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Water tower

A water tower is an elevated structure supporting a water tank constructed at a height sufficient to pressurize a distribution system for potable water, and to provide emergency storage for fire protection. Water towers often operate in conjunction with underground or surface service reservoirs, which store treated water close to where it will be used. Other types of water towers may only store raw (non-potable) water for fire protection or industrial purposes, and may not necessarily be connected to a public water supply.

Water towers are able to supply water even during power outages, because they rely on hydrostatic pressure produced by elevation of water (due to gravity) to push the water into domestic and industrial water distribution systems; however, they cannot supply the water for a long time without power, because a pump is typically required to refill the tower. A water tower also serves as a reservoir to help with water needs during peak usage times. The water level in the tower typically falls during the peak usage hours of the day, and then a pump fills it back up during the night. This process also keeps the water from freezing in cold weather, since the tower is constantly being drained and refilled.

History

Although the use of elevated water storage tanks has existed since ancient times in various forms, the modern use of water towers for pressurized public water systems developed during the mid-19th century, as steam-pumping became more common, and better pipes that could handle higher pressures were developed. In the United Kingdom, standpipes consisted of tall, exposed, N-shaped pipes, used for pressure relief and to provide a fixed elevation for steam-driven pumping engines which tended to produce a pulsing flow, while the pressurized water distribution system required constant pressure. Standpipes also provided a convenient fixed location to measure flow rates. Designers typically enclosed the riser pipes in decorative masonry or wooden structures. By the late 19th century, standpipes grew to include storage tanks to meet the ever-increasing demands of growing cities.

Many early water towers are now considered historically significant and have been included in various heritage listings around the world. Some are converted to apartments or exclusive penthouses. In certain areas, such as New York City in the United States, smaller water towers are constructed for individual buildings. In California and some other states, domestic water towers enclosed by siding (tankhouses) were once built (1850s–1930s) to supply individual homes; windmills pumped water from hand-dug wells up into the tank in New York.

Water towers were used to supply water stops for steam locomotives on railroad lines. Early steam locomotives required water stops every 7 to 10 mi.

Design and construction

A variety of materials can be used to construct a typical water tower; steel and reinforced or prestressed concrete are most often used (with wood, fiberglass, or brick also in use), incorporating an interior coating to protect the water from any effects from the lining material. The reservoir in the tower may be spherical, cylindrical, or an ellipsoid, with a minimum height of approximately 6 m and a minimum of 4 m in diameter. A standard water tower typically has a height of approximately 40 m.

Pressurization occurs through the hydrostatic pressure of the elevation of water; for every 102 mm of elevation, it produces 1 kPa of pressure. 30 m of elevation produces roughly 300 kPa, which is enough pressure to operate and provide for most domestic water pressure and distribution system requirements.

The height of the tower provides the pressure for the water supply system, and it may be supplemented with a pump. The volume of the reservoir and diameter of the piping provide and sustain flow rate. However, relying on a pump to provide pressure is expensive; to keep up with varying demand, the pump would have to be sized to meet peak demands. During periods of low demand, jockey pumps are used to meet these lower water flow requirements. The water tower reduces the need for electrical consumption of cycling pumps and thus the need for an expensive pump control system, as this system would have to be sized sufficiently to give the same pressure at high flow rates.

Very high volumes and flow rates are needed when fighting fires. With a water tower present, pumps can be sized for average demand, not peak demand; the water tower can provide water pressure during the day and pumps will refill the water tower when demands are lower.

Using wireless sensor networks to monitor water levels inside the tower allows municipalities to automatically monitor and control pumps without installing and maintaining expensive data cables.

Architecture

The adjacent image shows three architectural approaches to incorporating these tanks in the design of a building, one on East 57th Street in New York City. From left to right, a fully enclosed and ornately decorated brick structure, a simple unadorned roofless brick structure hiding most of the tank but revealing the top of the tank, and a simple utilitarian structure that makes no effort to hide the tanks or otherwise incorporate them into the design of the building.

The technology dates to at least the 19th century, and for a long time New York City required that all buildings higher than six stories be equipped with a rooftop water tower. Two companies in New York build water towers, both of which are family businesses in operation since the 19th century.

The original water tower builders were barrel makers who expanded their craft to meet a modern need as buildings in the city grew taller in height. Even today, no sealant is used to hold the water in. The wooden walls of the water tower are held together with steel cables or straps, but water leaks through the gaps when first filled. As the water saturates the wood, it swells, the gaps close and become impermeable. The rooftop water towers store 250000 to 50000 L of water until it is needed in the building below. The upper portion of water is skimmed off the top for everyday use while the water in the bottom of the tower is held in reserve to fight fire. When the water drops below a certain level, a pressure switch, level switch or float valve will activate a pump or open a public water line to refill the water tower.

Architects and builders have taken varied approaches to incorporating water towers into the design of their buildings. On many large commercial buildings, water towers are completely hidden behind an extension of the facade of the building. For cosmetic reasons, apartment buildings often enclose their tanks in rooftop structures, either simple unadorned rooftop boxes, or ornately decorated structures intended to enhance the visual appeal of the building. Many buildings, however, leave their water towers in plain view atop utilitarian framework structures.

Water towers are common in India, where the electricity supply is erratic in most places.

If the pumps fail (such as during a power outage), then water pressure will be lost, causing potential public health concerns. Many U.S. states require a "boil-water advisory" to be issued if water pressure drops below 20 psi. This advisory presumes that the lower pressure might allow pathogens to enter the system.

Some have been converted to serve modern purposes, as for example, the Wieża Ciśnień (Wrocław water tower) in Wrocław, Poland which is today a restaurant complex. Others have been converted to residential use.

Historically, railroads that used steam locomotives required a means of replenishing the locomotive's tenders. Water towers were common along the railroad. The tenders were usually replenished by water cranes, which were fed by a water tower.

Some water towers are also used as observation towers, and some restaurants, such as the Goldbergturm in Sindelfingen, Germany, or the second of the three Kuwait Towers, in the State of Kuwait. It is also common to use water towers as the location of transmission mechanisms in the UHF range with small power, for instance for closed rural broadcasting service, amateur radio, or cellular telephone service.

