.951  &  0.2477  &  0.004 \\
 &  Naive Regularization  &  -0.0112  &  0.895  &  0.2089  &  0.0037 \\
 &  Oracle OLS  &  0.0023  &  0.946  &  0.2173  &  0.0031 \\

\hline
0.3  &  New Approach  &  -0.0047  &  0.95  &  0.2751  &  0.0047 \\
 &  OLS  &  -0.0018  &  0.951  &  0.2808  &  0.0052 \\
 &  Naive Regularization  &  -0.0355  &  0.848  &  0.2293  &  0.0057 \\
 &  Oracle OLS  &  0.0026  &  0.946  &  0.2464  &  0.004 \\

\hline
0.5  &  New Approach  &  -3e-04  &  0.963  &  0.3345  &  0.0066 \\
 &  OLS  &  -0.0022  &  0.951  &  0.3323  &  0.0072 \\
 &  Naive Regularization  &  -0.0768  &  0.746  &  0.2631  &  0.012 \\
 &  Oracle OLS  &  0.0031  &  0.946  &  0.2915  &  0.0056 \\

\hline
0.7  &  New Approach  &  0.0084  &  0.964  &  0.4374  &  0.0113 \\
 &  OLS  &  0.0024  &  0.944  &  0.4303  &  0.0123 \\
 &  Naive Regularization  &  -0.1559  &  0.543  &  0.3292  &  0.0346 \\
 &  Oracle OLS  &  0.004  &  0.946  &  0.3764  &  0.0093 \\

\hline
0.9  &  New Approach  &  -0.004  &  0.972  &  0.7403  &  0.0292 \\
 &  OLS  &  0.0045  &  0.954  &  0.7469  &  0.0351 \\
 &  Naive Regularization  &  -0.4482  &  0.231  &  0.4779  &  0.2391 \\
 &  Oracle OLS  &  0.0069  &  0.946  &  0.6519  &  0.0278 \\
\hline
\end{tabular}
\end{center}
\caption{$\mathbf{n=100,p=30,k=3}$. $\kappa^{2}=0.05$. $\phi^{2}=0.7$. $\sigma_{\nu}^{2}=0.25$.}
\label{simtab1}
\end{table}

\begin{table}[H]
\begin{center}
\begin{tabular}{|l|l|c|c|c|c|}
\hline
$\rho^{2}$ & & Bias & Coverage & I.L. & MSE \\
\hline
0.1  &  New Approach  &  0.0082  &  0.918  &  0.3632  &  0.0105 \\
 &  OLS  &  -0.0017  &  0.944  &  0.4785  &  0.0144 \\
 &  Naive Regularization  &  -0.0068  &  0.835  &  0.2957  &  0.0097 \\
 &  Oracle OLS  &  -0.001  &  0.952  &  0.3235  &  0.0065 \\

\hline
0.3  &  New Approach  &  -1e-04  &  0.94  &  0.4203  &  0.0128 \\
 &  OLS  &  -0.002  &  0.944  &  0.5425  &  0.0186 \\
 &  Naive Regularization  &  -0.035  &  0.837  &  0.3191  &  0.0126 \\
 &  Oracle OLS  &  -0.0011  &  0.952  &  0.3668  &  0.0084 \\

\hline
0.5  &  New Approach  &  -0.0047  &  0.93  &  0.5183  &  0.0196 \\
 &  OLS  &  -0.0023  &  0.944  &  0.6419  &  0.026 \\
 &  Naive Regularization  &  -0.0869  &  0.738  &  0.3555  &  0.0222 \\
 &  Oracle OLS  &  -0.0014  &  0.952  &  0.434  &  0.0117 \\

\hline
0.7  &  New Approach  &  0.0056  &  0.937  &  0.6926  &  0.0341 \\
 &  OLS  &  0.0046  &  0.934  &  0.8204  &  0.0478 \\
 &  Naive Regularization  &  -0.189  &  0.539  &  0.4033  &  0.0565 \\
 &  Oracle OLS  &  -0.0018  &  0.952  &  0.5604  &  0.0195 \\

\hline
0.9  &  New Approach  &  -0.0772  &  0.959  &  1.1572  &  0.0804 \\
 &  OLS  &  -0.0156  &  0.931  &  1.4347  &  0.1402 \\
 &  Naive Regularization  &  -0.5419  &  0.102  &  0.4868  &  0.3297 \\
 &  Oracle OLS  &  -0.003  &  0.952  &  0.9706  &  0.0585 \\
\hline
\end{tabular}
\end{center}
\caption{$\mathbf{n=50,p=30,k=3}$. $\kappa^{2}=0.05$. $\phi^{2}=0.7$. $\sigma_{\nu}^{2}=0.25$.}
\label{simtab2}
\end{table}

\begin{table}[H]
\begin{center}
\begin{tabular}{|l|l|c|c|c|c|}
\hline
$\rho^{2}$ & & Bias & Coverage & I.L. & MSE \\
\hline
0.1  &  New Approach  &  -0.0053  &  0.93  &  0.4137  &  0.0126 \\
 &  OLS  &  -0.0031  &  0.951  &  0.4699  &  0.0145 \\
 &  Naive Regularization  &  -0.027  &  0.434  &  0.1528  &  0.0178 \\
 &  Oracle OLS  &  0.0044  &  0.946  &  0.4123  &  0.0111 \\

