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21 Best Things to Do in Fernandina Beach, Florida
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Amelia Island is one of the greater Jacksonville area's most beautiful barrier islands, located within the Sea Islands barrier chain, which stretches upward to the state's border with South Carolina. Visitors can explore state park military sites on the "Isle of Eight Flags," nicknamed for its eight ruling countries since the 16th century, or stroll through historic shops, restaurants, and art galleries in Fernandina Beach's lovely Old Town downtown district. Family-friendly beachfront access is offered at Main Beach Park, while area landmarks and natural wonders can be explored via a variety of kayak, charter boat, or airboat tours.
1.Marlin and Barrel Distillery
© Marlin and Barrel Distillery
Marlin and Barrel Distillery is Fernandina Beach's signature craft distillery, opened to the public in 2015 by Wyatt Griesmer. The high-quality microdistillery is known for its small-batch liquors crafted with regionally-produced citrus fruits, including variants emphasizing orange and grapefruit flavors. Signature vodkas in the company's Venture Vodka line include a craft variant prepared with Florida-produced molasses and a smoked pepper variants crafted with B&D Sauce Co. peppers. Rums produced as part of the company's Bearing Rum line include pineapple and vanilla bean espresso variants and Bearing 153, the highest-proof barrel-aged liquor produced in the state of Florida. Guests can tour the company's production facility and sample or purchase liquors at its public taproom, which also sells a wide variety of Marlin and Barrel branded merchandise.
115 S 2nd St a, Fernandina Beach, FL 32034, Phone: 904-556-3837
2.Fort Clinch State Park
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Fort Clinch State Park is anchored around the 19th-century Amelia Island fortress of the same name, which was originally constructed as a seaside fortification in 1847 and named after famed Seminole War general Duncan Lamont Clinch. The Fernandina Beach park, which was originally established in 1938 and sits adjacent to the Cumberland Sound, strives to protect the region's sensitive estuarine tidal marsh, maritime hammock, and sand dune ecosystems. Park visitors can enjoy a wide range of outdoor recreational activities, including opportunities for ocean fishing on the park's half-mile pier. A six-mile hiking trail is also offered, along with ample opportunities for cycling, marine wildlife watching, birding, surfcasting, and shelling. Guests can camp overnight at the nearby Amelia River Campground or attend monthly military reenactments honoring the park's fort history.
2601 Atlantic Ave, Fernandina Beach, FL 32034, Phone: 904-277-7274
3.Main Beach Park
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Main Beach Park is the premiere beachfront park and beach access area in Fernandina Beach, located along North Fletcher Avenue and offering waterfront access to the city's beautiful ocean shoreline. The park is known as the city's most family-friendly beach spot, offering a children's playground with safe play opportunities for kids of all ages and a large grassy lawn for families to enjoy picnics and pick-up sports games. Lifeguards are staffed on duty at the park between Memorial Day and Labor Day to ensure swimming safety. Two sand volleyball courts and a multipurpose sporting court are also offered, along with outdoor showers, picnic shelters, and a full skate park facility. Guests can explore the beach's marine welcome center for regional ecosystem and activity information or stroll along the waterfront on the park's beautiful wooden boardwalk. Ample public parking is available, with overflow beachfront facilities offered nearby at Peter's Point Park.
32 N Fletcher Ave, Fernandina Beach, FL 32034, Phone: 904-310-3350
4.The Amelia Island Museum of History
© The Amelia Island Museum of History
The Amelia Island Museum of History is the premiere cultural and historical museum of Amelia Island, housed within the city's historic Nassau County Jail building, constructed in 1878 and listed on the National Register of Historic Places. The museum, which is located within Fernandina Beach's core downtown district, showcases a variety of museum exhibits detailing the human history of the "Isle of Eight Flags" from indigenous times through the modern day. Visitors can peruse exhibits such as a recreated Timucuan indigenous village from the pre-Columbian era, which is showcased alongside displays about the region's Spanish Mission and railroading periods. A recreated jail cell exhibit is also presented, along with the interactive children's ship The Margery, which offers maritime skill play experiences for young visitors.
233 S 3rd St, Fernandina Beach, FL 32034, Phone: 904-261-7378
5.The Fernandina Beach Pinball Museum
© The Fernandina Beach Pinball Museum
The Fernandina Beach Pinball Museum showcases the history and technology of the arcade game industry from the creation of the first pinball machines in the 1930s through inventive 21st-century games. The museum showcases more than two dozen preserved pinball games dating back to the 1950s, which are available for ticketed game play by visitors of all ages during the evenings Wednesdays through Saturdays. Guests can play individual games for coin-operated game prices or purchase the museum's full daily admission passes, allowing unlimited play throughout operating hours. Try-it passes are also offered, giving visitors unlimited play time for an hour for a reduced rates. Special ticket rates are offered for young players under the age of 12, making the museum a perfect stop for family fun.
2106 Sadler Rd, Fernandina Beach, FL 32034, Phone: 904-435-8424
6.The Maritime Museum of Amelia Island
© The Maritime Museum of Amelia Island
The Maritime Museum of Amelia Island is Fernandina Beach's premiere maritime history museum, showcasing the rich historic maritime culture and industries of the Amelia Island region. The museum, which is housed within a converted ice house building in the city's downtown district, offers a variety of exhibits on the region's maritime culture underneath each of its eight national flags dating back to the 16th century. Visitors can view artifacts on display from the region's Spanish, Soviet, and United States rule eras, including Navy uniforms, Cold War diving suits, and buried pirate treasure. A substantial collection of maritime weaponry is also showcased, along with recovered shipwreck artifacts found within the region's surrounding waters. Guided tours are available for visitors of all ages, recounting tales of the region's maritime workers throughout the years in conjunction with exhibit information.
115 S 2nd St, Fernandina Beach, FL 32034, Phone: 904-548-7504
7.Blue Door Artists
© Blue Door Artists
Blue Door Artists showcases eight art studios used by Fernandina Beach area artists, which are open to the public for observation Mondays through Saturdays during the morning and afternoon hours. The collaboration-focused studios showcase the work of artists working in a wide variety of media, including watercolor and oil painting, sculpture, jewelry, photography, and mixed media artists. Visitors can meet and greet with the studio's artists as they work on current projects and hear stories about the inspiration behind pieces' creation. Fine contemporary art and crafts are available for purchase at the studio's store, including handcrafted home items and artisan pieces. Gallery visitors of all ages and ability levels can also participate in artist-run workshops or attend Second Saturday events, which showcase works by featured artists as part of public receptions.
205 Centre St, Fernandina Beach, FL 32034, Phone: 904-491-7733
8.Egan's Creek Park
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Egan's Creek Park protects 300 acres of lovely natural terrain in Fernandina Beach along Amelia Island's Egan's Creek, which were originally developed as a greenway and passive recreation site in 2000. The labyrinth-like park and trail system are open to island walkers, joggers, and cyclists throughout daylight hours, offering chances to observe a plethora of regional wildlife, including snakes, bobcats, alligators and a variety of native bird species. Interpretive displays along the park's trail system showcase information on the region's wildlife and ecosystems, with benches offered along the way for visitor relaxation. Young visitors can play at the children's Explorer's Dome playground, while water sports enthusiasts can make use of the park's kayak launch. Other amenities offered include a day-use picnic pavilion area and a quarter-mile fitness trail with outdoor fitness stations.
9.Fernandina Beach Marketplace
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Fernandina Beach Marketplace is Fernandina Beach's premiere weekly farmer's market, open to the public each Saturday morning year-round in all weather conditions. The market, which is hosted within the city's downtown district on North 7th Street, showcases booths from area farmers, producers, and bakers, selling a wide variety of seasonal food items and pantry goods to community customers. Regional food trucks and carts are also showcased, along with live music performances and appearances by area celebrities and media personalities. Visitors can bring well-behaved pets to the market or talk to representatives from the Nassau Humane Society about humane pet adoption. Every other Saturday, the Fernandina Beach Arts Market is held in conjunction with the event, showcasing the works of splendid area artists and crafters.
N 7th St, Fernandina Beach, FL 32034
10.George Crady Bridge Fishing Pier State Park
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George Crady Bridge Fishing Pier State Park is named in honor of Florida State Representative George Crady, known as a major advocate for the state's park system throughout his political career. The lovely park, which is accessible from Fernandina Beach via State Route A1A at the border of Duval and Nassau Counties, is open to the public daily between dawn and dusk, offering a mile-long pedestrian fishing bridge accessible to visitors of all mobility levels. Visitors can stretch out and observe the beautiful Nassau Sound landscape, enjoying some of the state's best fishing conditions for catching whiting, jack, drum, and tarpon. The pier is best known as the site of the world record for flounder catches, set in 1983 and measuring over 20 pounds in weight. Bicycle rentals are offered nearby at Kelly Seahorse Ranch, offering opportunities for beautiful waterfront cycling along the pier and surrounding areas.
9000 Heckscher Dr, Fernandina Beach, FL 32034, Phone: 904-251-2323
11.Fernandina Plaza Historic State Park
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Fernandina Plaza Historic State Park is the smallest state park in Florida, stretching for 0.8 acres along Amelia Island's northern end atop a bluff offering panoramic views of the Amelia River. The park is located on the plaza grounds of the former Fort San Carlos, constructed in 1816 and mostly preserved today within nearby Fort Clinch State Park. Visitors can learn about the development and history of the plaza through interpretive panels on the park's grounds, which explore the plaza's cultural use over the years. Unparalleled views of nearby Old Town Fernandina are offered from the park's grassy lawn area, which makes a great spot for picnics and recreation. Private special events can be held at the park with permits, including community gatherings, weddings, and business and school events.
715 San Fernando St, Fernandina Beach, FL 32034
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12.Amelia Island Lighthouse
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Amelia Island Lighthouse is the oldest preserved lighthouse in the state of Florida, located at the St. Mary's Entrance of Amelia Island along the beautiful Cumberland Sound and St. Mary's River waterfronts. The historic lighthouse was originally constructed in 1838 from recovered remains of the nearby Little Cumberland Island Light, which had been constructed nearly two decades earlier on Cumberland Island. Throughout its operational history, the lighthouse was expanded to its current size of 64 feet. Today, it is overseen by the Coast Guard Auxiliary, operating an automated third-order Barbier Bernard Fresnel lens and 15-inch reflector light. Lighthouse visitors can tour the historic facility each first and third Wednesday morning of the month with advance reservations.
215 O, Hagan Ln, Fernandina Beach, FL 32034
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13.Amelia Island Kayak Excursions
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Amelia Island Kayak Excursions is one of Fernandina Beach's top-rated kayaking rental and tour companies, striving to offer family-friendly and memorable kayaking experiences for tour participants of all ages and ability levels. The company specializes in two, three, and five-hour guided tours of the waters surrounding Amelia Island, as well as combination kayaking trips and overnight excursions to nearby mainland and island tourist destinations. All tour participants can rest assured that the company's kayaks and paddles are maintained to the highest quality possible safety standards. Class III personal flotation devices are available for all kayakers. Partial-day, full-day, and multi-day kayak rentals are also offered, along with a variety of private group boating experiences traveling to nearby sights such as St. Mary's, Fort Jekyll and George Islands, and the Cumberland Island National Seashore.
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14.Fish Amelia Island
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Fish Amelia Island offers high-quality chartered fishing tour experiences aboard the 22-foot Fin Finder center console boat, owned and operated by Jacksonville native Danny Flynn, best known as the 2010 Pro Redfish Tour winner. The company offers year-round fishing trips for anglers of all ages and ability levels, ranging from family excursions to professional competitors. Charter guests can enjoy opportunities to catch summer fish species such as coba, tarpon, bull redfish, and trout, along with autumn and winter species such as jacks, mangrove snappers, and sharks. Marine life watching opportunities are also offered aboard excursions, including chances to spot and photograph area populations of dolphins and shorebirds.
2103 Egret Ln, Fernandina Beach, FL 32034, Phone: 904-583-3131
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15.Amelia River Cruises and Charters
© Amelia River Cruises and Charters
Amelia River Cruises and Charters has been operated out of Fernandina Beach since 2000 by married duo Kevin and Cecilia McCarthy, offering a plethora of boating tours and excursions along the gorgeous waterways surrounding Cumberland and Amelia Islands. Tours explore regional landmarks by water, including beautiful Fort Clinch State Park, the historic Old Town district of Fernandina Beach, and the famed wild horses of Cumberland Island. Participants can explore the salt marsh habitats of Cumberland Island as part of afternoon-long Beach Creek Tours, which also showcase the remains of the historic Dungeness Mansion. Other tour options include an Adult Twilight BYOB cruise, a Shrimping Eco Tour, and family-friendly Sunset Tours. Up to 80 passengers can come aboard ADA-accessible private boating charters, which offer food and beverages aboard all trips.
1 N Front St Suite #3, Fernandina Beach, FL 32034, Phone: 904-261-9972
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16.Amelia Island Downtown Tasting Tours
© Amelia Island Downtown Tasting Tours
Amelia Island Downtown Tasting Tours are the best way to explore the culinary scene of downtown Fernandina Beach and the greater Amelia Island district, featuring meet-and-greet chef experiences, food and beverage tasting opportunities, and walking tours past the island's historic landmarks. Each tour experience shakes up locations and tasting offerings to ensure a brand-new experience for repeat tour guests and a complete overview of the region's culinary offerings. Tour packages include two-hour downtown cocktail tours, which sample the island's best spirits and cocktails, or Booze and Bites Tours which offer meet-and-greets with top area chefs and mixologists. Wedding and birthday party groups can book private tour options, with add-on packages to make special days even more memorable.
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17.Florida Adventure Sports
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Florida Adventure Sports is one of Amelia Island's best ways to explore Amelia Island by air, offering 20-, 40-, and 60-minute excursions for tour participants of all ages led by licensed pilots. All excursions take place aboard the company's Airborne XT-912 aircraft line, which are noted for their high safety standards and smooth flights. More than 5,000 flight hours have been logged since the company's founding, traveling over top attractions within the Nassau Sound and Jacksonville regions. Tour participants receive a bird's-eye view of the region's gorgeous natural scenery, including its preserved salt marsh and coastal maritime ecosystems. Guests can choose from morning or twilight flight options, which are offered alongside motorized hang-gliding expeditions and lessons.
3990 Centurion Way #5202, Fernandina Beach, FL 32034, Phone: 904-430-3800
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18.Espana Restaurant and Tapas
© Espana Restaurant and Tapas
Espana Restaurant and Tapas is a charming Spanish and Portuguese-style restaurant in Fernandina Beach's historic downtown district, offering indoor and outdoor seating in intimate dining room and outdoor garden seating spaces. The restaurant showcases the traditional flavors of the Iberian peninsula, including high-quality paellas prepared with authentic recipes and imported ingredients. A full menu of Spanish-style tapas is served up throughout the day, including picadinho pork tenderloin, patatas bravas, chorizo encebollado, and drunken clams tossed in white wine sauce. Land and sea entrees range from Alentejo-style pork and clams and pollo con limon y ajo to Lisbon-style grouper, scallops with salsa verde, and Spanish-style shellfish casserole. Homemade Iberian-style desserts are complemented by a full menu of fine wines and house-prepared sangrias.
22 S. 4th Street, Fernandina Beach, FL 32034, Phone: 904-261-7700
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19.T-Rays Burger Station
© T-Rays Burger Station
T-Rays Burger Station has been named as one of USA Today's 50 Best Burger Joints in America, originally opened in 1988 within an Exxon gas station in Fernandina Beach's downtown district. The restaurant retains its historic character today, with no formal outdoor sign or advertising beyond word-of-mouth reputation. Excellent diner-style fare is served up at breakfast and lunch daily, including mouthwatering omelettes, homestyle biscuits and gravy, and full breakfast platters with eggs, sausage, pancakes, or hashbrowns. At lunch, diners can enjoy the restaurant's signature Little T and Big T burgers or choose from a selection of classic American sandwich favorites, ranging from Cajun fish sandwiches and traditional BLTs to vegetarian options like grilled portobello mushroom sandwiches. Crab cake dinners are also served up, along with fresh-caught fried shrimp and fish platters.
202 S 8th St, Fernandina Beach, FL 32034, Phone: 904-261-6310
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20.Four Seasons Bistro
© Four Seasons Bistro
Four Seasons Bistro was originally opened in 2012 by married duo Stephanie Christopher and Michael Walden, serving up hearty, casual Americana and Mediterranean-style entrees in a laid-back dining environment. The restaurant is themed to reflect the seasonal bounty of the Florida region, offering quarterly-rotating menus and seasonal ambiance. Diners can enjoy hearty pastas and land and sea entrees at dinner seven nights a week, including options such as shrimp and grits, gyro platters, Italian-style fettuccine alfredo, and the restaurant's signature meat lasagna. At brunch, seafood frittatas and bistro French toast platters take center stage, along with cauliflower or pita crust pizzas and health-conscious wrap sandwiches.
474305 E State Road 200, Fernandina Beach, FL 32034, Phone: 904-310-6821
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21.Bosque Bello Cemetery
© Lucian Milasan/stock.adobe.com
Bosque Bello Cemetery, commonly referred to as Beautiful Woods, is Fernandina Beach's oldest historic cemetery, believed to have been originally established by the island's Spanish colonizers in the late 18th century. The cemetery, which is located within the city's Old Town district, has lost most of its earliest gravestones to erosion and theft over the years, but preserves historic gravestones dating back as far as 1813. Notable area citizens buried at the cemetery include Alice Youngblood, Dr. Elisha Graham Johnson, Nathaniel Borden, Salvatore Versaggi, and Mike Salvador. Several sections also inter regional soldiers who fought in the Vietnam and Spanish-American Wars and area convent members from the Sisters of St. Joseph. Guided tours of the cemetery are offered each Saturday morning between October and May with advance reservations.
1320 N 14th St, Fernandina Beach, FL 32034, Phone: 904-310-3356
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Marlin and Barrel Distillery, Photo: Marlin and Barrel Distillery
Fort Clinch State Park, Photo: steveottie/stock.adobe.com
Main Beach Park, Photo: Sved Oliver/stock.adobe.com
The Amelia Island Museum of History, Photo: The Amelia Island Museum of History
The Fernandina Beach Pinball Museum, Photo: The Fernandina Beach Pinball Museum
The Maritime Museum of Amelia Island, Photo: The Maritime Museum of Amelia Island
Blue Door Artists, Photo: Blue Door Artists
Egan's Creek Park, Photo: Pamela/stock.adobe.com
Fernandina Beach Marketplace, Photo: Brian Yarvin/stock.adobe.com
George Crady Bridge Fishing Pier State Park, Photo: gadzius/stock.adobe.com
Fernandina Plaza Historic State Park, Photo: danflcreativo/stock.adobe.com
Amelia Island Lighthouse, Photo: Christopher Boswell/stock.adobe.com
Amelia Island Kayak Excursions, Photo: Mat Hayward/stock.adobe.com
Fish Amelia Island, Photo: John/stock.adobe.com
Amelia River Cruises and Charters, Photo: Amelia River Cruises and Charters
Amelia Island Downtown Tasting Tours, Photo: Amelia Island Downtown Tasting Tours
Florida Adventure Sports, Photo: smuki/stock.adobe.com
Espana Restaurant and Tapas, Photo: Espana Restaurant and Tapas
T-Rays Burger Station, Photo: T-Rays Burger Station
Four Seasons Bistro, Photo: Four Seasons Bistro
Bosque Bello Cemetery, Photo: Lucian Milasan/stock.adobe.com
Cover Photo: jovannig/stock.adobe.com
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Murder for TV Ratings! See the True Story of Brazilian TV Show Turned Murderer in "Killer Ratings"
By MaFt On May 16, 2019 In Documentaries, Netflix Originals, News, Release Dates, Trailers	Documentaries, Netflix Originals, News, Release Dates, Trailers 2
Netflix has just released the trailer for an upcoming docuseries, "Killer Ratings", about a Brazilian TV host who literally murdered to get his TV ratings up. Wallace Souza hosted the TV show 'Canal Livre' from 1989 through to 2009 – a show that used investigative journalism to fight crime and often had camera crews following police during raids. The popularity of the show resulted in him getting into politics as well. In 2009, however, Souza was arrested after accusations were made that he ordered a number of murders just to get better ratings on the TV show… "Killer Ratings" arrives on May 31st, only on Netflix.
Killer Ratings: A series about a TV celebrity, murder, political power, and a criminal empire. It tells the true-life story of a Brazilian TV host who who was accused of literally killing for ratings and using his show to cover up the truth.
The CW Has Not Renewed Their Deal with Netflix
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About MaFt
Film and TV fan, creator of New On Netflix (UK, USA, Australia and Canada), dad of two amazing children, code geek and passionate about autism.
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Online Programs Applied Innovation & Entrepreneurship Staff
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Meet the University of Adelaide Staff
Dedicated to the career success of all students in the project management program, the University of Adelaide staff are a devoted team of professors, operating in the Entrepreneurship, Commercialisation and Innovation Centre (ECIC) combines industry insight with an interactive online curriculum, ensuring that students receive a unique, relevant education.
The ECIC staff provides high-quality teaching and research in strategically chosen areas of study. Members of this staff are highly regarded by employers and have been successful in a variety of areas. The academic staff are leaders in their disciplines, who actively contribute to solving real-world problems through consulting and advanced fundamental and applied research.
Paul Steffens
Professor Paul Steffens is Professor of Entrepreneurship at the ECIC, The University of Adelaide. His core areas of expertise are entrepreneurship and diffusion of innovations. Paul has been a lead investigator for research projects totalling more than AUD 1.5 million and has published over fifty academic publications, including journal articles in leading entrepreneurship, marketing and technology management journals.
Previous appointments include Director of the Technology and Innovation Management Centre, University of Queensland, academic positions at University of Queensland and Monash University and visiting positions at University of Kiel, Penn State University and Southern Illinois University. He holds a Bachelor of Engineering (Honours I) and a PhD in diffusion of innovations, both from the University of Queensland.
Paul has also consulted for a wide variety of both business and public agencies. His clients include Energex, Waitrose (UK), Agribuys (USA), IPAA, Queensland Department of Primary Industries, Monsanto, Bunge, Franklins, Solutions Research and the Meat Research Corporation.
The Entrepreneurship, Commercialisation And Innovation Centre (ECIC)
The ECIC is a centre for excellence, where imagination, creativity and knowledge produce innovative outcomes.
The ECIC's mission is to foster innovation through entrepreneurship, commercialisation and project management research, teaching and community and industry engagement activities.
The ECIC is all about helping individuals make a difference – to create wealth in the form of economic growth and social advancement for companies and communities.
The ECIC's programs and courses are designed to assist individuals in achieving personal growth and professional goals. Experienced Academic Directors within the ECIC deliver a comprehensive suite of flexible programs in the following disciplines:
Project Management – Project Systems
The ECIC is committed to research excellence and has cultivated a strong and diverse research program with a range of options for people wishing to undertake research-based projects or PhD studies.
Industry and Community Engagement
Our industry and community engagement initiatives include our student business planning competition the Entrepreneurs' Challenge (eChallenge), and our Thebarton Student Entrepreneurial Incubator (ThIncLab).