In hilly regions, local topography can be substituted for structures to elevate the tanks. These tanks are often nothing more than concrete cisterns terraced into the sides of local hills or mountains, but function identically to the traditional water tower. The tops of these tanks can be landscaped or used as park space, if desired.

Spheres and spheroids
The Chicago Bridge and Iron Company has built many of the water spheres and spheroids found in the United States. The website World's Tallest Water Sphere describes the distinction between a water sphere and water spheroid thus: "A water sphere is a type of water tower that has a large sphere at the top of its post. The sphere looks like a golf ball sitting on a tee or a round lollipop. A cross section of a sphere in any direction (east-west, north-south, or top-bottom) is a perfect circle. A water spheroid looks like a water sphere, but the top is wider than it is tall. A spheroid looks like a round pillow that is somewhat flattened. A cross section of a spheroid in two directions (east-west or north-south) is an ellipse, but in only one direction (top-bottom) is it a perfect circle. Both spheres and spheroids are special-case ellipsoids: spheres have symmetry in 3 directions, spheroids have symmetry in 2 directions. Scalene ellipsoids have 3 unequal length axes and three unequal cross sections."

The Union Watersphere is a water tower topped with a sphere-shaped water tank in Union, New Jersey, and characterized as the World's Tallest Water Sphere.

A Star Ledger article suggested a water tower in Erwin, North Carolina completed in early 2012, 219.75 ft tall and holding 500000 gal, had become the World's Tallest Water Sphere. However, photographs of the Erwin water tower revealed the new tower to be a water spheroid.

The water tower in Braman, Oklahoma, built by the Kaw Nation and completed in 2010, is 220.6 ft tall and can hold 350000 gal. Slightly taller than the Union Watersphere, it is also a spheroid.

Another tower in Oklahoma, built in 1986 and billed as the "largest water tower in the country", is 218 ft tall, can hold 500000 gal, and is located in Edmond.

The Earthoid, a perfectly spherical tank located in Germantown, Maryland is 100 ft tall and holds 2000000 gal of water. The name is taken from it being painted to resemble a globe of the world.

The golf ball-shaped tank of the water tower at Gonzales, California is supported by three tubular legs and reaches about 125 ft high.

The Watertoren (or Water Towers) in Eindhoven, Netherlands contain three spherical tanks, each 10 m in diameter and capable of holding 500 m3 of water, on three 43.45 m spires were completed in 1970.

Decoration

Water towers can be surrounded by ornate coverings including fancy brickwork, a large ivy-covered trellis or they can be simply painted. Some city water towers have the name of the city painted in large letters on the roof, as a navigational aid to aviators and motorists. Sometimes the decoration can be humorous. An example of this are water towers built side by side, labeled HOT and COLD. Cities in the United States possessing side-by-side water towers labeled HOT and COLD include Granger, Iowa; Canton, Kansas; Pratt, Kansas, and St. Clair, Missouri. Eveleth, Minnesota at one time had two such towers, but no longer does.

Many small towns in the United States use their water towers to advertise local tourism, their local high school sports teams, or other locally notable facts. A "mushroom" water tower was built in Örebro, Sweden and holds almost two million gallons of water.

Alternatives
Alternatives to water towers are simple pumps mounted on top of the water pipes to increase the water pressure. This new approach is more straightforward, but also more subject to potential public health risks; if the pumps fail, then loss of water pressure may result in entry of contaminants into the water system. Most large water utilities do not use this approach, given the potential risks.

Australia

 * Bankstown Reservoir, Sydney

Austria

 * Wasserturm Amstetten
 * Wolfersberg Water Tower (Water tower with transmission antenna)

Belgium

 * Mechelen-Zuid Watertoren

Brazil

 * Nave Espacial de Varginha in Varginha

Canada

 * Guaranteed Pure Milk bottle in Montreal, Quebec

Croatia

 * Vukovar water tower in Vukovar.

Denmark

 * Svaneke water tower

Germany

 * Lüneburg Water Tower
 * Heidelberg TV Tower (TV tower with water reservoir)
 * Mannheim Water Tower (built 1886–1889)

Kuwait
Kuwait Towers, which include two water reservoirs, and Kuwait Water Towers (Mushroom towers in Kuwait City.

India

 * Tala tank in Kolkata

Italy

 * Ginosa Water Tower, 122 m tall

Netherlands

 * Amsterdamsestraatweg Water Tower in Utrecht
 * Eindhoven Water Towers in Eindhoven
 * Poldertoren in Emmeloord
 * Water Tower Simpelveld in Simpelveld
 * Water Tower Hellevoetsluis in Hellevoetsluis

Poland

 * Wrocław Water Tower
 * Old Water Tower, Bydgoszcz

Romania

 * Fabric Water Tower
 * Iosefin Water Tower
 * Oltenița Water Tower
 * Turnu Măgurele Water Tower

Slovakia

 * Water Tower in Komárno
 * Water Tower in Trnava

Slovenia

 * Brežice Water Tower in Brežice

Sweden

 * Vanadislundens water reservoir (Stockholm)

United Kingdom

 * Cardiff Central Station Water Tower
 * Dock Tower in Grimsby
 * House in the Clouds in Thorpeness, Suffolk
 * Jumbo in Colchester, Essex
 * Norton Water Tower in Norton, Cheshire
 * Tilehurst Water Tower in Reading
 * Tower Park in Poole, Dorset
 * Cranhill, Garthamlock and Drumchapel in Glasgow, and Tannochside just outside the city

United States

 * Brooks Catsup Bottle Water Tower near Collinsville, Illinois
 * Chicago Water Tower in Chicago, Illinois
 * Florence Y'all Water Tower in Florence, Kentucky
 * Lawson Tower in Scituate, Massachusetts
 * Leaning Water Tower in Groom, Texas
 * North Point Water Tower in Milwaukee, Wisconsin
 * Peachoid next to I-85 on the edge of Gaffney, South Carolina
 * Show Place Arena water tower in Upper Marlboro, Maryland
 * Union Watersphere in Union Township, New Jersey
 * Volunteer Park Water Tower in Capitol Hill, Seattle, Washington
 * Warner Bros. Water Tower in Burbank, California (In the animated TV series Animaniacs, it was used to incarcerate the characters Yakko, Wakko, and Dot, as well as to serve as their home.)
 * Weehawken Water Tower in Weehawken, New Jersey
 * Ypsilanti Water Tower in Ypsilanti, Michigan (Winner of the Most Phallic Building contest in 2003)