\hline
0.3  &  New Approach  &  -0.0101  &  0.933  &  0.472  &  0.0176 \\
 &  OLS  &  -0.0035  &  0.951  &  0.5329  &  0.0186 \\
 &  Naive Regularization  &  -0.075  &  0.373  &  0.1625  &  0.0256 \\
 &  Oracle OLS  &  0.005  &  0.946  &  0.4675  &  0.0143 \\

\hline
0.5  &  New Approach  &  -0.001  &  0.933  &  0.5751  &  0.0245 \\
 &  OLS  &  -0.0041  &  0.951  &  0.6305  &  0.026 \\
 &  Naive Regularization  &  -0.1407  &  0.304  &  0.1751  &  0.0411 \\
 &  Oracle OLS  &  0.0059  &  0.946  &  0.5532  &  0.02 \\

\hline
0.7  &  New Approach  &  0.0044  &  0.95  &  0.7509  &  0.0368 \\
 &  OLS  &  -0.0049  &  0.953  &  0.8156  &  0.0394 \\
 &  Naive Regularization  &  -0.265  &  0.134  &  0.1801  &  0.0918 \\
 &  Oracle OLS  &  0.0076  &  0.946  &  0.7141  &  0.0333 \\

\hline
0.9  &  New Approach  &  -0.01  &  0.939  &  1.2784  &  0.1131 \\
 &  OLS  &  -0.0022  &  0.942  &  1.416  &  0.1345 \\
 &  Naive Regularization  &  -0.6114  &  0.002  &  0.1841  &  0.3983 \\
 &  Oracle OLS  &  0.0132  &  0.946  &  1.2369  &  0.0999 \\
\hline
\end{tabular}
\end{center}
\caption{$\mathbf{n=100,p=30,k=3}$. $\kappa^{2}=0.05$. $\phi^{2}=0.05$. $\sigma_{\nu}^{2}=0.9$.}
\label{simtab3}
\end{table}

\begin{table}[H]
\begin{center}
\begin{tabular}{|l|l|c|c|c|c|}
\hline
$\rho^{2}$ & & Bias & Coverage & I.L. & MSE \\
\hline
0.1  &  New Approach  &  0.0021  &  0.919  &  0.6073  &  0.0306 \\
 &  OLS  &  -0.0119  &  0.93  &  0.8888  &  0.0528 \\
 &  Naive Regularization  &  -0.0291  &  0.443  &  0.2207  &  0.0352 \\
 &  Oracle OLS  &  -0.0019  &  0.952  &  0.6138  &  0.0234 \\

\hline
0.3  &  New Approach  &  -0.0033  &  0.909  &  0.6918  &  0.0421 \\
 &  OLS  &  -0.0038  &  0.944  &  1.0294  &  0.0668 \\
 &  Naive Regularization  &  -0.0651  &  0.402  &  0.237  &  0.0428 \\
 &  Oracle OLS  &  -0.0022  &  0.952  &  0.696  &  0.0301 \\

\hline
0.5  &  New Approach  &  -0.011  &  0.894  &  0.8191  &  0.064 \\
 &  OLS  &  0.0071  &  0.927  &  1.2041  &  0.103 \\
 &  Naive Regularization  &  -0.1354  &  0.349  &  0.233  &  0.0577 \\
 &  Oracle OLS  &  -0.0026  &  0.952  &  0.8235  &  0.0421 \\

\hline
0.7  &  New Approach  &  -0.028  &  0.904  &  1.0842  &  0.105 \\
 &  OLS  &  -8e-04  &  0.938  &  1.5533  &  0.1603 \\
 &  Naive Regularization  &  -0.2752  &  0.217  &  0.2474  &  0.1163 \\
 &  Oracle OLS  &  -0.0033  &  0.952  &  1.0632  &  0.0702 \\