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Guardians Extend Spring Training Invites to 3 MLB Veterans
The Cleveland Guardians have extended Spring Training invites to three MLB veterans. / (AP Photo/Ron Schwane, File)
The Cleveland Guardians will continue to fine-tune their roster for the upcoming season. Now officially in 2023, Cleveland must decide on what players it wants a closer look at with Spring Training rapidly approaching.
On Wednesday, the team extended camp invites to three MLB veterans for the spring.
The Guardians have signed four players to minor-league deals that include invites to big-league spring training:
OF Roman Quinn
RHP Touki Toussaint
C Cam Gallagher
RHP Michael Kelly
— Zack Meisel (@ZackMeisel) January 4, 2023
Outfielder Roman Quinn, pitchers Touki Toussaint and Michael Kelly, and catcher Cam Gallagher have all signed minor-league contracts with the Guardians. These deals include Spring Training invitations, bringing some experienced options to fight for roster spots.
Quinn is the biggest name of this lot. He saw a significant run with the Philadelphia Phillies from 2016-22, logging 201 appearances. Injuries prevented him from ever developing any real consistent production, though. The 2011 draftee finished his Phillies tenure with 8 home runs and 41 RBIs before spending the second half of last season with the Tampa Bay Rays.
Touissant also sticks out here. He was the No. 16 overall selection in the 2014 draft and viewed as one of the best prospects in the league at one point. However, his early dominance in the minors never translated to the majors even after switching to a bullpen role. The 6-foot-3 righty owns a career 4.62 ERA across 170.1 innings so far.
Gallagher has served in a backup role for the Kansas City Royals since the 2017 season. He appeared in 171 games for KC over the past six years, hitting .240 with 7 home runs and 39 RBIs combined.
Kelly, meanwhile, just made his MLB debut in 2022 at 29 years old. He saw four appearances for the Philadelphia Phillies, allowing 1 earned run in four innings.
This is a group of intriguing names and backgrounds. It remains to be seen if any are able to crack the Opening Day roster for Cleveland.