Standpipe
A standpipe is a water tower which is cylindrical (or nearly cylindrical) throughout its whole height, rather than an elevated tank on supports with a narrower pipe leading to and from the ground. There were originally over 400 standpipe water towers in the United States, but very few remain today, including:

 * Addison Standpipe, in Addison, Michigan
 * Belton Standpipe in Belton, South Carolina (also in Allendale and Walterboro)
 * Belton Standpipe in Belton, Texas
 * Bellevue Standpipe (actually a water tank, not a tower), in Boston, Massachusetts
 * Chicago Water Tower, in Chicago, Illinois Chicago Water tower.jpg]]
 * Cochituate standpipe, in Boston, Massachusetts
 * Craig, Nebraska standpipe
 * Eden Park Stand Pipe, in Cincinnati
 * Evansville Standpipe (a steel tower), in Evansville, Wisconsin
 * Fall River Waterworks, in Fall River, Massachusetts
 * Forbes Hill Standpipe, in Quincy, Massachusetts
 * Louisville Water Tower, in Louisville, Kentucky
 * North Point Water Tower, in Milwaukee, Wisconsin
 * Reading Standpipe (demolished in 1999 and replaced by a modern steel tower), in Reading, Massachusetts
 * Roxbury High Fort contains the Cochituate Standpipe
 * St. Louis, Missouri has three standpipe water towers which are on the National Register of Historic Places.
 * Bissell Tower (also known as the Red Tower)
 * Compton Hill Tower
 * Grand Avenue Water Tower
 * Thomas Hill Standpipe, in Bangor, Maine
 * Ypsilanti Water Tower, in Ypsilanti, Michigan
 * Bremen Water Tower, in Bremen, Indiana

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Telluride

Telluride may refer to:

 * Telluride, Colorado, the county seat of San Miguel County in southwest Colorado
 * Telluride Ski Resort, a ski resort located in Mountain Village, Colorado
 * Telluride Film Festival, a film festival that takes place in Telluride, Colorado
 * Telluride (chemistry), the tellurium anion and its derivatives
 * Telluride mineral, any mineral that has the telluride ion as its main component
 * Telluride Association, an educational non-profit organization in the United States
 * Telluride House, a Cornell University residential society and one of the Telluride Association's programs
 * "Telluride" (Tim McGraw song), 2001; covered by Josh Gracin, 2008
 * "Telluride" (Jade Eagleson song), 2023
 * Kia Telluride, a mid-size crossover SUV made by Kia Motors

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Telluride


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Drake Hotel

Drake Hotel may refer to:

 * in Canada
 * Drake Hotel (Toronto), Ontario

 * in the United States (by state)
 * Drake Hotel (Chicago, Illinois), listed on the National Register of Historic Places (NRHP)
 * Drake Hotel (Gallup, New Mexico), NRHP-listed in McKinley County
 * Drake Hotel (New York City), New York
 * Drake Hotel (Philadelphia, Pennsylvania), NRHP-listed

title: Drake Hotel


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U.S. National Ice Center

The U.S. National Ice Center (USNIC) is a tri-agency operational center and echelon V command of the Naval Oceanographic Office (NAVOCEANO), whose mission is to provide worldwide navigational ice analyses for the armed forces of the United States, allied nations, and U.S. government agencies.

It is represented by the United States Navy (Department of Defense); the National Oceanic and Atmospheric Administration (Department of Commerce); and the United States Coast Guard (Department of Homeland Security). The U.S. National Ice Center is a subordinate command of the Naval Oceanographic Office (NAVOCEANO). Originally known as the Navy/NOAA Joint Ice Center, which was established on December 15, 1976 in a memorandum of agreement between the U.S. Navy and NOAA, the National Ice Center was formed in 1995 when the U.S. Coast Guard became a partner. The U.S. National Ice Center produces global sea ice charts and various cryospheric GIS products. They also name and track Antarctic icebergs if greater than 10 nmi on its longest axis.

Icebergs must be a minimum of 19 kilometers in length to be tracked by the USNIC.

title: U.S. National Ice Center


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Stefano Benni

Stefano Benni (born 12 August 1947) is an Italian satirical writer, poet and journalist. His books have been translated into around 20 foreign languages and scored notable commercial success. 2.5 million copies of his books have been sold in Italy.

Biography
Benni has written many successful novels and anthologies, among which are Bar Sport, Elianto, Terra!, La compagnia dei celestini, Baol, Comici spaventati guerrieri, Saltatempo, Margherita Dolcevita and Il bar sotto il mare. He has also worked with the weekly magazines L'espresso and Panorama, and with the satirical Cuore and Tango, the monthly magazines Linus and Il Mago (where he began and published in installments part of Bar Sport), and the newspapers La Repubblica and il manifesto.

He also wrote television sketches for Beppe Grillo at the beginning of Grillo's career, and one of these sketches – called "Pietro Longo=P2" – caused the Psdi to ask the RAI commission to remove Grillo from RAI TV; their request was denied.

In 1989, with Umberto Angelucci, Benni directed the film Musica per vecchi animali, adapted from his book Comici spaventati guerrieri, with the actors Dario Fo, Paolo Rossi, and Viola Simoncioni. Two years before that he was also the screenwriter of another film, Topo Galileo by Francesco Laudadio, starring his friend Beppe Grillo and with music by Fabrizio De André and Mauro Pagani.

With the jazz musician Umberto Petrin, he wrote ''Misterioso. Viaggio nel silenzio di Thelonius Monk''.

His novels contain, within imaginary worlds and situations, a strong satire of Italian society over the last few decades. His writing style includes many puns, neologisms and parodies of other literary styles.

In January and February 2010, he presented the event Bennac, a meeting between himself and the French writer Daniel Pennac.