\hline
0.9  &  New Approach  &  -0.1078  &  0.948  &  1.8128  &  0.2303 \\
 &  OLS  &  0.0291  &  0.942  &  2.6708  &  0.4893 \\
 &  Naive Regularization  &  -0.6045  &  0.015  &  0.2576  &  0.4096 \\
 &  Oracle OLS  &  -0.0058  &  0.952  &  1.8415  &  0.2106 \\
\hline
\end{tabular}
\end{center}
\caption{$\mathbf{n=50,p=30,k=3}$. $\kappa^{2}=0.05$. $\phi^{2}=0.05$. $\sigma_{\nu}^{2}=0.9$.}
\label{simtab4}
\end{table}

\subsection{Dense case}
In tables \ref{densetable1} and \ref{densetable2}, the same simulation study as before is run with $\{p = 30,\hspace{1mm} k = 10\}$ and $\{p = 30, \hspace{1mm} k = 30\}$, respectively.  For the $k = 10$ case, $\beta_c$ and $\beta_d$ each have 20 nonzero entries and 10 zero entries and are sparse with respect to our transformed model \ref{themodel}.  However, $\beta$ itself is dense. For the $k=30$ case, we abuse our simulation construction slightly and construct both $\beta_c$ and $\beta_d$ (and thus $\beta$) as fully dense vectors with all $p=30$ components nonzero.  In both cases, note that OLS and Oracle OLS are identical methods. Two salient patterns emerge from this simulation. First, the new method performs essentially on par with OLS; there is no benefit for the bet on sparsity, but their is no major penalty either. Second, the naive response-only regularized regression continues to exhibit dismal performance.

\begin{table}[H]
\begin{center}
\begin{tabular}{|l|l|c|c|c|c|}
\hline
$\rho^{2}$ & & Bias & Coverage & I.L. & MSE \\
\hline
0.1  &  New Approach  &  -0.0038  &  0.939  &  0.2484  &  0.0043 \\
 &  OLS  &  -0.0014  &  0.944  &  0.2497  &  0.0041 \\
 &  Naive Regularization  &  -0.0094  &  0.948  &  0.241  &  0.0039 \\
 &  Oracle OLS  &  -0.0014  &  0.944  &  0.2497  &  0.0041 \\

\hline
0.3  &  New Approach  &  -0.0051  &  0.94  &  0.2895  &  0.0057 \\
 &  OLS  &  0.0029  &  0.929  &  0.2827  &  0.0057 \\
 &  Naive Regularization  &  -0.0268  &  0.921  &  0.2638  &  0.0055 \\
 &  Oracle OLS  &  0.0029  &  0.929  &  0.2827  &  0.0057 \\

\hline
0.5  &  New Approach  &  -0.012  &  0.966  &  0.351  &  0.007 \\
 &  OLS  &  -0.001  &  0.946  &  0.3327  &  0.007 \\
 &  Naive Regularization  &  -0.0715  &  0.85  &  0.2964  &  0.0103 \\
 &  Oracle OLS  &  -0.001  &  0.946  &  0.3327  &  0.007 \\

\hline
0.7  &  New Approach  &  -0.0105  &  0.96  &  0.4614  &  0.0126 \\
 &  OLS  &  -1e-04  &  0.946  &  0.4279  &  0.0124 \\
 &  Naive Regularization  &  -0.1587  &  0.563  &  0.3489  &  0.0341 \\
 &  Oracle OLS  &  -1e-04  &  0.946  &  0.4279  &  0.0124 \\

\hline
0.9  &  New Approach  &  -0.0496  &  0.963  &  0.7862  &  0.0351 \\
 &  OLS  &  -0.012  &  0.953  &  0.748  &  0.0369 \\
 &  Naive Regularization  &  -0.5131  &  0.01  &  0.4303  &  0.2764 \\
 &  Oracle OLS  &  -0.012  &  0.953  &  0.748  &  0.0369 \\
 \hline
 \end{tabular}
\end{center}
\caption{$\mathbf{n=100,p=30,k=10}$. $\kappa^{2}=0.05$. $\phi^{2}=0.7$. $\sigma_{\nu}^{2}=0.25$.}
\label{densetable1}
\end{table}

\begin{table}[H]
\begin{center}
\begin{tabular}{|l|l|c|c|c|c|}
\hline
$\rho^{2}$ & & Bias & Coverage & I.L. & MSE \\
\hline
0.1  &  New Approach  &  0.0023  &  0.942  &  0.2563  &  0.0045 \\
 &  OLS  &  0.0013  &  0.945  &  0.249  &  0.0041 \\
 &  Naive Regularization  &  -0.0025  &  0.947  &  0.2525  &  0.004 \\
 &  Oracle OLS  &  0.0013  &  0.945  &  0.249  &  0.0041 \\

\hline
0.3  &  New Approach  &  -0.0048  &  0.954  &  0.2996  &  0.0057 \\
 &  OLS  &  -0.0039  &  0.956  &  0.2841  &  0.0052 \\
 &  Naive Regularization  &  -0.0215  &  0.937  &  0.28  &  0.0056 \\
 &  Oracle OLS  &  -0.0039  &  0.956  &  0.2841  &  0.0052 \\