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text:
Elsbeth Tucher (auch: Elisabeth Tucher, geborene Pusch; geboren 1473; gestorben im September 1517) war eine Angehörige des Nürnberger Patriziergeschlechts der Tucher. Bekannt wurde sie durch ihre Abbildung auf einem Gemälde Albrecht Dürers, das sich in der Gemäldegalerie Alte Meister (Kassel) befindet.

Leben
Elisabeth Pusch stammte aus Verhältnissen bescheidenen Wohlstands. Ihr Vater Hans Pusch war Zeugmeister der Artillerie, nach Angaben des Familienbuchs der Tucher aber auch verschuldet. Ihre Mutter war eine geborene Elisabeth Zollner. Im Jahr 1492 heiratete sie Nikolaus Tucher (1464–1527), der aus einer reichen Patrizierfamilie stammte. Die Ehe blieb kinderlos. Elisabeth Tucher gehört zu den Personen, über die im Familienbuch der Tucher am wenigsten zu erfahren ist.

Porträt und Vergleich

Im Jahr 1499 fertigte Albrecht Dürer ein Eheporträt von Nikolaus und Elsbeth Tucher in Form eines Diptychons aus Lindenholz an. Elsbeth, 26 Jahre alt, wurde durch ein ihr zugewandtes Porträt ihres Ehemanns Nikolaus Tucher ergänzt, das den verschollenen linken Flügel des Ehepaar-Diptychons bildete.

Ein weiteres Diptychon im selben Stil fertigte er im selben Jahr von Nikolaus' Bruder Hans Tucher und dessen Ehefrau Felicitas, geb. Rieter, an. Getreu mittelalterlicher und frühneuzeitlicher Symbolik ist der Ehemann jeweils links, die Ehefrau rechts abgebildet, sodass sich die Eheleute gegenseitig ansehen. Der Hintergrund ist in diesem Fall jeweils ein mittiges Fenster mit vergleichsweise schematischer Landschaft sowie bei den Damen ein dekoratives Brokattuch, das über die Schulter hängt.

In dem detailgetreuen und lebensnahen Porträt Elsbeths zeigt sich im Vergleich zu den ähnlich gestalteten Werken früherer Künstler – etwa Dürers Meister Michael Wolgemut, der 1478 das Porträt von Ursula Tucher geb. Harsdörffer und 1481 das ihres Mannes Hans Tucher (1428–1491) gemalt hatte – die stilistische Fortentwicklung in der Porträtkunst. Zu den auffallenden Details des Gemäldes zählen der lose zwischen den Fingern gehaltene Ehering in Elsbeths Hand unten links, die große Haube für ihr langes Haupthaar und verschiedene Inschriften sowie abgekürzte Initialen (MH MNSK). Auch Hans der Jüngere (bei Dürer) und Hans der Ältere (bei Wohlgemut) halten jeweils einen Ring in der Hand, ersterer einen schlichten Silberring, letzterer einen wohl seiner Frau zu überreichenden Ring, der mit Edelsteinen besetzt ist, während diese eine Blume hält. Der zu überreichende Ehering blieb auch im 16. Jahrhundert ein künstlerisches Motiv, etwa bei den bemalten Terrakotta-Büsten der Eheleute Willibald Imhoff und Anna Imhoff von Johan Gregor van der Schardt (1570).

Rezeption

Das Gemälde gelangte in der ersten Hälfte des 18. Jahrhunderts in die Sammlungen der Landgrafen von Hessen-Kassel. In der Folgezeit kopierten und zitierten weitere Künstler das Werk. 1951 erschien ein Roman von Fridel Marie Kuhlmann mit Tucher als Bezugsperson, welches auch das bekannte Gemälde referenzierte: Der Ring der Elisabeth Tucher.

Besondere Bekanntheit erlangte das Porträt Elisabeth Tuchers durch die Abbildung auf der Vorderseite des deutschen 20-Mark-Scheins in der Zeit von 1961 bis 1992 – diese Banknote der Dritten Serie war die am längsten im Umlauf befindliche Banknote in der Geschichte der Bundesrepublik; die Auswahl dieses Porträts lenkte das öffentliche Interesse entsprechend stark auf die porträtierte Person. Bei der Banknote wurde aber das netzförmige Dekor des haubenartigen Kopftuches zu einer Art echter Netzstruktur verstärkt und hängt nicht nur auf einer, sondern auf beiden Seiten herunter.

Einzelnachweise

Weblinks

Person (Nürnberg)
Geboren 1473
Gestorben 1517
Frau
Elsbeth

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}

Missing data are ubiquitous in  applications of statistics, including survey sampling, healthcare, public policy and bioinformatics.
The effort needed for modeling and analyzing observations in these situations crucially depends on the mechanism that induces the missing data as well as on the mode of inference \citep{littlebook}.

Here, we work within the inferential framework outlined in \citet{rubi:2004}.
The inferential target of interest is generally a function of  observed and missing data.
One key concept is that of  a ``missing at random'' missing data mechanism, for which the conditional probability distribution of the missingness indicators is  a function only of the observed data, and the related concept of ignorable missing data \citep{Rubin1976,rubin1978aa,Mealli:2015aa}.
For Bayesian inference, ignorability implies that the posterior distribution of the target is conditionally independent
of the observation indicators, given the observed data, and so
there is no need to specify a model for observation indicators to achieve valid Bayesian or likelihood-based inference. When the posterior distribution of the
inferential target depends on these indicators, valid Bayesian
 inference requires specifying a joint model for the data and
the  indicators.

With missing data, there are two basic
approaches to specify the the joint distribution of the complete data and missing
data indicators given parameters, and the implied likelihood, as a product of marginal and conditional
distributions.
The first basic approach \citep{rubin1974aa} is to posit a standard model for the complete data, and
then specify an explicit model that selects observed data from the complete data, called the missingness mechanism \citep{Rubin1976}.
The second basic approach is to specify separate distributions for the observed data and the missing data, thus eschewing explicit assumptions about the missingness mechanism \citep{Rubin1977,little1993}.
 The fundamental challenge with these two basic approaches is that
 some assumptions about the missingness mechanism, whether explicit or implicit, are typically not testable from the observed data.
 As a result, some literature on inference in the presence of missing data  centers on assessing sensitivity to different model specifications \citep{Rubin1977,Scharfstein1999,andrea2001,littlebook}.

\subsection{Contributions}

Here, we develop an alternative approach to modeling data possibly missing not at random, evidentially originally proposed by Tukey and discussed by Hartigan and Rubin, reported by \citet{holland-notes}.
The key insight is to represent the joint distribution of the complete data and  missing-data indicators as proportional to factors that involve only  observed values and the missingness mechanism.  Assumptions about these factors are either testable, or typically allow the incorporation of substantive knowledge about the problem at hand, thereby offering a clear path to eliciting a realistic portrayal of the sources of variation in the data.

In Section \ref{sec:mnar}, we review the two basic  model specifications for  missing data, before formally introducing Tukey's representation.  In Section \ref{sec:bayesian}, we discuss technical issues involved when using Tukey's representation, and provide a full characterization for exponential-family models. In Section \ref{sec:results}, we then illustrate the use of these models on simulated data when the model is correctly specified,  when the model is incorrectly specified, and on an application in  biology. In Section \ref{sec:disc}, we offer  theoretical insights and use them to state formal results.

\section{Basic models for  missing data}

\label{sec:mnar}

Discussion of  models using basic factorizations for missing data can be found in a variety of places, including \citet{glynn} and \citet{handbook}.