Works in English

 * Timeskipper, translator Antony Shugaar, Europa Editions, New York, 2008. ISBN 9781933372440
 * Margherita Dolce Vita, translator Antony Shugaar, Europa Editions, New York, 2008. ISBN 9781933372204
 * The story of Cyrano de Bergerac, Pushkin Children's Books, London, 2014. ISBN 9781782690214

title: Stefano Benni


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Colorado Mineral Belt

The Colorado Mineral Belt (CMB) is an area of ore deposits from the La Plata Mountains in Southwestern Colorado to near the middle of the state at Boulder, Colorado, and from which over 25 million troy ounces (778 t) of gold were extracted beginning in 1858. The belt is a "northeast-striking zone defined by: a Proterozoic shear zone system (McCoy, 2001); a suite of Laramide-aged plutons and related ore deposits (Tweto and Sims, 1963); a major gravity low (Isaacson and Smithson, 1976); low-crustal velocities; and high heat flow (Decker et al., 1988)." Mining districts include:

 * Central City-Idaho Springs district
 * Leadville mining district, named for Leadville, Colorado
 * Sneffels-Red Mountain-Telluride district

The belt lies within a zone that has been geologically active at intervals beginning from near the time of crustal accretion in central Colorado at least 1.6 billion years ago until the present. Parts of the CMB follow shear zones of Precambrian age and the Paleozoic and Mesozoic. Igneous rocks intruded about 60 to 70 million years ago during the Laramide orogeny are associated with the belt and once were thought to be responsible for most of the ore deposits. Now many of the important ore deposits are thought to be genetically related to younger magmatism, some at least as young as about 25 million years.

title: Colorado Mineral Belt


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Likelihood function

The likelihood function (often simply called the likelihood) is the joint probability mass (or probability density) of observed data, but viewed as a function of the parameters of a statistical model. That is, the likelihood function $\mathcal{L}(\theta \mid x)$, which gives the likelihood of a vector of parameters $\theta$ under the assumption that a set of observed data $x$  is true, is numerically the same as the probability function $f(x \mid \theta)$ , which gives the probability of a set of data $x$  under the assumption that a vector of parameters $\theta$  is true.

In maximum likelihood estimation, the arg max (over the parameter $\theta$ ) of the likelihood function serves as a point estimate for $\theta$, while the Fisher information (often approximated by the likelihood's Hessian matrix) is an indication of the estimate's precision.

In contrast, in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, via a posterior probability $f(\theta \mid x)$ calculated via Bayes' rule.

Definition
The likelihood function, parameterized by a (possibly multivariate) parameter $\theta$, is usually defined differently for discrete and continuous probability distributions (a more general definition is discussed below). Given a probability density or mass function

$$x\mapsto f(x \mid \theta),$$

where $x$ is a realization of the random variable $X$, the likelihood function is $$\theta\mapsto f(x \mid \theta),$$ often written $$\mathcal{L}(\theta \mid x). $$

In other words, when $f(x\mid\theta)$ is viewed as a function of $x$  with $\theta$  fixed, it is a probability density function, and when viewed as a function of $\theta$  with $x$  fixed, it is a likelihood function. In the frequentist paradigm, the notation $f(x\mid\theta)$ is often avoided  and instead $f(x;\theta)$  or $f(x,\theta)$  are used to indicate that $\theta$  is regarded as a fixed unknown quantity rather than as a random variable being conditioned on.

The likelihood function does not specify the probability that $\theta$ is the truth, given the observed sample $X = x$. Such an interpretation is a common error, with potentially disastrous consequences (see prosecutor's fallacy).

Discrete probability distribution
Let $X$ be a discrete random variable with probability mass function $p$  depending on a parameter $\theta$. Then the function

$$\mathcal{L}(\theta \mid x) = p_\theta (x) = P_\theta (X=x), $$

considered as a function of $\theta$, is the likelihood function, given the outcome $x$ of the random variable $X$. Sometimes the probability of "the value $x$ of $X$  for the parameter value $\theta$ 20%" is written as $P(X = x | θ)$ or $P(X = x; θ)$. The likelihood is the probability that a particular outcome $x$ is observed when the true value of the parameter is $\theta$, equivalent to the probability mass on $x$ ; it is not a probability density over the parameter $\theta$. The likelihood, $\mathcal{L}(\theta \mid x) $, should not be confused with $P(\theta \mid x)$ , which is the posterior probability of $\theta$ given the data $x$.

Example
Consider a simple statistical model of a coin flip: a single parameter $p_\text{H}$ that expresses the "fairness" of the coin. The parameter is the probability that a coin lands heads up ("H") when tossed. $p_\text{H}$ can take on any value within the range 0.0 to 1.0. For a perfectly fair coin, $p_\text{H} = 0.5$.

Imagine flipping a fair coin twice, and observing two heads in two tosses ("HH"). Assuming that each successive coin flip is i.i.d., then the probability of observing HH is

$$P(\text{HH} \mid p_\text{H}=0.5) = 0.5^2 = 0.25.$$

Equivalently, the likelihood of observing "HH" assuming $p_\text{H} = 0.5$ is

$$\mathcal{L}(p_\text{H}=0.5 \mid \text{HH}) = 0.25.$$

This is not the same as saying that $P(p_\text{H} = 0.5 \mid HH) = 0.25$, a conclusion which could only be reached via Bayes' theorem given knowledge about the marginal probabilities $P(p_\text{H} = 0.5)$  and $P(\text{HH})$.

Now suppose that the coin is not a fair coin, but instead that $p_\text{H} = 0.3$. Then the probability of two heads on two flips is

$$P(\text{HH} \mid p_\text{H}=0.3) = 0.3^2 = 0.09.$$

Hence

$$\mathcal{L}(p_\text{H}=0.3 \mid \text{HH}) = 0.09.$$

More generally, for each value of $p_\text{H}$, we can calculate the corresponding likelihood. The result of such calculations is displayed in Figure 1. The integral of $\mathcal{L}$ over [0, 1] is 1/3; likelihoods need not integrate or sum to one over the parameter space.

Continuous probability distribution
Let $X$ be a random variable following an absolutely continuous probability distribution with density function $f$  (a function of $x$ ) which depends on a parameter $\theta$. Then the function

$$\mathcal{L}(\theta \mid x) = f_\theta (x), $$

considered as a function of $\theta$, is the likelihood function (of $\theta$ , given the outcome $X=x$ ). Again, $\mathcal{L}$ is not a probability density or mass function over $\theta$, despite being a function of $\theta$  given the observation $X = x$.