\hline
0.5  &  New Approach  &  0.003  &  0.965  &  0.3653  &  0.0078 \\
 &  OLS  &  9e-04  &  0.952  &  0.3334  &  0.0071 \\
 &  Naive Regularization  &  -0.0411  &  0.905  &  0.3171  &  0.0091 \\
 &  Oracle OLS  &  9e-04  &  0.952  &  0.3334  &  0.0071 \\

\hline
0.7  &  New Approach  &  0.0042  &  0.954  &  0.4813  &  0.0149 \\
 &  OLS  &  3e-04  &  0.929  &  0.4328  &  0.014 \\
 &  Naive Regularization  &  -0.1147  &  0.772  &  0.3854  &  0.025 \\
 &  Oracle OLS  &  3e-04  &  0.929  &  0.4328  &  0.014 \\

\hline
0.9  &  New Approach  &  -0.0329  &  0.965  &  0.8018  &  0.0363 \\
 &  OLS  &  -0.0053  &  0.942  &  0.7433  &  0.0375 \\
 &  Naive Regularization  &  -0.4212  &  0.155  &  0.5178  &  0.2052 \\
 &  Oracle OLS  &  -0.0053  &  0.942  &  0.7433  &  0.0375 \\
 \hline
 \end{tabular}
\end{center}
\caption{$\mathbf{n=100,p=30,k=30}$. $\kappa^{2}=0.05$. $\phi^{2}=0.7$. $\sigma_{\nu}^{2}=0.25$.}
\label{densetable2}
\end{table}

\subsection{$p>n$ case}

In order to explore the behavior of our proposal in a $p>n$ set-up,  we extend the first simulation analysis of \cite{wang2012bayesian}.  Now, the true model for the data is: $Y_{i} = \alpha Z_{i} + \beta_{1}X_{1i} + \beta_{2}X_{2i} + \epsilon_{i}$, where $i = 1, ..., 30$ and $\epsilon_{i} \sim N(0,0.04)$.  The vector of treatment and covariates is distributed as $(Z_{i},X_{1i},X_{2i}) \sim N(0,\Sigma)$ where $\Sigma_{kk} = 1$ for $k=1,2,3$, $\Sigma_{12} = \Sigma_{21} = \rho$, and $\Sigma_{13}=\Sigma_{31}=\Sigma_{23}=\Sigma_{32}=0$.  The potential confounders are $(X_{1}, X_{2})$ with 33 additional independent random variables drawn from a standard normal, for a total of 35 control variables. We set the parameters as $\rho=0.7$ and $\alpha=\beta_{1}=\beta_{2}=0.1$ and generate 1000 data sets for analysis.

In the $p>n$ setting it is helpful to return to a variable-selection model. Specifically, we employ normal $g$-priors \citep{zellner1986assessing} on both $\beta_c$ and $\beta_d$ with point-masses at zero. We define $g$ through {\em local empirical Bayes} \citep{liang2012mixtures} with model probabilities are defined by $p(\mathcal{M})\propto {{p}\choose{2}} ^{-1} \mathbf{1}_{p< p_{\mbox{max}}}$ where $p_{\mbox{max}}$ defines the maximum number of non-zero elements in both $\beta_c$ and $\beta_d$ (separately).

The primary reason for adopting this model in this setting is that it allows to directly handle exact-sparsity via the $p_{\mbox{max}}$ parameter; we can examine how the method behaves as this parameter changes relative to the true level of sparsity (two non-zero elements out of 35, in this case). A secondary reason is that the elliptical slice sampler we use for the continuous prior model would require special modification for the $p > n$ setting, because the maximum likelihood estimate is not well-defined. As a side benefit, this simulation allows us to demonstrate and emphasize that the benefits of the new parameterization are fundamentally prior-agnostic; it is not the specific choice of prior that matter, rather it is the ability to specify the prior in terms of $\beta_c$ and $\beta_d$.  It is worth noting that the local empirical Bayes approach can be quite slow when $p_{\mbox{max}}$ is large; when $p_{\mbox{max}} = 1000$ it will take more than a dozen minutes to obtain ten thousand samples (whereas the horseshoe implementation discussed above would take approximately one minute).