Throughout, let $Y = (Y_1, Y_2, ..., Y_N)^\intercal$ represent the
complete data and $R=(R_1,R_2,...,R_N)^\intercal$ represent the response
indicators for $Y$, which are ``missing'' when $R_i$=0 and
``observed'' when $R_i=1$.  The joint distribution, assuming $(Y_i,R_i)$ are i.i.d., is then
$$P(Y,R \mid \theta) = \prod_i f(Y_i,R_i\mid\theta),$$
where $\theta$ is a  parameter vector.
For simplicity,  we focus on the the case without covariates.

\subsection{The selection factorization}
\label{sec:selfact}

The selection approach \citep{rubin1974aa} factors the joint distribution of $(Y,R)$ as
\begin{equation}
\label{eq:selfac}
P(Y,R \mid \theta) = \prod_i^N f(Y_i|\theta_Y)f(R_i|Y_i,\theta_{R|Y}),
\end{equation}
using the  distribution of the complete data, $P(Y|\theta_Y)$, and the missingness mechanism, or selection function, $P(R|Y,\theta_{R|Y})$, which controls
which data are actually observed, where the parameters $(\theta_Y, \theta_{R|Y})$,  are functions of $\theta$.
Models for $f(Y_i|\theta_Y)$ include the normal, with $\theta_Y$ the mean and variance of the normal, or the Bernoulli with $\theta_Y$ the probability of success of the Bernoulli.  Typical models for $f(R_i|Y_i,\theta_{R|Y})$ include the logistic and probit models \citep[e.g., see][]{Gelman2004}.

\subsection{The pattern-mixture factorization}
\label{sec:patfact}

The pattern-mixture approach \citep{Rubin1977,little1993,littlebook} is the  alternative basic factorization; the complete data distribution is specified as a mixture of observed
and missing data components,
\begin{eqnarray}
\label{eq:patmix}
P(Y,R | \theta) &=& \prod_{i=1}^N f(Y_i|R_i, \theta_{Y|R})f(R_i|\theta_R) \nonumber \\
&=& \prod_{i=1}^N\prod_{r=0}^1 \left[f(Y_i|R_i=r, \theta_{Y|R=r})f(R_i=r|\theta_R)\right]^{I(R_i=r)},
\end{eqnarray}
where $(\theta_{Y|R},\theta_R)$ are functions of $\theta$, leading to a mixture likelihood, which for a single  observation is
$$ P(Y_i \mid \theta) = \sum_{r=0}^1 f(Y_i|R_i=r, \theta_{Y|R=r})f(R_i=r|\theta_R).$$
The  model for $f(R_i|\theta_R)$ is a Bernoulli distribution with parameter $\theta_R$. The model for $f(Y_i|R_i=1)$ is typically chosen to fit the observed data well, whereas the model for $f(Y_i|R_i=0)$ is commonly chosen to be a location shift or scale change of $f(Y_i|R_i=1)$ \citep[e.g., see][]{Gelman2004,littlebook}.

\subsection{Tukey's representation}
\label{sec:tukey}

John W.\ Tukey, in a discussion of  \citet{glynn},
suggested an alternative factorization of the joint distribution for
$(Y,R)$ in terms of conditional distributions \citep[recorded in][]{holland-notes}, which he refers to as the {\em simplified
  selection model}, with parameters $\theta_{Y|R=1}$ and $\theta_{R|Y}$,
\begin{equation}
\label{tukeysFactorization}
P(Y,R \mid \theta ) \propto \prod_{i=1}^N f(Y_i\mid R_i=1, \theta_{Y|R=1}) \cdot \frac{f(R_i\mid Y_i, \theta_{R|Y})}{f(R_i=1\mid Y_i,\theta_{R|Y})},
\end{equation}
with normalizing constant $\prod_{i=1}^N f(R_i=1\mid\theta_{R|Y},\theta_{Y|R=1})$ ensuring integrability.
\revs{
As Holland notes, a main advantage of this factorization is
that it only involves the observed data density, $f(Y_i \mid R_i=1, \theta_{Y|R=1})$, which can be
estimated directly, and the missingness mechanism, $f(R_i\mid Y_i, \theta_{R|Y})$, which can be easy to elicit in the context of a specific application.}

Tukey's representation can be obtained through a simple application of Brook's
lemma, which equates the ratio of joint distributions to the ratio of their full conditionals \citep{brook64,besag74}.
Although this lemma is most commonly referenced in the theory of
spatial autoregressive models \citep{cressie}, its connection to
Tukey's representation is relevant because it immediately reveals some theoretical insights.
Importantly, Brook's Lemma is only applicable
when the so-called \emph{positivity condition} is satisfied \citep{hammersley1971aa}, which for Tukey's representation  means that
\begin{align*}
&\text{If: } P(R_i=r|\theta)>0 \text{ and } P(Y_i=y|\theta)>0 \\
&\text{Then: } P(R_i=r,Y_i=y|\theta) > 0
\end{align*}
for all  pairs of values $(r,y)$.
This condition is not trivially satisfied in  missing data problems.  For instance, Tukey's representation {cannot} be applied to models where $P(R_i=1|Y_i<c,\theta_{R|Y}) = 0$, deterministically, for some cutoff $c$, as when the complete data model is normal and the observed data model is truncated normal.  Consequently, here we focus on problems where $P(R_i=1|Y_i,\theta_{R|Y}) > 0$, that is, where the support of the missing data is contained in the support of the observed data.

In addition, the  conditional distributions specified in Equation \ref{tukeysFactorization}
must imply an integrable joint density \citep{besag74}.  With
Tukey's representation, the \emph{integrability condition} constrains the rate
at which the tails of the distribution for observed data decrease relative to the rate
at which the odds of a missing value increase. This condition is illustrated in Section \ref{sec:sim}, and discussed in Section \ref{sec:integrability}.

\section{Modeling and inference using Tukey's representation}
\label{sec:bayesian}

Let $P(Y_i = y_i \mid R_i=1,\theta_{Y_i|R_i=1})$ be denoted $f^{\rm obs}(y_i\mid\theta_{Y|R})$, for simplicity.
Using Equation \ref{tukeysFactorization}, we can write the density for $(y_i,r_i)$ as
\begin{eqnarray}
\label{lik:comp}
f(y_i,r_i \mid \theta_{R|Y},\theta_{Y|R})
&\propto& \begin{cases} f^{\rm obs}(y_i\mid\theta_{Y|R}) &\mbox{if } r_i=1 \nonumber \\
\frac{f(r_i=0\mid y_i, \theta_{R|Y})}{f(r_i=1\mid y_i, \theta_{R|Y})} f^{\rm obs}(y_i\mid\theta_{Y|R}) & \mbox{if } r_i=0\\
\end{cases} \\
&=& f(r_i=1\mid y_i, \theta_{R|Y})^{r_i-1}f(r_i=0\mid y_i, \theta_{R|Y})^{1-r_i}f^{\rm obs}(y_i\mid\theta_{Y|R})
\end{eqnarray}
with normalizing constant
\begin{eqnarray}
\label{eqn:normconst}
Q (\theta_{Y|R},\theta_{R|Y})
 &=& \left(1+\int \frac{f(r_i=0\mid y_i, \theta_{R|Y})}{f(r_i=1\mid y_i, \theta_{R|Y})} f^{\rm obs}(y_i \mid \theta_{Y|R}) ~dy_i\right)^{-1},
\end{eqnarray}
which ensures the integral of Equation \ref{lik:comp} over  random variables $R$ and $Y$ times $Q$ is unity.

The normalizing constant $Q$  is generally difficult to compute. As  a consequence, the missing data density   cannot easily be expressed.    Below, we introduce a class of models for which computation of  the normalizing constant is tractable and which also implies simple  distributional forms for the missing data and complete data  densities.

\subsection{Exponential family models}
\label{sec:ef}

Assume that the observed data distribution belongs to an exponential family and that
the logit of the missingness mechanism is linear in the sufficient statistics of that family, which is related to the
exponential tilt pattern-mixture models introduced by
\citet{Birmingham2003}.
Formally, let $f^{\rm obs}(y_i \mid \theta_{Y|R=1})$ be an exponential family distribution with natural parameter $\theta_{Y|R=1}$.  Then,
\begin{equation}
\label{eq:exp-log-obs}
f^{\rm obs}(y_i \mid \theta_{Y|R=1}) = h(y_i)g(\theta_{Y|R=1})e^{T(y_i) ' \theta_{Y|R=1}},
\end{equation}
where $g(\theta_{Y|R=1})$ is the normalizing constant and $T(y_i)$ is the natural sufficient statistic, possibly multivariate.
The missingness mechanism
\begin{equation}
\label{eq:exp-log-sel}
f(r_i = 1 \mid y_i,\theta_{R|Y}) = \hbox{logit }(T(y_i)'\theta_{R|Y}) = \frac{1}{1+e^{- \theta_{R|Y}}}
\end{equation}
implies that
\begin{equation}
\nonumber \frac{f(r_i = 0 \mid y_i,\theta_{R|Y})}{f(r_i = 1 \mid y_i,\theta_{R|Y})} = e^{- T(y_i)\theta_{R|Y}}.
\end{equation}
Then the normalizing constant $Q$ in Equation \ref{eqn:normconst} can be written as a simple function of the normalizing constant $g(\cdot)$ in the exponential family formulation of $f^{\rm obs}$,
\begin{eqnarray}
\label{eqn:exfam}
  Q(\theta_{Y|R=1},\theta_{R|Y})
  &=& \frac{g({\theta_{Y|R=1}}+{\theta_{R|Y}})}{g({\theta_{Y|R=1}}+{\theta_{R|Y}})+g(\theta_{Y|R=1})}.
\end{eqnarray}

For the class of exponential-logistic models defined by Equations \ref{eq:exp-log-obs}--\ref{eq:exp-log-sel}, the missing data distribution, as specified in Equation \ref{miss_dist_ef}, is from the same exponential family as the
observed data with natural parameter\footnote{In the exponential-logistic models we consider in this section, the two parameter vectors $(\theta_{Y|R=1},\theta_{R|Y})$ always have the same dimensionality. }
$\theta_{Y|R=0} = \theta_{Y|R=1} + \theta_{R|Y}$,
\begin{equation}
\label{miss_dist_ef}
f^{\rm mis}(y \mid \theta_{Y|R=1},\theta_{R|Y}) = h(y) g(\theta_{Y|R=0}) e^{{T(y)}'\theta_{Y|R=0}}.
\end{equation}
Thus, missing data imputation with this class of models is straightforward.

In Tukey's representation,
 the main source of uncertainty about any inferential target is due to the missingness mechanism, and the prior distribution on $\theta_{R|Y}$.

\subsection{Estimation and inference}
\label{sec:est:inf}

The primary estimands of interest are typically functions of the
parameters specifying the complete-data distribution, $(\theta_{Y|R=1},\theta_{R|Y})$.
Because the observed and missing data densities for exponential-logistic models are exponential families, see Equations \ref{eq:exp-log-obs} and \ref{miss_dist_ef},  the complete-data distribution is a mixture of exponential
families.  Further, analytic expressions for the normalizing constant
$Q$ and the likelihood are available, and thus standard Markov Chain
Monte Carlo methods are applicable \citep{RoberCase}.