Relationship between the likelihood and probability density functions
The use of the probability density in specifying the likelihood function above is justified as follows. Given an observation $x_j$, the likelihood for the interval $[x_j, x_j + h]$ , where $h > 0$ is a constant, is given by $ \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) $. Observe that $$ \mathop\operatorname{arg\,max}_\theta \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) = \mathop\operatorname{arg\,max}_\theta \frac{1}{h} \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) ,$$ since $ h $ is positive and constant. Because $$ \mathop\operatorname{arg\,max}_\theta \frac 1 h \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) = \mathop\operatorname{arg\,max}_\theta \frac 1 h \Pr(x_j \leq x \leq x_j + h \mid \theta) = \mathop\operatorname{arg\,max}_\theta \frac 1 h \int_{x_j}^{x_j+h} f(x\mid \theta) \,dx, $$

where $ f(x\mid \theta) $ is the probability density function, it follows that

$$ \mathop\operatorname{arg\,max}_\theta \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) = \mathop\operatorname{arg\,max}_\theta \frac{1}{h} \int_{x_j}^{x_j+h} f(x\mid\theta) \,dx .$$

The first fundamental theorem of calculus provides that $$ \lim_{h \to 0^{+}} \frac 1 h \int_{x_j}^{x_j+h} f(x\mid\theta) \,dx = f(x_j \mid \theta). $$

Then $$ \begin{align} & \mathop\operatorname{arg\,max}_\theta \mathcal{L}(\theta\mid x_j) = \mathop\operatorname{arg\,max}_\theta \left[ \lim_{h\to 0^{+}} \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) \right] \\[4pt] = {} & \mathop\operatorname{arg\,max}_\theta \left[ \lim_{h\to 0^{+}} \frac{1}{h} \int_{x_j}^{x_j+h} f(x\mid\theta) \,dx \right] = \mathop\operatorname{arg\,max}_\theta f(x_j \mid \theta). \end{align} $$

Therefore, $$ \mathop\operatorname{arg\,max}_\theta \mathcal{L}(\theta\mid x_j) = \mathop\operatorname{arg\,max}_\theta f(x_j \mid \theta), $$ and so maximizing the probability density at $ x_j $ amounts to maximizing the likelihood of the specific observation $ x_j $.

In general
In measure-theoretic probability theory, the density function is defined as the Radon–Nikodym derivative of the probability distribution relative to a common dominating measure. The likelihood function is this density interpreted as a function of the parameter, rather than the random variable. Thus, we can construct a likelihood function for any distribution, whether discrete, continuous, a mixture, or otherwise. (Likelihoods are comparable, e.g. for parameter estimation, only if they are Radon–Nikodym derivatives with respect to the same dominating measure.)

The above discussion of the likelihood for discrete random variables uses the counting measure, under which the probability density at any outcome equals the probability of that outcome.

Likelihoods for mixed continuous–discrete distributions
The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete probability masses $p_k (\theta)$ and a density $f(x\mid\theta)$, where the sum of all the $p$ 's added to the integral of $f$  is always one. Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with in the manner shown above. For an observation from the discrete component, the likelihood function for an observation from the discrete component is simply $$\mathcal{L}(\theta \mid x )= p_k(\theta), $$ where $k$ is the index of the discrete probability mass corresponding to observation $x$, because maximizing the probability mass (or probability) at $x$  amounts to maximizing the likelihood of the specific observation.

The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and the probability mass) arises from the way in which the likelihood function is defined up to a constant of proportionality, where this "constant" can change with the observation $x$, but not with the parameter $\theta$.

Regularity conditions
In the context of parameter estimation, the likelihood function is usually assumed to obey certain conditions, known as regularity conditions. These conditions are in various proofs involving likelihood functions, and need to be verified in each particular application. For maximum likelihood estimation, the existence of a global maximum of the likelihood function is of the utmost importance. By the extreme value theorem, it suffices that the likelihood function is continuous on a compact parameter space for the maximum likelihood estimator to exist. While the continuity assumption is usually met, the compactness assumption about the parameter space is often not, as the bounds of the true parameter values might be unknown. In that case, concavity of the likelihood function plays a key role.

More specifically, if the likelihood function is twice continuously differentiable on the k -dimensional parameter space $ \Theta $ assumed to be an open connected subset of $ \mathbb{R}^{k} \,,$  there exists a unique maximum $\hat{\theta} \in \Theta$  if the matrix of second partials $$ \mathbf{H}(\theta) \equiv \left[\, \frac{ \partial^2 L }{\, \partial \theta_i \, \partial \theta_j \,} \,\right]_{i,j=1,1}^{n_\mathrm{i},n_\mathrm{j}} \;$$ is negative definite for every $\, \theta \in \Theta \,$ at which the gradient $\; \nabla L \equiv \left[\, \frac{ \partial L }{\, \partial \theta_i \,} \,\right]_{i=1}^{n_\mathrm{i}} \;$  vanishes, and if the likelihood function approaches a constant on the boundary of the parameter space, $\; \partial \Theta \;,$ i.e., $$ \lim_{\theta \to \partial \Theta} L(\theta) = 0 \;,$$ which may include the points at infinity if $ \, \Theta \, $ is unbounded. Mäkeläinen and co-authors prove this result using Morse theory while informally appealing to a mountain pass property. Mascarenhas restates their proof using the mountain pass theorem.