Table \ref{pgreater}  shows the results of this study, where the same prior is used for $\beta$ (the naive approach) versus separately for both $\beta_c$ and $\beta_d$ in the new parametrization. The new model performs best when $p_{\mbox{max}}$ is smaller (closer to the true number of non-zero coefficients), according to mean squared error. In all cases except $p_{\mbox{max}} = 20$, the MSE of the new model is lower than the naive model. In every case, the new model has better coverage. The naive model has smaller posterior credible intervals, but greater bias.

\begin{table}[H]
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
$p=35$, $n=30$. $p_{\mbox{max}}=3$&Bias & Coverage&I.L.& MSE\\
\hline
New Approach  &  0.055  &  0.87 &  0.301  &  0.008 \\
Naive Regularization  &  0.093  &  0.64  &  0.239  &  0.012 \\
\hline
$p=35$, $n=30$. $p_{\mbox{max}}=5$&Bias & Coverage&I.L.& MSE\\
\hline
New Approach  &  0.056  &  0.88  &  0.319  &  0.010 \\
Naive Regularization  &  0.097  &  0.60  &  0.239  &  0.013 \\
\hline
$p=35$, $n=30$. $p_{\mbox{max}}=10$&Bias & Coverage&I.L.& MSE\\
\hline
New Approach  &  0.059 &  0.88  &  0.335  &  0.010 \\
Naive Regularization  &  0.099  &  0.63  &  0.255  &  0.013 \\
\hline
$p=35$, $n=30$. $p_{\mbox{max}}=20$&Bias & Coverage&I.L.& MSE\\
\hline
New Approach  &  0.068 &  0.86  &  0.435  &  0.016 \\
Naive Regularization  &  0.103  &  0.65  &  0.255  &  0.015 \\
\hline
\end{tabular}
\end{center}
\caption{A variable selection prior used in the $p > n$ setting still reveals the benefit of the new parameterization over the naive response-only model. In this simulation the true data generating process had only two non-zero regression coefficients; accordingly, the model performs better when $p_{\mbox{max}}$ is smaller, according to mean squared error (MSE). In all cases except $p_{\mbox{max}} = 20$, the MSE is lower than the naive model. In every case, the new model has better coverage. The naive model has smaller posterior credible intervals, but greater bias.}\label{pgreater}
\end{table}

\section{Empirical illustration: abortion and crime}\label{levitt}
In this section, we consider the relationship between legalized abortion and crime rates, using data first analyzed in \cite{donohue2001impact} and widely publicized in the popular book \cite{levitt2010freakonomics}.  \cite{donohue2001impact} propose that their data tell an intriguing story: unwanted children are more likely to grow up to be criminals, so legalized abortion, which leads to fewer unwanted children, leads to lower levels of crime in society. They conduct three analyses, one each for three different types of crime: violent crime, property crime, and murders.

Here, in the spirit of the similar reanalysis of \cite{belloni2014inference}, we reanalyze the \cite{donohue2001impact} data using a substantially more elaborate model, and observe the impact regularization has on the resulting conclusions. Specifically, we will compare four estimation approaches: one using the original control variables and OLS, one using an expanded covariate set (which includes many interactions) fit with OLS, one using the expanded covariate set fit with a naively regularized Bayesian regression, and one using the expanded covariate set fit with a regularized Bayesian model using our new parametrization.

The response variable, $Y$, is per capita crime rates (violent crime, property crime, and murders) by state, from 1985 to 1997 (inclusive). The treatment variable, $Z$, is the ``effective'' abortion rate. This metric is an averaged abortion rate, weighted by criminal age at the time of arrest (to account for the fact that crimes committed by criminals should be associated with abortion rates at the time of their births).

As control variables, $\mbox{X}$, \cite{donohue2001impact} include a host of state and year specific attributes that could otherwise contribute to the observed crime rates:
\begin{itemize}
 \item prisoners per capita (log),
\item  police per capita (log),
\item state unemployment rate,
\item state income per capita (log),
\item percent of population below the poverty line,
\item generosity of Aid to Families with Dependent Children (lagged by fifteen years),
\item concealed weapons law,
\item beer consumption per capita.
 \end{itemize}
Including state and year dummy variables brings the total number of control variables to 66. For additional details concerning how these attributes are defined and where they were obtained, see the original paper \citep{donohue2001impact}.

Our expanded model includes the following additional control variables:
\begin{itemize}
\item  interactions between the original eight controls and year,
\item interactions between the original eight controls and year squared,
\item interactions between state effects and year,
\item interactions between state effects and year squared.
\end{itemize}

These additional variables allow the impact of the original eight covariates on crime rate to change flexibly across time (according to a quadratic trend) and allows for the state specific crime rates to likewise change over time (in terms of an offset from overall state and year rates according to a quadratic trend). When allowing for this degree of flexibility, estimation becomes quite challenging, with just $n = 624$ observations and $p = 176$ control variables.

Tables \ref{LevittTable1} and \ref{LevittTable2} show our posterior inference compared to OLS and naive regularization. First, we note that the reported OLS results on the original covariate set are very similar to the results given in \cite{donohue2001impact}, although they used weighted least squares to adjust for differing state populations.