We take a simple approach to inference via Markov Chain Monte Carlo, which is computationally less
demanding than with samplers that explicitly characterize the geometry of the solution space, as discussed in Section \ref{sec:disc}.
Consider a simple Normal-logistic model as an illustration, with  $\theta_{R|Y} = (\beta_0,\beta_1) = \beta
\in R^2$ with $\beta_0$ corresponding to the intercept and $\beta_1$
the rate at which the odds of selection change in $T(y)=y$:
\begin{eqnarray}
 f(r_i = 1 \mid y_i,\beta) &=& \hbox{logit }(\beta_0+\beta_1 y_i)
                            = \bigm(1+\exp\{- \beta_0-\beta_1 y_i\}\bigm)^{-1} \nonumber \\
 f^{\rm obs}(y_i)          &=& \hbox{Normal }(0,1). \nonumber
\end{eqnarray}
Rather than specifying a
prior distribution on  $(\beta_0, \beta_1)$, we specify a prior distribution on $Q$ and
$\beta_1$, but not on $\beta_0$. We then invert Equation \ref{eqn:exfam} and solve for $\beta_0$ to get,
\begin{equation}
\label{eqn:invert}
\beta_0(Q,\eta,\beta_1) =
\text{log}\left(\frac{g(\eta+\beta_1)(1-Q)}{g(\eta)}\right).
\end{equation}
In this Normal-logistic example,
the standard Normal has natural parameters $(\eta_1,\eta_2)=(0,-1/2)$, and  $g(\eta) =
-\frac{\eta_1^2}{4\eta_2} - \frac12\ln(-2\eta_2)$.  Thus, Equation \ref{eqn:exfam} becomes,
\[
 Q(\beta_0,\beta_1) = \frac{\beta_1^2}{\beta_1^2-2e^{\beta_0}},
\]
and solving Equation \ref{eqn:invert} we get
\[
\beta_0 = \text{log}\left(\frac{\beta_1^2(1-Q)}{2}\right).
\]
Next, we use this inferential strategy to demonstrate the utility of Tukey's representation.

\section{Numerical examples and an application in biology}
\label{sec:results}

In Section \ref{sec:sim}, we explore selection, pattern-mixture, and Tukey representations for missing data using simulation studies, where $f^{\rm obs}(y_i\mid r_i=1,\theta_{Y|R=1})$ is a mixture of exponential family distributions.
In this example, we discuss how both the selection and the pattern mixture factorizations require difficult modeling choices in practice, and illustrate how Tukey's representation can be employed with relative ease.
In Section \ref{sec:robustness} we explore the robustness of logistic-exponential family missing-data models when the true generating process cannot be expressed as a finite mixture of exponential families.
In Section \ref{sec:app} we demonstrate the utility of Tukey's factorization in an application to biological data \citep{franks2014}.

\subsection{Illustration with semicontinuous data}
\label{sec:sim}

Let us posit that the complete data observations are drawn from a mixture of discrete and continuous distributions, which we term a semicontinuous distribution.  Specifically, we assume that the continuous component is a mixture of normals and that the missing data mechanism is logistic in the data.  Here, we specify the full data generating process using Tukey's representation.  First, we assume the observed data density has form
\[
 f^{\rm obs}(y_i \mid r_i=1, \theta_{Y|R=1}) = \alpha\left(\sum_{k=1}^K w_k \cdot \hbox{Normal }(y_i; \mu_k,\sigma_k^2)\right)+(1-\alpha)\left(\sum_{m=1}^M \, p_m \cdot \delta_{\{\gamma_m\}}(y_i)\right)
 \]
 where $\theta_{Y|R=1} = (\alpha, w_{1:k,}, \mu_{1:k}, \sigma^2_{1:k}, p_{1:m}, \gamma_{1:m})$, and $\delta_{\{\gamma\}}(y)$ is the Dirac delta function that is one when $y=\gamma$, and zero elsewhere.  The parameter $\alpha$ captures the fraction of observations attributable to the continuous component, and the parameters $p_{1:M}$ specify the probabilities for the point masses at $M$ locations $\gamma_{1:M}$.  Recalling the notation introduced in Section \ref{sec:ef}, for the $k$-th Normal mixture component, the natural parameter is $\theta_{Y|R=1} = (\mu_k,\sigma^2_k)$ and the corresponding sufficient statistics is $T(y_i) = (y_i,y_i^2)$. For the discrete distribution, $\theta_{Y|R=1} = p_{1:M}$ and $T(y_i) = 1$ if $y_i$ equals one of the $M$ discrete locations, zero otherwise.

The missingness mechanism specifies that the logit of the selection probabilities are
quadratic in $y_i$, but importantly, linear as a function of the sufficient statistics of the normal components:
\begin{eqnarray}
\label{sim:mech}
f(r_i=1 \mid y_i,\theta_{R|Y})
   &=& \hbox{logit}^{-1}(-(\beta_0 + (y_i-\beta_1)^2\beta_2)) \\
   &=& \bigm(1+\exp\{\beta_0 + y_i^2\beta_2 - 2 \beta_1 \beta_2 y_i + \beta_1^2 \beta_2\}\bigm)^{-1} \nonumber
\end{eqnarray}
where $\theta_{R|Y} = (\beta_0,\beta_1,\beta_2)$.  Here, $\beta_1$ corresponds to the value at which an observation is most likely to occur, whereas $\beta_2$ controls how quickly the observation probability decays with the distance of $y_i$ to $\beta_1$.  Non-monotonicity in the probabilities of selection occurs in practice when extreme values, both small and large, are less likely to be observed.

Using Equation \ref{miss_dist_ef}, we can see that  the missing
data distribution for the continuous component is also a  mixture of Normals.  The missing data has density
\begin{eqnarray}
\label{sim:miss}
 f^{\rm mis}(y_i \mid -)

  &=& \alpha^*\left(\sum_{k=1}^K \, w_k^* \cdot \hbox{Normal }(y_i ; ~\frac{\mu_k - 2 \beta_1 \beta_2 \,\sigma_k^2}{1 - 2\beta_2 \,\sigma_k^2}, ~\frac{\sigma_k^2}{1 - 2 \beta_2 \, \sigma_k^2})\right) ~+ \nonumber \\
  &+& (1-\alpha^*)\left(\sum_{m=1}^M \,p^*_k \cdot \delta_{\{\gamma_j\}} (y_i)\right).

\end{eqnarray}
This density, with specifications for the mixture
weights $\alpha^*$ and $w^*_{1:K}$, is derived in Appendix
\ref{app:sec:sim}.  The observed data and missing data densities specify
the complete data as semi-continuous mixture, from which all relevant
estimands can be computed.

In order to simulate data from this complex mixture, we fix $K=3$ components for the continuous portion of the observed data mixture, with parameters  set as follows,
\begin{align}
\mu_{1:3} & =(-2,0,3)\\
w_{1:3} & =(0.3,0.4,0.3)\\
\sigma_{1:3} & =(1,1,1)~.
\end{align}
We set $\alpha=0.8$, that is, 80\% of the observed data comes from  the
continuous component. We and fix the discrete component of the observed data to be uniform on $\gamma_{1:9} = \{-4,-3,...,3,4\}$, where $M=9$.
Finally, for the
missingness mechanism, we assume that $\beta_2 = .06$, $\beta_1=-2$ and
$Q = 0.5$, that is, $50\%$ of the complete data is missing.  We then use
Equation \ref{eqn:invert} to derive that $\beta_0 \approx -0.85$.
The full derivation is provided in Appendix \ref{app:sec:sim}.

The top panel of Figure
\ref{fig:sim_dens} shows the true missingness mechanism used to
simulate data, as a bold black line.
The bottom panel shows a
histogram of draws from the observed data as well as the true
observed, missing and complete data densities for the continuous
component.  For this simulation study, we will assume that the
relevant estimands are the complete data mean and complete data standard deviation.

\begin{figure}[t!]
     \centering
     \includegraphics[width=0.7\textwidth]{Figs/sim_densities}
     \caption{\onehalfspacing Top: the true missingness mechanism (black) and 10
       samples from the prior distribution specified in Section
       \ref{sec:tukeyanalysis} (gray).  Bottom: the observed data
       histogram (gray, $N=10,000$) with observed (red) and missing
       data (blue) densities for the continuous component.  Ten
       percent of the observed data comes from a discrete distribution
       on \{-4,-3,...,0,...,3,4\}.}

\label{fig:sim_dens}
\end{figure}

\subsubsection{Analysis using Tukey's representation}
\label{sec:tukeyanalysis}

Here we use Tukey's representation to estimate the
posterior distribution of the complete data mean and standard
deviation, from observations and observation indicators generated from the
model  above.  In order to do so, we  specify prior distributions for the
parameters of the missingness mechanism, $\beta$, the fraction of
missing data, $Q$, and a model for the observed data, $f^{\rm obs}$.

We posit the following prior distributions on the parameters of the missingness mechanism,
\begin{eqnarray}
\label{sim:selprior}
\nonumber \beta_1 &\sim& \text{Normal }(-2;2)\\
\nonumber \beta_2 &\sim& 0.08 \cdot \text{Beta }(3;1)\\
Q &\sim& \text{Uniform }(0,1)
\end{eqnarray}
These prior distribution are centered around the parameters of the true missingness mechanism, $(\beta_1,\beta_2)$, with some variability to reflect a  degree of uncertainty.   Importantly, neither $\beta_1$ nor $\beta_2$ can be estimated from the observed data, and thus the uncertainty in this prior specification will propagate to the posterior, regardless of the amount of observed data.
The top panel of Figure \ref{fig:sim_dens}  shows a number of draws from the above prior distributions, in gray.

Next, we posit the following prior distributions for the observed data parameters, $\theta_{Y|R=1}$:
\begin{eqnarray}
\label{sim:selprior}
\nonumber p_i &\sim& \text{Dirichlet}(1, 1, \ldots, 1) \hbox{ for } i=1,\dots,M=9\\
\nonumber \mu_{k} &\sim& \text{Normal}(0, 10) \hbox{ for } k=1,\dots,K=3\\
\nonumber w_{k} & \sim& \text{Dirichlet}(1, 1, 1) \\
 \sigma_{k} & \sim& \text{Uniform}(0, 2)
\end{eqnarray}
Unlike the prior specification for the parameters of the missingness mechanism, all of the observed data parameters are  estimable.  Thus, the results are only sensitive to the prior specifications of $\theta_{Y|R=1}$ for smaller  observed data sample sizes.

Note that by the integrability condition stated in Section \ref{sec:ef}, in order for the continuous mixture components of the complete data to all have positive variance, it must hold that $\beta_2 < \underset{k=1 \dots K}{\text{min}} ~\frac{1}{2\sigma_k^2} ~.$  This inequality is due to Equation \ref{sim:mis_moms} in  Appendix \ref{app:sec:sim}.
To ensure the integrability condition is satisfied, we also bound the
prior distributions on $\sigma_{1:3}$ above by 2, because $0.08 <
\frac{1}{8}$, where $0.08$ is the maximum value of $\beta_2$ under our
prior distribution.