In the proofs of consistency and asymptotic normality of the maximum likelihood estimator, additional assumptions are made about the probability densities that form the basis of a particular likelihood function. These conditions were first established by Chanda. In particular, for almost all $x$, and for all $\, \theta \in \Theta \,,$ $$\frac{\partial \log f}{\partial \theta_r} \,, \quad \frac{\partial^2 \log f}{\partial \theta_r \partial \theta_s} \,, \quad \frac{\partial^3 \log f}{\partial \theta_r \, \partial \theta_s \, \partial \theta_t} \,$$ exist for all $\, r, s, t = 1, 2, \ldots, k \,$ in order to ensure the existence of a Taylor expansion. Second, for almost all $x$ and for every $\, \theta \in \Theta \,$  it must be that $$ \left| \frac{\partial f}{\partial \theta_r} \right| < F_r(x) \,, \quad \left| \frac{\partial^2 f}{\partial \theta_r \, \partial \theta_s} \right| < F_{rs}(x) \,, \quad \left| \frac{\partial^3 f}{\partial \theta_r \, \partial \theta_s \, \partial \theta_t} \right| < H_{rst}(x) $$ where $H$ is such that $\, \int_{-\infty}^{\infty} H_{rst}(z) \mathrm{d}z \leq M < \infty \;.$  This boundedness of the derivatives is needed to allow for differentiation under the integral sign. And lastly, it is assumed that the information matrix, $$\mathbf{I}(\theta) = \int_{-\infty}^{\infty} \frac{\partial \log f}{\partial \theta_r}\ \frac{\partial \log f}{\partial \theta_s}\ f\ \mathrm{d}z $$ is positive definite and $\, \left| \mathbf{I}(\theta) \right| \,$ is finite. This ensures that the score has a finite variance.

The above conditions are sufficient, but not necessary. That is, a model that does not meet these regularity conditions may or may not have a maximum likelihood estimator of the properties mentioned above. Further, in case of non-independently or non-identically distributed observations additional properties may need to be assumed.

In Bayesian statistics, almost identical regularity conditions are imposed on the likelihood function in order to proof asymptotic normality of the posterior probability, and therefore to justify a Laplace approximation of the posterior in large samples.

Likelihood ratio
A likelihood ratio is the ratio of any two specified likelihoods, frequently written as: $$\Lambda(\theta_1:\theta_2 \mid x) = \frac{\mathcal{L}(\theta_1 \mid x)}{\mathcal{L}(\theta_2 \mid x)}.$$

The likelihood ratio is central to likelihoodist statistics: the law of likelihood states that degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio.

In frequentist inference, the likelihood ratio is the basis for a test statistic, the so-called likelihood-ratio test. By the Neyman–Pearson lemma, this is the most powerful test for comparing two simple hypotheses at a given significance level. Numerous other tests can be viewed as likelihood-ratio tests or approximations thereof. The asymptotic distribution of the log-likelihood ratio, considered as a test statistic, is given by Wilks' theorem.

The likelihood ratio is also of central importance in Bayesian inference, where it is known as the Bayes factor, and is used in Bayes' rule. Stated in terms of odds, Bayes' rule states that the posterior odds of two alternatives, $A_1$ and $A_2$, given an event $B$, is the prior odds, times the likelihood ratio. As an equation: $$O(A_1:A_2 \mid B) = O(A_1:A_2) \cdot \Lambda(A_1:A_2 \mid B).$$

The likelihood ratio is not directly used in AIC-based statistics. Instead, what is used is the relative likelihood of models (see below).

In evidence-based medicine, likelihood ratios are used in diagnostic testing to assess the value of performing a diagnostic test.

Relative likelihood function
Since the actual value of the likelihood function depends on the sample, it is often convenient to work with a standardized measure. Suppose that the maximum likelihood estimate for the parameter $θ$ is $\hat{\theta}$. Relative plausibilities of other $θ$ values may be found by comparing the likelihoods of those other values with the likelihood of $\hat{\theta}$. The relative likelihood of $θ$ is defined to be  $$R(\theta) = \frac{\mathcal{L}(\theta \mid x)}{\mathcal{L}(\hat{\theta} \mid x)}.$$ Thus, the relative likelihood is the likelihood ratio (discussed above) with the fixed denominator $ \mathcal{L}(\hat{\theta})$. This corresponds to standardizing the likelihood to have a maximum of 1.

Likelihood region
A likelihood region is the set of all values of $θ$ whose relative likelihood is greater than or equal to a given threshold. In terms of percentages, a $p$% likelihood region for $θ$ is defined to be

$$ \left\{\theta : R(\theta) \ge \frac p {100} \right\}. $$

If $θ$ is a single real parameter, a $p$% likelihood region will usually comprise an interval of real values. If the region does comprise an interval, then it is called a likelihood interval.

Likelihood intervals, and more generally likelihood regions, are used for interval estimation within likelihoodist statistics: they are similar to confidence intervals in frequentist statistics and credible intervals in Bayesian statistics. Likelihood intervals are interpreted directly in terms of relative likelihood, not in terms of coverage probability (frequentism) or posterior probability (Bayesianism).

Given a model, likelihood intervals can be compared to confidence intervals. If $θ$ is a single real parameter, then under certain conditions, a 14.65% likelihood interval (about 1:7 likelihood) for $θ$ will be the same as a 95% confidence interval (19/20 coverage probability). In a slightly different formulation suited to the use of log-likelihoods (see Wilks' theorem), the test statistic is twice the difference in log-likelihoods and the probability distribution of the test statistic is approximately a chi-squared distribution with degrees-of-freedom (df) equal to the difference in df's between the two models (therefore, the $e$−2 likelihood interval is the same as the 0.954 confidence interval; assuming difference in df's to be 1).

Likelihoods that eliminate nuisance parameters
In many cases, the likelihood is a function of more than one parameter but interest focuses on the estimation of only one, or at most a few of them, with the others being considered as nuisance parameters. Several alternative approaches have been developed to eliminate such nuisance parameters, so that a likelihood can be written as a function of only the parameter (or parameters) of interest: the main approaches are profile, conditional, and marginal likelihoods. These approaches are also useful when a high-dimensional likelihood surface needs to be reduced to one or two parameters of interest in order to allow a graph.

Profile likelihood
It is possible to reduce the dimensions by concentrating the likelihood function for a subset of parameters by expressing the nuisance parameters as functions of the parameters of interest and replacing them in the likelihood function. In general, for a likelihood function depending on the parameter vector $\mathbf{\theta}$ that can be partitioned into $\mathbf{\theta} = \left( \mathbf{\theta}_{1} : \mathbf{\theta}_{2} \right)$, and where a correspondence $\mathbf{\hat{\theta}}_{2} = \mathbf{\hat{\theta}}_{2} \left( \mathbf{\theta}_{1} \right)$  can be determined explicitly, concentration reduces computational burden of the original maximization problem.