Second, using the original covariate set, the results of our new method are broadly in agreement with OLS. Already in this case we observe signs of the naive regularization approach being biased.

Finally, using the augmented covariate set, we observe that OLS no longer identifies the originally reported negative effect. However, the interval it returns is not tight about zero, indicating that there is not enough signal in the data to determine the impact of abortion on crime rates. Our new approach, by comparison, has much smaller credible interval, although they also include zero. Notably, the asymmetry (with respect to zero) of the interval of OLS and our approach coincide, while naive regularization is off-centered in the opposite direction. In fact, naive regularization excludes zero in the case of property and violent crime, and reports the reverse of the effect in the original study. This relationship between the three methods bears out the patterns observed in our simulation studies and suggests that the naively regularized method is misestimating the treatment effect as a result of misallocated shrinkage.

\begin{table}
\begin{tabular}{l|cc||cc||cc||}
\cline{2-7}
&\multicolumn{2}{c||}{Property Crime}&\multicolumn{2}{c||}{Violent Crime}&\multicolumn{2}{c||}{Murder}\\
\cline{2-7}
&2.5\% &97.5\%&2.5\% &97.5\%&2.5\% &97.5\%\\
\hline
OLS   &-0.110 & -0.072& -0.171 &-0.090& -0.221& -0.040\\
new approach &-0.113 & -0.073&-0.182&-0.098&-0.222 &-0.039  \\
naive regularization &-0.075 & -0.010&0.079& 0.301&-0.186 &0.085  \\
\hline
\end{tabular}
\caption{\label{LevittTable1}Credible/confidence intervals (95\%) for the Donohue and Levitt (2001) example with original controls ($p = 66$, $n = 624$). On the smaller set of original controls, our new approach gives similar credible intervals as the OLS confidence interval. In this case, already the naive regularization approach shows signs of bias, although the impact is minor.}
\end{table}

\begin{table}
\begin{tabular}{l|cc||cc||cc||}
\cline{2-7}
&\multicolumn{2}{c||}{Property Crime}&\multicolumn{2}{c||}{Violent Crime}&\multicolumn{2}{c||}{Murder}\\
\cline{2-7}
&2.5\% &97.5\%&2.5\% &97.5\%&2.5\% &97.5\%\\
\hline
OLS & -0.226 & 0.019  & -0.374 &0.336&-0.125 & 1.763 \\
new approach  &-0.038 & 0.014& -0.114 &0.053&-0.081 &0.279 \\
naive regularization  & 0.007 &0.129 & 0.011 &0.412 &-0.227 &0.116\\
\hline
\end{tabular}
\caption{\label{LevittTable2}Credible/confidence intervals (95\%) for the Donohue and Levitt (2001) example with augmented controls ($p = 176$, $n = 624$). With the enlarged set of control variables, the new approach and OLS show notable differences, specifically our new regularized Bayesian approach has markedly smaller credible intervals. The naive regularization approach disagrees on the directionality of the effect compared to the other two methods, consistent with the bias observed in our simulation studies.}
\end{table}

\section{Discussion}
In this paper, we have documented the perhaps counterintuitive fact that naively applied shrinkage priors can dramatically corrupt inference concerning treatment effects and have developed a regularized Bayesian regression model that avoids this pitfall, while still boasting the usual advantages of shrinkage estimation.

In this section we conclude with additional discussion concerning the mechanism by which this parametrization improves estimation. Specifically, while it is explained above that the new parametrization is designed to be approximately unbiased for $\alpha$ (as a function of $\beta$), it is perhaps less clear that shrinkage priors on $\beta_c$ and $\beta_d$ are not conferring some additional advantage. For example, adjusting for variables that only associate with $Z$, but not with $Y$, is widely understood to decrease precision in estimates of $\alpha$ (relative to the model that omits these variables from the regression altogether). This phenomenon can be understood concretely through the lens of the new parametrization. First, such variables have a direct parametric interpretation: $\beta_d = 0$ and $\beta_c \neq 0$. Now, suppose that someone informs the analysts that a certain $\beta_d = 0$ {\em a priori}; in this event, one is better off running a regression to estimate $\alpha$ excluding the variable $X_j$ from the model because, intuitively, larger variation in the implied residuals gives more heterogeneity to estimate $\alpha$.