We present the simulation results for different  sample sizes in
Figure \ref{fig:tukey_sim}.  With a reasonably informative prior distribution for the missingness mechanism, these results show that the analysis with the Tukey's representation accurately recovers the posterior distributions for the relevant estimands.  As the sample size increases, our posterior uncertainty shrinks, but never disappears.  This is because even with perfect knowledge of the observed data density\footnote{\onehalfspacing We simulate this situation  by setting the parameters underlying the observed data distribution to their true values, and we label the corresponding results in Figure \ref{fig:tukey_sim} with $N=\infty$.}, there is no information in the data about the parameters of the missingness mechanism, as mentioned above. Therefore uncertainty about the missingness mechanism propagates to the uncertainty about the complete-data quantities, and is the main source of uncertainty in the posterior when enough data is available. This fact facilitates the sensitivity analysis about the missingness mechanism when using Tukey's representation.

\begin{figure}[t!]
     \centering
     \includegraphics[width=0.95\textwidth]{Figs/tukey_sim}
     \caption{\onehalfspacing Posterior estimates of the complete-data mean (left), complete-data standard deviation (middle) and maximum absolute difference between the true and inferred discrete probability masses (right).  Dashed red line represents the true complete-data value.  As we get more data, estimates of the parameters underlying the observed-data distribution improve, and the posterior concentrates around the true values.  However,  information about the parameters underlying the selection mechanism does not accumulate, thus posterior uncertainty about the complete-data parameters remains.}
    \label{fig:tukey_sim}
\end{figure}

Above, we illustrated how data analysis using Tukey's representation can be applied in a straightforward manner to a complex data generating process. It
does not require any explicit assumptions about the complete-data
distribution; rather it only requires a reasonable model for the observed data,
and plausible assumptions about the selection mechanism.
In contrast, there are often significant challenges for data analysis when applying the
selection factorization, or pattern mixture factorization, as we now discuss.

\subsubsection{Remarks on the use of the selection factorization}

Inference under selection factorization
models typically require numerical or Monte Carlo integration of the
complete data distribution, and hence is computationally and
analytically more demanding than both pattern mixture models and Tukey's representation.
For instance, often the missing data density does not take on a simple form, and
thus missing data imputation is non-trivial.  Finally, it is not
 obvious that the specified selection factorization model
will imply a reasonable fit to the this observed data.  This is a
fundamental challenge associated with any approach that does not
directly model the observed data \citep{rubi:2004,littlebook,lunagomez2014bayesian,Mealli:2015aa}. The focus on modeling the observed data explicitly, rather than implicitly via assumptions on the data generating process, is one of the strengths of analyses based on the Tukey's representation.

Even if we knew the complete data distribution corresponded to a six
component mixture of Normal with discrete masses at certain values, it
would still be difficult to specify appropriately informed prior
distribution for the relevant parameters. Note that in this example,
the parameters of the missing data density are a function of the
selection parameters (Equation \ref{sim:miss}).  For this reason,
specifying independent prior information for the mixture components of
the complete data density implicitly adds additional prior information
about the parameters of the missingness mechanism.  By parameterizing the
model in terms of complete data parameters $\theta_Y$, we are
combining what we don't have information about ($\theta_{Y|R=0}$) with
what we do have information about ($\theta_{Y|R=1}$)
\citep{holland-notes}.

\subsubsection{Remarks on the use of the pattern-mixture factorization}

The true data generating process described at the beginning of Section \ref{sec:sim} is specified as a mixture. Thus, at least in principle, the pattern mixture factorization, described in Section \ref{sec:patfact}, provides an appropriate way to model the complete data.
However, the implied missing data distribution in this example does not correspond to a simple location or scale change of the observed data distribution.  Thus, as is common with pattern mixture models, the missing data distribution for this data cannot be represented with a simple between-components location and scale change \citep[as discussed, e.g., in][]{Daniels2000}.

More generally, when the missing data distribution is complicated, it can be difficult to propose plausible, scientifically justifiable prior distributions for the parameters of the missing data density, $\theta_{Y|R=0}$.
Perhaps even more problematic is that these parameters often imply smoothness and monotonicity of the selection mechanism, which is not explicitly specified in analyses that use the pattern-mixture factorization.

\subsection{Robustness to Misspecification}
\label{sec:robustness}

In Section \ref{sec:tukeyanalysis}, we performed the analysis using Tukey's representation using a family of models that included the true data generating process.  In practice, finite mixtures of exponential families can only be expected to approximate the observed data density.  Here, we evaluate the robustness of Tukey's approach in such cases.

We consider  data generated under a simple selection factorization model.  We assume the complete data is a standard normal, i.e. $\theta_Y = (\mu, \sigma) = (0, 1)$, and that the selection mechanism logistic, linear in the data $y_i$.  Under this data generating process, the observed data density is
\[
 f^{\rm obs}(y_i \mid r_i=1, \theta_{Y}, \theta_{R|Y}) \propto \frac{\exp\left(\frac{(y_i-\mu)^2}{2\sigma^2}\right)}{\sigma(1+\exp(-\beta_0-\beta_1y_i))},
 \]
which cannot
be represented as a finite mixture of exponential families.

For the analysis, we again assume the goal is to estimate the complete data mean and standard deviation.
We use a mixture of normals to approximate the observed data density but take the parameters of the logistic missingness mechanism, $\theta_{R|Y} = (\beta_0, \beta_1)$ to be known exactly.
We estimate the posterior distributions for $\mu$ and $\sigma$ for increasing sample sizes and for different values of the selection parameters, $\beta_1$.
More in detail, in each simulation $\beta_0$ is fixed to 1/2, the parameter $\beta_1$ takes values 1, 2 and 5, and we  replicate the analysis for 100 and 1,000 data points.
To increase the realism of the data analysis, we fit a mixture model with 3 components to 100 data points, but we increase the flexibility of the model for the larger sample size by fitting  a mixture model with 5 components to 1,000 data points.
\begin{figure}[b!]
\centering
\subfigure[Complete Data Mean]{
 \label{fig:robust-mean}
 \includegraphics[width=0.6\textwidth]{Figs/mean-robustness}} \\
\subfigure[Complete Data Standard Deviation]{
 \label{fig:robust-sd}
 \includegraphics[width=0.6\textwidth]{Figs/sd-robustness}} \caption{\onehalfspacing Posterior estimates of the complete data mean and complete data standard deviation as a function of  sample size and the slope of the missingness mechanism.  Blue dots represent the empirical observed data estimates and the dashed red line corresponds to the true complete data quantity.  As the slope, $\beta_1$ increases, the observed data density approaches a truncated normal, for which Tukey's representation is not applicable.}
\label{fig:robustness}
\end{figure}

Figure \ref{fig:robustness} shows box plots of the posterior distributions for the complete data mean (top panels) and complete data standard deviation (bottom panels) for the different values of $\beta_1$ and increasing sample sizes.  To better appreciate the results, note that, as $\beta_1$ increases, the selection probability approaches zero for all $y_i < \beta_0$ and the observed data density approaches a truncated normal model. Because of the positivity condition, such a model cannot be estimated using Tukey's representation.  The results  in Figure \ref{fig:robustness} show that even when the positivity condition holds, in theory, when the selection probabilities are very close to zero, the complete data estimates (blue dots)  become unstable, increasing the bias of the complete data estimates. Thus, in this example, increasing sample size only partially offsets the failure of inference procedures based only on the observed data.

More generally, instability occurs when the odds of selection are  small in regions where the inferred observed data density is non-zero.  This insight can be formalized  by examining the interaction of the observed data density and the selection probabilities in the Equation \ref{miss_dist}.  As a consequence, the fit of the observed data distribution in the tails of the distribution can be very important for accurately inferring complete data quantities.  In practice, caution is needed when using Tukey's representation in cases where the observed data sample size is small or where the selection probabilities diverge rapidly.

\subsection{Analysis of transcriptomic and proteomic data}
\label{sec:app}

In experiments involving measurements of transcriptomic and proteomic
data, mRNA transcripts and proteins which occur at low levels are less
likely to be observed \citep{Walther2010,Soon2013}.  This makes it challenging to infer
normalizing constants for absolute protein levels
\citep{karpievitch12}, cluster genes into functionally related sets
\citep{olga01}, infer the degree of coordination between transcription
and translation \citep{franks2014}, and determine the ratio of dynamic
range inflation from transcript to protein levels \citep{Csardi2015}.
Here, we demonstrate how data analysis with Tukey's representation can be used to
investigate some these issues by assessing the sensitivity of
estimands to different assumptions about the missingness mechanism.

In
this analysis, we use transcriptomic data from \citet{pelechano10}, with 14\% missing data, and protein abundance data from \citet{ghaem03}, with 34\% missing data. For illustrative purposes we treat the complete
data mean and variance as the estimands of interest in the analysis.

It is standard to assume that both mRNA and protein levels are
log-normally distributed \citep{beng2005,lu2007}, although this
assumption may not be justified \citep{Lu09,marko2012}.  Here, we
instead model the observed data as a mixture of normals and specify a
prior distribution for the parameters of the logistic missingness mechanism.  Together these
assumptions imply a more flexible prior distribution over complete data densities.

Further, as noted in \citet{Karpievitch2009}, missing values can occur for
multiple reasons, at different stages of the data collection process.
They find that a small fraction of missing proteomic
data collected using mass spectrometry is missing completely at
random.  Consistent with this, we again generalize the results of Section
\ref{sec:ef} to allow the selection mechanism to have a logistic
form that asymptotes at some value less than one.  The observed data
distribution and missingness mechanism together define the joint
distribution:
\begin{align}
\label{app:obs}
f^{\rm obs}(y_i | r_i=1,\beta,\kappa,\mu_k,\sigma_k) &\sim \sum_{k=1}^K
                                         w_kN(y_i;\mu_k,\sigma_k^2)\\
f(r_i=1|y_i,\theta_{R|Y}) &= \frac{\kappa e^{\beta_1y_i+\beta_0}}{1+e^{\beta_1y_i+\beta_0}}
\end{align}
with $\theta_{Y|R=1} = (\mu,\sigma,w)$ and
$\theta_{R|Y} = (\beta_0,\beta_1,\kappa)$.  Here, $0 < (1-\kappa) < 1$
corresponds the fraction of data that is missing completely at random, and
$\beta_0$ and $\beta_1$ describe the odds of a missing value, with
$\beta_1$ parametrizing the rate at which the odds of a missing value
change in $y_i$.  Under this model, the implied missing data
distribution is
\begin{align}
\label{app:miss}
\nonumber f^{\rm miss}(y_i|r_i=0,\beta,\kappa,\mu_k,\sigma_k) &= (1-\kappa^*)
f^{\rm obs}(y_i|r_i=1,\mu_k,\sigma_k) \\
& + \kappa^*\left(\sum_{k=1}^K w_k^*N(y_i;\mu_k+
\beta_1\sigma_k^2,\sigma_k^2)\right)
\end{align}
The full derivation of the mixture weights
$w^*_k$ and $\kappa^*$ is given in Appendix \ref{app:sec:app}.