For instance, in a linear regression with normally distributed errors, $\mathbf{y} = \mathbf{X} \beta + u$, the coefficient vector could be partitioned into $\beta = \left[ \beta_{1} : \beta_{2} \right]$ (and consequently the design matrix $\mathbf{X} = \left[ \mathbf{X}_{1} : \mathbf{X}_{2} \right]$ ). Maximizing with respect to $\beta_{2}$ yields an optimal value function $\beta_{2} (\beta_{1}) = \left( \mathbf{X}_{2}^{\mathsf{T}} \mathbf{X}_{2} \right)^{-1} \mathbf{X}_{2}^{\mathsf{T}} \left( \mathbf{y} - \mathbf{X}_{1} \beta_{1} \right)$. Using this result, the maximum likelihood estimator for $\beta_{1}$ can then be derived as $$\hat{\beta}_{1} = \left( \mathbf{X}_{1}^{\mathsf{T}} \left( \mathbf{I} - \mathbf{P}_{2} \right) \mathbf{X}_{1} \right)^{-1} \mathbf{X}_{1}^{\mathsf{T}} \left( \mathbf{I} - \mathbf{P}_{2} \right) \mathbf{y}$$ where $\mathbf{P}_{2} = \mathbf{X}_{2} \left( \mathbf{X}_{2}^{\mathsf{T}} \mathbf{X}_{2} \right)^{-1} \mathbf{X}_{2}^{\mathsf{T}}$ is the projection matrix of $\mathbf{X}_{2}$. This result is known as the Frisch–Waugh–Lovell theorem.

Since graphically the procedure of concentration is equivalent to slicing the likelihood surface along the ridge of values of the nuisance parameter $\beta_{2}$ that maximizes the likelihood function, creating an isometric profile of the likelihood function for a given $\beta_{1}$, the result of this procedure is also known as profile likelihood. In addition to being graphed, the profile likelihood can also be used to compute confidence intervals that often have better small-sample properties than those based on asymptotic standard errors calculated from the full likelihood.

Conditional likelihood
Sometimes it is possible to find a sufficient statistic for the nuisance parameters, and conditioning on this statistic results in a likelihood which does not depend on the nuisance parameters.

One example occurs in 2×2 tables, where conditioning on all four marginal totals leads to a conditional likelihood based on the non-central hypergeometric distribution. This form of conditioning is also the basis for Fisher's exact test.

Marginal likelihood
Sometimes we can remove the nuisance parameters by considering a likelihood based on only part of the information in the data, for example by using the set of ranks rather than the numerical values. Another example occurs in linear mixed models, where considering a likelihood for the residuals only after fitting the fixed effects leads to residual maximum likelihood estimation of the variance components.

Partial likelihood
A partial likelihood is an adaption of the full likelihood such that only a part of the parameters (the parameters of interest) occur in it. It is a key component of the proportional hazards model: using a restriction on the hazard function, the likelihood does not contain the shape of the hazard over time.

Products of likelihoods
The likelihood, given two or more independent events, is the product of the likelihoods of each of the individual events: $$\Lambda(A \mid X_1 \land X_2) = \Lambda(A \mid X_1) \cdot \Lambda(A \mid X_2).$$ This follows from the definition of independence in probability: the probabilities of two independent events happening, given a model, is the product of the probabilities.

This is particularly important when the events are from independent and identically distributed random variables, such as independent observations or sampling with replacement. In such a situation, the likelihood function factors into a product of individual likelihood functions.

The empty product has value 1, which corresponds to the likelihood, given no event, being 1: before any data, the likelihood is always 1. This is similar to a uniform prior in Bayesian statistics, but in likelihoodist statistics this is not an improper prior because likelihoods are not integrated.

Log-likelihood
Log-likelihood function is the logarithm of the likelihood function, often denoted by a lowercase $l$ or $\ell$, to contrast with the uppercase $L$ or $\mathcal{L}$ for the likelihood. Because logarithms are strictly increasing functions, maximizing the likelihood is equivalent to maximizing the log-likelihood. But for practical purposes it is more convenient to work with the log-likelihood function in maximum likelihood estimation, in particular since most common probability distributions—notably the exponential family—are only logarithmically concave, and concavity of the objective function plays a key role in the maximization.

Given the independence of each event, the overall log-likelihood of intersection equals the sum of the log-likelihoods of the individual events. This is analogous to the fact that the overall log-probability is the sum of the log-probability of the individual events. In addition to the mathematical convenience from this, the adding process of log-likelihood has an intuitive interpretation, as often expressed as "support" from the data. When the parameters are estimated using the log-likelihood for the maximum likelihood estimation, each data point is used by being added to the total log-likelihood. As the data can be viewed as an evidence that support the estimated parameters, this process can be interpreted as "support from independent evidence adds", and the log-likelihood is the "weight of evidence". Interpreting negative log-probability as information content or surprisal, the support (log-likelihood) of a model, given an event, is the negative of the surprisal of the event, given the model: a model is supported by an event to the extent that the event is unsurprising, given the model.

A logarithm of a likelihood ratio is equal to the difference of the log-likelihoods: $$\log \frac{\mathcal{L}(A)}{\mathcal{L}(B)} = \log \mathcal{L}(A) - \log \mathcal{L}(B) = \ell(A) - \ell(B).$$

Just as the likelihood, given no event, being 1, the log-likelihood, given no event, is 0, which corresponds to the value of the empty sum: without any data, there is no support for any models.

Graph
The graph of the log-likelihood is called the support curve (in the univariate case). In the multivariate case, the concept generalizes into a support surface over the parameter space. It has a relation to, but is distinct from, the support of a distribution.

The term was coined by A. W. F. Edwards in the context of statistical hypothesis testing, i.e. whether or not the data "support" one hypothesis (or parameter value) being tested more than any other.

The log-likelihood function being plotted is used in the computation of the score (the gradient of the log-likelihood) and Fisher information (the curvature of the log-likelihood). Thus, the graph has a direct interpretation in the context of maximum likelihood estimation and likelihood-ratio tests.