In fact, this intuition can be made more precise. Without loss of generality consider the case of only one potential confounder, $X$. If $\beta_c$ and $\beta_d$ were both known, consider estimating $$\alpha = \mbox{E}\left((Y - X\beta_d)(Z - X\beta_c)\right)$$ from a sample $(Y_i, Z_i, X_i)$, $i = 1...n$. Some straightforward manipulation shows that $$\alpha = \mbox{E}(YZ) - E(X^2)\beta_c\beta_d.$$ In the Gaussian linear regression model, the sample moments $n^{-1}\sum_i Y_iZ_i$ and $n^{-1}\sum_i X_i^2$ are sufficient statistics. From the above expression we observe that knowing that $\beta_d = 0$ annihilates the second term involving data $n^{-1}\sum_i X_i^2$; the associated estimator has less sample variability because it is unaffected by sampling variation in $n^{-1}\sum_i X_i^2$. However, any model that must estimate $\beta_d$ necessarily incorporates $n^{-1}\sum_i X_i^2$ and pays the price in precision. Therefore, the more prior mass about $\beta_c\beta_d = 0$, the more limited will be the sampling variation due to $n^{-1}\sum_i X_i^2$; independent zero-centered priors over $\beta_c$ and $\beta_d$ achieve that, while the use of fat-tailed priors allows the data to speak.

At the same time, better estimate of $\beta_c$ is naturally obtained by incorporating the sampling model for $Z$. Indeed, consider the case where $\beta_d$ is known and non-zero; one need not use $Z$ to obtain a consistent estimate of $\beta_c$ and $\alpha$, but discarding the $Z$ model (presuming it is correctly specified) is simply throwing away available information, as $\beta_c$ appears there. This is true especially if the signal-to-noise ratio in the selection equation is much more favorable than that of the response equation ($\sigma_{\nu} \gg \sigma_{\epsilon}$). This is precisely why our new parametrization pays dividends, because the naive parametrization implies that the $Z$ model is ignored. In other words, the new parametrization has an advantage over single-equation approaches in terms of estimating $\beta_c$, but not in terms of estimating $\beta_d$; for which no essentially new data is being brought to bear, merely a strongly zero-biased prior. (The extent to which this zero-bias is beneficial will presumably depend on the true data generating process; this is the subject of ongoing investigation.)

Ongoing work looks at adapting the ideas in this paper to the nonlinear regression models for treatment effect estimation; preliminary results are promising.

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redpajama_set_name: RedPajamaArXiv


Shown here is The Porcelain Garden Fairy Moon Lithophane Porcelain Night Light. Measuring approximately 3 1/2" by 4 1/2" these night light's make a wonderful gift. Gift boxed with history and care literature.