In the analysis of the observations, to poist a good model for $f^{\rm obs}$ in  Tukey's representation, we
found that $K=3$ components were enough to approximate the
observed data well.  Furthermore, we chose the following prior
distributions for Q and $\theta_{R|Y}$,
\begin{eqnarray}
\label{app:selprior}
 \beta_1  &\sim&  {\rm Beta}(1,3) \nonumber\\
  Q       &\sim&  {\rm Uniform}(0,1) \nonumber\\
 \kappa   &\sim&  1-(1-Q){\rm Beta}(2,1), \quad \kappa \geq Q
\end{eqnarray}
Note that $\kappa$ must be greater than $Q$ because the selection
probabilities cannot be  less than the population fraction
of observed data. Draws from this prior distribution are shown in Figure \ref{fig:seldist2}, top-left panel, in grey, around the prior mean, in black. We
implemented the sampler for the making inference with this model
using  STAN \citep{stan}.

Figure \ref{fig:seldist2}, bottom-left panel, shows the fit to the protein data \citep{ghaem03}, when $\beta_1$ is set to its median posterior value.
For comparison, the bottom-right panel, shows the fit of the selection factorization model published in \citet{franks2014}, which  assumes the complete data  is distributed according to a lognormal and the missingness mechanism is logistic, with the mean linear in in $y_i$.
The black, red and blue lines, in both  bottom panels, correspond to the estimated densities of complete data, missing data and observed data, respectively.
Lack-of-fit to the observed data is the analysis with the selection model.
Figure \ref{fig:seldist1} shows the corresponding results for the transcript data set \citep{pelechano10}, where the lack-of-fit using the selection factorization is less pronounced.

\begin{figure}[ht!]
\centering
 \includegraphics[width=0.425\textwidth]{Figs/tukey_2}
 \includegraphics[width=0.425\textwidth]{Figs/selection_normal_2}
\caption{\onehalfspacing Model fit to proteomic data from
  \citet{ghaem03} data using two different approaches: Tukey's
  representation (left) and the selection factorization (right).  The
  grey lines in the top-left panel represent draws of the selection
  mechanism from the prior distribution provided in Equation \ref{app:selprior}. The black, red and blue lines in the bottom panels correspond to the estimated densities of complete data, missing data and observed data, respectively.}
\label{fig:seldist2}
\end{figure}
\begin{figure}[ht!]
\centering
 \includegraphics[width=0.425\textwidth]{Figs/tukey_1}
 \includegraphics[width=0.425\textwidth]{Figs/selection_normal_1}
\caption{\onehalfspacing Model fit to mRNA data from
  \citet{pelechano10} data using two different approaches: Tukey's
  representation (left) and the selection factorization (right). The
  grey lines in the top-left panel represent draws of the selection
  mechanism from the prior distribution provided in Equation \ref{app:selprior}. The black, red and blue lines in the bottom panels correspond to the estimated densities of complete data, missing data and observed data, respectively.}
\label{fig:seldist1}
\end{figure}

In Figures \ref{fig:comparisons2}--\ref{fig:comparisons1}, we compare the estimated complete data posterior means and standard deviations  for the protein and transcript data sets, respectively, using three different missing data models:
 Tukey's representation model implied by Equations \ref{app:obs} and \ref{app:selprior};
 a selection factorization model that assumes log-normality of the complete data and a logistic selection mechanism \citep{franks2014,Csardi2015}; and
 a missing complete at random model.
For both data sets, the estimates obtained with  the MCAR model and with the selection factorization models bookend the estimates obtained with  the model specified with Tukey's representation.
 Under the selection
factorization model, the complete data standard deviation is large and
the mean is small relative to the estimates from Tukey's
factorization.  This is likely because the strong parametric assumptions
associated with the  selection factorization models overly constrain the fit
to the observed data.
 Table \ref{app:table} reports exact numerical estimates of the two estimands of interest, using the three competing models.
\begin{figure}[t!]
     \centering
     \includegraphics[width=0.85\textwidth]{Figs/scm_comparisons_2}
     \caption{\onehalfspacing Posterior distributions of the complete data mean (right) and
       complete data standard deviation (left) for protein data
       \citep{ghaem03}.  The MCAR estimates (red) and an estimate
       assuming normality of the complete data (blue) are shown as
       vertical lines for comparison.  Under the prior distribution in Equation \ref{app:selprior}, estimates using the MCAR
       and the selection factorization models are at opposite ends of these posterior distributions. }
    \label{fig:comparisons2}
\end{figure}
\begin{figure}[t!]
     \centering
     \includegraphics[width=0.85\textwidth]{Figs/scm_comparisons_1}
     \caption{\onehalfspacing Posterior distributions of the complete data mean (right) and
       complete data standard deviation (left) for mRNA data
       \citep{pelechano10}.  The MCAR estimates (red) and an estimate
       assuming normality of the complete data (blue) are shown as
       vertical lines for comparison.  Under the prior distribution in Equation \ref{app:selprior}, estimates using the MCAR
       and the selection factorization models are at opposite ends of these posterior distributions. }
    \label{fig:comparisons1}
\end{figure}

Recent published analyses of data using the selection factorization found that translational regulation widens the dynamic range of protein expression \citep{franks2014,Csardi2015}. One way to quantify the relative dynamic ranges of mRNA and protein is by computing the ratio of the standard deviations between log-mRNA and log-protein levels, a protein-specific quantity often referred to as the ``dynamic range ratio''.  A value of this ratio less then one  suggests that the dynamic range of protein levels is smaller than that of mRNA, and is taken as evidence of a suppressive role of translational regulation.  A value greater than one is  taken as evidence of amplification, as claimed by the recent analyses.

We used posterior estimates of the complete data standard deviations, obtained from the three competing models fit to both protein and mRNA data sets \citep{pelechano10,ghaem03} to estimate the distribution of the dynamic range ratios, in Figure \ref{fig:ratio}.  The results obtained with Tukey's representation are consistent with those reported by \citet{Csardi2015}, suggesting that translational regulation reflects amplification of protein levels.
 Table \ref{app:table} also reports  numerical estimates of the the dynamic range ratios.

\begin{table}[t!]
\centering
\begin{tabular}{l|c|c|c|l}
 \belowstrut{0.10in}
 Estimand & Tukey's representation & Selection factorization  & MCAR & Data set \\ \hline
 \abovestrut{0.20in}
      Mean & 7.42 ~(7.18, 7.73) & 6.84 & 7.82 & Prot. \\
 \belowstrut{0.10in}
 Std. Dev. & 1.66 ~(1.52, 1.94) & 2.01 & 1.55 & Prot. \\ \hline
 \abovestrut{0.20in}
      Mean & 0.51 ~(0.44, 0.59) & 0.35 & 0.60 & mRNA \\
 \belowstrut{0.10in}
 Std. Dev. & 1.13 ~(1.07, 1.23) & 1.23 & 1.08 & mRNA \\ \hline
 \abovestrut{0.20in}
    Ratios & 1.48 ~(1.28, 1.73) & 1.62 & 1.43 & Both \\
\end{tabular}
\caption{\onehalfspacing Estimates for the quantities of interest obtained with different models, from protein and mRNA data. The dynamic range ratios are computed using both data sets.  We report  maximum likelihood point estimates for both the MCAR and selection models. We report posterior medians and 95\% posterior intervals (in parentheses) for Tukey's representation.}
\label{app:table}
\end{table}

\begin{figure}[t!]
     \centering
     \includegraphics[width=0.5\textwidth]{Figs/ratio_comparison}
     \caption{\onehalfspacing Dynamic range ratios obtained using Tukey's representation (histogram), the selection factorization model  (blue) of \citet{Csardi2015}, and an MCAR model (red).}
    \label{fig:ratio}
\end{figure}

These results demonstrate the relative ease of applied data analysis  with
Tukey's representation models, and the increased flexibility of models
specified using this full conditional specification.  By directly modeling the observed
data, we avoid the need for Monte Carlo integration of the missing
data, and do not require parametric specifications for the complete data
density as is typical for selection models.  By modeling the selection
function directly, we are also able to express uncertainty about the
missing data density beyond the simple location and scale changes
typical in pattern-mixture model sensitivity analyses.

\section{Discussion}
\label{sec:disc}
Tukey's representation provides a powerful alternative for specifying missing data models.
It allows analysts to eschew some difficult questions about identifiability in models for non-ignorable missing  data \citep{miao2015} by factoring the joint distribution of the complete-data, $Y$, and missing-data indicators, $R$, in such a way that the missingness mechanism is the only component that must rely on assumptions unassailable using observed data.

\subsection{Theoretical insights}

Thus far we largely worked with exponential-family models. Here, we make formal statements about exponential family, and give results that hold  in greater generality.

\begin{theorem}
The normalizing constant $Q$ is the population fraction of observed data.
\end{theorem}
The proof is straightforward because
\begin{eqnarray}
\label{q:er}
 \mathbb{E} [ r_i \mid \theta_{Y|R},\theta_{R|Y}]
 &=& f(r_i=1\mid \theta_{Y|R},\theta_{R|Y})  \\
 &=& \int f(y_i, r_i=1\mid \theta_{Y|R},\theta_{R|Y}) ~dy_i \nonumber \\
 &=& Q (\theta_{Y|R},\theta_{R|Y}) ~\int f^{\rm obs}(y_i\mid \theta_{Y|R}) ~dy_i \nonumber\\
 &=& Q \nonumber
\end{eqnarray}

The density of the observed values, $y^{\rm obs}$ and $r$, can be written in terms of the population fraction of the observed data, $Q$, in full generality,
\begin{eqnarray}
\label{lik:obs}
f(y^{\rm obs},r \mid \theta_{R|Y},\theta_{Y|R})
 &=&        \prod_{\{i : r_i=1\}} f^{\rm obs} (y_i \mid \theta_{Y|R}) ~Q \times \nonumber \\
 & & \times \prod_{\{i : r_i=0\}} \int \frac{f(r_i=0\mid y_i, \theta_{R|Y})}{f(r_i=1\mid y_i, \theta_{R|Y})} f^{\rm obs} (y_i \mid \theta_{Y|R}) ~Q ~dy_i \nonumber \\
 &=&        Q^{\sum_i r_i} (1-Q)^{N-\sum_i r_i} \prod_{\{i : r_i=1\}} f^{\rm obs} (y_i \mid \theta_{Y|R}),
\end{eqnarray}
using the observation that $\frac{\int \dots \,dy_i}{1+\int \dots \,dy_i} = 1-(1+\int \dots \,dy_i)^{-1} = 1-Q$.
And the missing-data density can  be expressed as a function of the observed-data density in full generality.
\begin{theorem}
The missing-data density can be expressed a function of the observed-data density and the odds of missingness,
\begin{equation}
\label{miss_dist}
f^{\rm mis}(y_i \mid \theta_{R|Y}, \theta_{Y|R}) =
 \frac{Q (\theta_{Y|R},\theta_{R|Y})}{1-Q (\theta_{Y|R},\theta_{R|Y})}
 \frac{f(r_i=0\mid y_i, \theta_{R|Y})}{f(r_i=1\mid y_i, \theta_{R|Y})}
 ~f^{\rm obs} (y_i \mid \theta_{Y|R}).
\end{equation}
\end{theorem}
This result
can  be derived from from the complete-data likelihood and the formula for $Q$ in Equations \ref{lik:comp} and \ref{eqn:normconst}.