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Queer & Now
21st century life from a queer perspective
March 24, 2011 April 9, 2014 admin
Coming Out Kinky
Recently on Fearless Press, Viola wrote a post called Pissing on the Oyster about the idea that kinky people should come out as kinky to increase acceptance of kinky sexualities. Viola does a great job in that post of covering the legal and lifestyle ramifications of coming out as kinky, and I wanted to add a voice to the chorus and talk a little bit about why I have a problem with the idea of "coming out" in the first place.
Coming out can be an empowering experience. I know many queer people for whom coming out was a blessing, a way to find community, and a way to feel more comfortable in their own skin. Throughout my own teenage years and early twenties, I found coming out (as bisexual, then as lesbian, as feminist, as queer) to be important because my gender and sexuality were huge parts of how I identified as well as huge parts of my activism.
Coming out can also be problematic, however. This is true whatever you're coming out as–queer, poly, kinky, asexual, anything. Personally, I'm out to different degrees in different spaces. I'm out as queer everywhere. I explain genderqueer to some, but not to all, because it might affect my ability to get a job and because it's frankly complicated to explain. I did tell my mother, but she completely missed the significance of that identifier in my life. I'm out as poly only to particular people I trust, and not to my family. I'm out as kinky only to certain friends that I met online and of course on this blog, which doesn't list my full name but maybe wouldn't be that hard to find if you knew what you were looking for.
I don't believe that anyone has a responsibility to come out. I'm generally wary of the personal responsibility rhetoric. I think that the drive to come out often is wrapped up in essentialist notions of identity. "Come out as gay!" they say, and then tell you what gay is supposed to mean. Once you come out, it's expected that your identity is unchanging. The notion of a constant identity is very important to some queer people, and many of us are shamed or kicked out of certain communities when our identities shift. I believe that my gender and sexuality are a living thing, and I don't assume that any identity I claim now will exist tomorrow. As I learn, question, grow, get to know myself, experience sexual and romantic relationships, and move through my life, change happens. I welcome this. If I come out in grand fashion today, I may have to adjust my declaration five years down the line.
The same is true of kink. "Kinky" doesn't mean the same thing for all people. I might disagree that kink is necessarily about our sex lives (for me, for example, a huge part of submission is a non-sexual emotional component) or that if it is, sex should necessarily be something to keep in private (I believe that our society is extremely sexually repressed and that we should work towards opening it up). But I do think that like coming out as genderqueer or poly, coming out kinky is something that you have to explain. People have preconceived notions, but they aren't notions that we can dispel with a single swoop. It would be a disservice to kinky people to try to wrap BDSM up in a neat little package to present to vanilla people in order to show them "what kink is." Kink is many things. Kink is everything.
I do think that we need to figure out ways to increase acceptance. How do we do that? Through increased awareness and acceptance of sexuality as a legitimate topic for conversation and education, and as a legitimate identifier. Through an increased understanding of the fluidity of gender and sexuality, and the power of the individual to create hir own narrative. Through supporting laws that increase privacy and autonomy. Through direct confrontation of the way shame operates around sexuality in our society. Through education about different relationship and family styles, and support of laws that allow for differing family structures. Through patient, dedicated effort, community building, and respect for one another.
In the end, isn't that sort of what it's all about?
This post was originally published on the blog Sex Positive Activism, which has now merged to become the sex & relationships section of Queer & Now.
fearless press
Previous Does Sexual Identity Make Physical Attractiveness Less Important?
Next Children and Adult Sexuality in an Activist Context
Desmond Ravenstone says:
Avory: One thing we also need to consider is that many folks need to put a human face to an issue before they can actually deal with it. That's what many in the GLBT community discovered — all well and good to talk about human and civil rights on a theoretical level, but a sizable percentage of people are not able to move from the theoretical to the real until they know a flesh-and-blood person who is affected by this.
You mentioned "supporting laws that increase privacy and autonomy" — that can also mean changing how assault laws are interpreted and enforced, for example. But police, prosecutors and legislators are less likely to do that on behalf of an abstract minority, and more likely to listen to actual constituents, whether they are kinksters or people who know and care about them.
Yes, coming out kinky is indeed an individual decision, and one which has to be decided contextually. One of those important contexts is whether and how it can increase awareness and support for the kinds of changes we need to deal with discrimination and stigmatization. So rather than label the coming-out process as "problematic" and giving a vague to-do list, how about coming up with practical tools for explaining the basics of BDSM/kink to vanilla people — particularly why they shouldn't be frightened of us or for us?
How You Can Respond When Your Family Disapproves of Your Chosen Lifestyle/Sexuality | Submissive Guide says:
[…] Coming Out Kinky […]
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Stand Up Paddle Board Rentals are a great way to explore the area and Lake Coeur d'Alene! Looking for a 360 degree view of the lake? Spend the day touring on an SUP! Stand up paddleboard rentals are available through us May – September.
-get some exercise or do yoga!
All Stand Up Paddle Board Rentals include pfd, one paddle and whistle for each board.
Two-hour, Half-day (4 Hours), Full-Day (8 Hours) and Multiple Day Rentals are available.
We provide complimentary delivery to the North Shore of Coeur d'Alene Lake free of charge. You can paddle around the North Shore area, North Idaho College area, Tubbs Hill, and up the Spokane River if you like. Parking is sometimes a hassle along the lake and you are welcome to take advantage of our service!
Wait for your confirmation and directions to the downtown shop. Show up 10 minutes prior to departure.
We suggest our customers pack a snack, water bottle, and a lunch, sunscreen, hat/sunglasses.
Our usual and best drop off point is at the corner of W. Lakeshore/Hubbard along the Lake Coeur d'Alene shore.


text:
Алемор () — село у повіті Сібіу в Румунії. Входить до складу комуни Лоамнеш.

Село розташоване на відстані 233 км на північний захід від Бухареста, 19 км на північний захід від Сібіу, 99 км на південь від Клуж-Напоки, 128 км на захід від Брашова.

Населення
За даними перепису населення 2002 року у селі проживали  особи.

Національний склад населення села:

Рідною мовою назвали:

Примітки

Села повіту Сібіу

redpajama_set_name: RedPajamaWikipedia


text:
Blackmoorfoot är en ort i Storbritannien.   Den ligger i grevskapet Kirklees och riksdelen England, i den centrala delen av landet,  km nordväst om huvudstaden London. Blackmoorfoot ligger  meter över havet och antalet invånare är . Den ligger vid sjön Blackmoorfoot Reservoir.

Terrängen runt Blackmoorfoot är kuperad åt sydväst, men åt nordost är den platt. Den högsta punkten i närheten är  meter över havet,  km söder om Blackmoorfoot. Runt Blackmoorfoot är det tätbefolkat, med  invånare per kvadratkilometer. Närmaste större samhälle är Huddersfield,  km nordost om Blackmoorfoot. Trakten runt Blackmoorfoot består i huvudsak av gräsmarker.

Kustklimat råder i trakten. Årsmedeltemperaturen i trakten är  °C. Den varmaste månaden är augusti, då medeltemperaturen är  °C, och den kallaste är januari, med  °C.

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