When the missingness mechanism, $f(r_i \mid y_i, \theta_{R|Y})$ does not depend on $y_i$, the distribution of the missing data is directly proportional to the distribution of the observed data.
Equation \ref{miss_dist} can help assess the plausibility of different missingness mechanisms---{\em not at random}, {\em completely at random}, and {\em at random}  \citep{Mealli:2015aa}---by  viewing them  as functions of the odds of a missing value, $\frac{f(r_i=0\mid y_i, \theta_{R|Y})}{f(r_i=1\mid y_i, \theta_{R|Y})}$.  For instance, when the odds  have low variance, it may be reasonable to assume the missing data mechanism is completely at random, or at random.

Equation \ref{miss_dist} also leads to a general understanding of the main result of Section \ref{sec:ef}, which can be crystallized in the following statement.
\begin{theorem}
If the observed-data distribution, $f^{\rm obs}$, belongs to an exponential family, and the log-odds of a missing value are linear in the natural sufficient statistics of that observed-data distribution, then the  missing-data distribution, $f^{\rm mis}$, must have the same exponential family as the observed data distribution.
\end{theorem}
This result can be immediately extended to mixtures.
\begin{corollary}
If the observed-data distribution, $f^{\rm obs}$, can be expressed as a $K$-component mixture of an exponential family distribution, and the log-odds of a missing value are linear in the natural sufficient statistics of that observed-data distribution, then the implied missing-data distribution, $f^{\rm mis}$, must be a $K$-component mixture of the same exponential family.
\end{corollary}
We conjecture that a similar relation might hold outside exponential family models.

\subsection{A note on the integrability condition}
\label{sec:integrability}

Not all integrable specifications for
$f^{\rm obs}(y_i \mid\theta_{Y|R=1})$ and $f(r_i\mid y_i,\theta_{R|Y})$ imply a
proper  distribution for $f^{\rm mis}(y_i \mid \theta_{Y|R=1},\theta_{R|Y})$.  In the setting we consider in Section \ref{sec:bayesian}, the integrability condition requires the sum $\theta_{Y|R=1}+\theta_{R|Y}$ to lie in the natural parameter space of the exponential family.
In practice, analysts may want to consider  missing data mechanisms that involve a richer set of parameters, $\tilde\theta_{R|Y}$, such as including an intercept, as we illustrate in the context of a biology application, in Section \ref{sec:app}. In such cases, $\theta_{R|Y}$ is taken to denote the subset of parameters in $\tilde\theta_{R|Y}$ that multiply the sufficient statistics of $f^{\rm obs}$.
The derivations in Section \ref{sec:bayesian} are simple to repurpose for this situation, accordingly.

For example, assume that the natural parameter $\theta_{Y|R=1} =
\eta$, and that the missing data mechanism is logistic with extended parameter vector $\tilde\theta_{R|Y}
= (\beta_0,\beta_1) = (\beta_0,\theta_{R|Y})$ and $f(r_i = 1 \mid y_i,\beta) =
(1+e^{-(\beta_0+T(y_i) \beta_1)})^{-1}$. Then, Equations \ref{eqn:exfam} and \ref{miss_dist_ef} become
\begin{align}
\label{eqn:intercept}
Q(\eta,\beta) &=
  \frac{g(\eta+\beta_1)}{g(\eta+\beta_1)+g(\eta)e^{\beta_0}}\\
f^{\rm mis}(y \mid \eta,\beta) &= h(y) g(\eta + \beta_1) e^{{T(y)}'(\eta+\beta_1)}.
\end{align}

The class of exponential-logistic models defined in Equations \ref{eq:exp-log-obs}--\ref{eq:exp-log-sel} can be further generalized in two useful ways while maintaining its desirable properties.
For instance, generalizing $f^{\rm obs}$ to be a mixture of
exponential families as in Section \ref{sec:sim} is straightforward, and does not increase
computation substantially.  Relaxing assumptions about the missingness
mechanism can be more difficult.  Still, it is possible to model
$f(r_i \mid y_i,\theta_{R|Y})$ with a mixture of logistic functions, including a  missingness mechanism  where a fraction of the data is missing completely at random as in Section \ref{sec:app}.

\subsection{Inference strategies}

Recall the simple Normal-logistic model of Section \ref{sec:est:inf},
\begin{eqnarray}
 f(r_i = 1 \mid y_i,\beta) &=& \hbox{logit }(\beta_0+\beta_1 y_i)
                            = \bigm(1+\exp\{- \beta_0-\beta_1 y_i\}\bigm)^{-1} \nonumber \\
 f^{\rm obs}(y_i)          &=& \hbox{Normal }(0,1). \nonumber
\end{eqnarray}
The inferential strategy  proposed was to posit prior distributions on $\beta_1$ and $Q$, and solve for $\beta_0$ at each iteration of the Markov Chain Monte Carlo sampler.
A conceptually simpler approach to  inference would be to place a prior distribution on all the parameters of the missingness mechanism, and solve for the implied $Q$ at each iteration of the sampler.

In situations where the number of missing values is itself missing, as with truncated data, specifying a   prior distribution for all the parameters of the missingness mechanism would lead to an implied prior distribution for the unknown number of missing values, or equivalently, the population fraction of observed data Q.
\begin{figure}[b!]
 \centering
  \includegraphics[width=78mm]{Figs/contour_100}
  \includegraphics[width=78mm]{Figs/contour_1000}
\caption{\onehalfspacing The region of positive support for the likelihood, restricted to the parameters of the missingness mechanism, is increasingly constrained as
  the population fraction of missing data $Q$ is estimated  with
  increasingly high precision. This intuition is illustrated by the
  width of the ridge, which is a function of the amount of information
  about the fraction of missing data $Q$. We simulated data from a
  standard Normal distribution and a logistic missingness mechanism. The parameters $(\beta_0,\beta_1)$ were set to get 90 percent missing data. The sample size determines the amount of information: $N=100$ (left) and $N=1000$ (right).}
    \label{fig:contour1}
\end{figure}

In situations where the number of missing values is known, however, as with censored data, and therefore $Q$ can be estimated from observed data, the support of the likelihood is a constrained parameter space, and a number of choices for the prior distribution on $\beta = \theta_{R|Y}$ would lead to a posterior distribution that is challenging to explore using Monte Carlo  methods.
Specifically, Equation \ref{q:er} induces a moment constraint that restricts the region where the parameters of the missingness mechanism have positive support to a lower dimensional ridge.
Figure \ref{fig:contour1} illustrates this phenomenon for the simple Normal-logistic model, and increasing sample size.

Sequential Monte Carlo and other specialized Monte Carlo methods that  exploit the geometry of the support of posterior distribution may provide a  solution in this situation \citep{DFG2001aa, liu2008aa, Girolami2011, airohaas:2010}.

\subsection{Concluding remarks}

In
this paper, we used  logistic-exponential family models
to illustrate how Tukey's representation can be used to encode non-monotonicity in the
missingness mechanism,
and to model data with complex
distributional forms.  These models could also be used to facilitate tipping
point analyses \citep{Liublinska2014}, or to incorporate
subjective model uncertainty via prior distributions on the missingness mechanism \citep{rubi:2004}.

Tukey's representation is most useful when positing
reasonable prior distributions on the selection mechanism is feasible.  Translating expert
knowledge or expectations into a functional form can be
challenging, in general, and a logistic missingness mechanism is
not always a good choice.  In practice, Tukey's representation should be used
in concert with strategies for expert prior elicitation
\citep{ohagan06,kynn05,paddock09}.
Nevertheless,  prior elicitation for Tukey's representation is simpler than for other factorizations, since it involves only the  set of parameters $\theta_{R|Y}$.
In contrast, the selection factorization requires  additional assumptions about the complete data density.

In many settings, like the example presented in Section
\ref{sec:app}, we may be able to collect data which partially informs
the specification for the selection mechanism.  As such, when
possible, we can design experiments to learn about the functional form
of $f(r_i|y_i,\theta_{R|Y})$ as well as to further refine prior distributions for $\theta_{R|Y}$.
Along these lines, Tukey's representation may be useful in the context
of multiphase inference, which is intimately related to problems in
missing data \citep{blocker2013}.  In these problems, when
preprocessing data, it is often the case that we have strong knowledge
(or control) of the missingness mechanism yet a weaker understanding of
the underlying scientific model.

All in all, we argue that Tukey's representation, which
represents a hybrid of the selection and pattern mixture models is an
under-researched yet promising alternative for modeling non-ignorable
missing data.

\bibliographystyle{asa}

redpajama_set_name: RedPajamaArXiv


A Russian su-57 in many respects superior to the American planes F-22 Raptor and F-35 Lightning II, said the chief designer-Director of "Sukhoi" Mikhail Sagittarius.
"Load the plane, which he can take to his weight… the higher the ratio, the higher the quality of the aircraft from the point of view of the media. The same applies to the volume and weight of item weapons, placed in the inner compartments. And I must say that according to this indicator, the su-57 is unique among aircraft of the fifth generation," — said the expert.
The designer said that the plane was originally created as a multi-role fighter, whereas the F-22 Raptor and F-35 Lightning II is able to efficiently solve only specific tasks.
"The F-22 was originally designed as an aircraft to gain air superiority. And then the Americans realized that it was fundamentally wrong to design a plane just for placing of rockets "air — air", and attempted to place aviation assets, "air — surface" to the existing configuration of the compartments. But the configuration of the compartments is not allowed to place larger loads," explained Archer.
The specialist noted that it is designed to meet the challenges of the destruction of ground targets the F-35 Lightning II for its maneuverability, acceleration characteristics inferior to the su-57 and Russian fourth generation fighters.
In November, the TV station published videos flight of the su-57, testing composite wing of the fighter and the aisle of such aircraft under the radar.
In October the former commander in chief of Russian air force, Chairman of the Federation Council Committee on defense and security, Colonel-General Viktor Bondarev made the emergence of the unmanned version, the su-57.
In July, the Deputy Prime Minister on questions military-industrial complex Yuri Borisov stated that "the su-35 is now considered one of the best aircraft in the world", so "it makes no sense to speed up the work for the mass production of fifth generation aircraft".
In April, The Drive made that the suspension by India of the joint project with Russia the FGFA (Fifth Generation Fighter Aircraft), involving the creation on the basis of the su-57 India's first fifth generation fighter, will trigger the cancellation of the previously announced plans for mass production of this aircraft. In the same month in The National Interest expressed doubts about the successful implementation of the project su-57.