Likelihood equations
If the log-likelihood function is smooth, its gradient with respect to the parameter, known as the score and written $s_{n}(\theta) \equiv \nabla_{\theta} \ell_{n}(\theta)$, exists and allows for the application of differential calculus. The basic way to maximize a differentiable function is to find the stationary points (the points where the derivative is zero); since the derivative of a sum is just the sum of the derivatives, but the derivative of a product requires the product rule, it is easier to compute the stationary points of the log-likelihood of independent events than for the likelihood of independent events.

The equations defined by the stationary point of the score function serve as estimating equations for the maximum likelihood estimator. $$s_{n}(\theta) = \mathbf{0}$$ In that sense, the maximum likelihood estimator is implicitly defined by the value at $\mathbf{0}$ of the inverse function $s_{n}^{-1}: \mathbb{E}^{d} \to \Theta$, where $\mathbb{E}^{d}$  is the d -dimensional Euclidean space, and $\Theta$  is the parameter space. Using the inverse function theorem, it can be shown that $s_{n}^{-1}$ is well-defined in an open neighborhood about $\mathbf{0}$  with probability going to one, and $\hat{\theta}_{n} = s_{n}^{-1}(\mathbf{0})$  is a consistent estimate of $\theta$. As a consequence there exists a sequence $\left\{ \hat{\theta}_{n} \right\}$ such that $s_{n}(\hat{\theta}_{n}) = \mathbf{0}$  asymptotically almost surely, and $\hat{\theta}_{n} \xrightarrow{\text{p}} \theta_{0}$. A similar result can be established using Rolle's theorem.

The second derivative evaluated at $\hat{\theta}$, known as Fisher information, determines the curvature of the likelihood surface, and thus indicates the precision of the estimate.

Exponential families
The log-likelihood is also particularly useful for exponential families of distributions, which include many of the common parametric probability distributions. The probability distribution function (and thus likelihood function) for exponential families contain products of factors involving exponentiation. The logarithm of such a function is a sum of products, again easier to differentiate than the original function.

An exponential family is one whose probability density function is of the form (for some functions, writing $\langle -, - \rangle$ for the inner product):

$$ p(x \mid \boldsymbol \theta) = h(x) \exp\Big(\langle \boldsymbol\eta({\boldsymbol \theta}), \mathbf{T}(x)\rangle -A({\boldsymbol \theta}) \Big).$$

Each of these terms has an interpretation, but simply switching from probability to likelihood and taking logarithms yields the sum:

$$ \ell(\boldsymbol \theta \mid x) = \langle \boldsymbol\eta({\boldsymbol \theta}), \mathbf{T}(x)\rangle - A({\boldsymbol \theta}) + \log h(x).$$

The $\boldsymbol \eta(\boldsymbol \theta)$ and $h(x)$  each correspond to a change of coordinates, so in these coordinates, the log-likelihood of an exponential family is given by the simple formula:

$$ \ell(\boldsymbol \eta \mid x) = \langle \boldsymbol\eta, \mathbf{T}(x)\rangle - A({\boldsymbol \eta}).$$

In words, the log-likelihood of an exponential family is inner product of the natural parameter $\boldsymbol\eta$ and the sufficient statistic $\mathbf{T}(x)$, minus the normalization factor (log-partition function) $A({\boldsymbol \eta})$. Thus for example the maximum likelihood estimate can be computed by taking derivatives of the sufficient statistic $T$ and the log-partition function $A$.

Example: the gamma distribution
The gamma distribution is an exponential family with two parameters, $\alpha$ and $\beta$. The likelihood function is

$$\mathcal{L} (\alpha, \beta \mid x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}.$$

Finding the maximum likelihood estimate of $\beta$ for a single observed value $x$  looks rather daunting. Its logarithm is much simpler to work with:

$$\log \mathcal{L}(\alpha,\beta \mid x) = \alpha \log \beta - \log \Gamma(\alpha) + (\alpha-1) \log x - \beta x. \, $$

To maximize the log-likelihood, we first take the partial derivative with respect to $\beta$ :

$$\frac{\partial \log \mathcal{L}(\alpha,\beta \mid x)}{\partial \beta} = \frac{\alpha}{\beta} - x.$$

If there are a number of independent observations $x_1, \ldots, x_n$, then the joint log-likelihood will be the sum of individual log-likelihoods, and the derivative of this sum will be a sum of derivatives of each individual log-likelihood:

$$ \begin{align} & \frac{\partial \log \mathcal{L}(\alpha,\beta \mid x_1, \ldots, x_n)}{\partial \beta} \\ = {} & \frac{\partial \log \mathcal{L}(\alpha,\beta \mid x_1)}{\partial \beta} + \cdots + \frac{\partial \log \mathcal{L}(\alpha,\beta \mid x_n)}{\partial \beta} = \frac{n \alpha} \beta - \sum_{i=1}^n x_i. \end{align} $$

To complete the maximization procedure for the joint log-likelihood, the equation is set to zero and solved for $\beta$ :

$$\widehat\beta = \frac{\alpha}{\bar{x}}.$$

Here $\widehat\beta$ denotes the maximum-likelihood estimate, and $\textstyle \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$  is the sample mean of the observations.

Historical remarks
The term "likelihood" has been in use in English since at least late Middle English. Its formal use to refer to a specific function in mathematical statistics was proposed by Ronald Fisher, in two research papers published in 1921 and 1922. The 1921 paper introduced what is today called a "likelihood interval"; the 1922 paper introduced the term "method of maximum likelihood". Quoting Fisher:

[I]n 1922, I proposed the term 'likelihood,' in view of the fact that, with respect to [the parameter], it is not a probability, and does not obey the laws of probability, while at the same time it bears to the problem of rational choice among the possible values of [the parameter] a relation similar to that which probability bears to the problem of predicting events in games of chance. . . . Whereas, however, in relation to psychological judgment, likelihood has some resemblance to probability, the two concepts are wholly distinct. . . ."

The concept of likelihood should not be confused with probability as mentioned by Sir Ronald Fisher

I stress this because in spite of the emphasis that I have always laid upon the difference between probability and likelihood there is still a tendency to treat likelihood as though it were a sort of probability. The first result is thus that there are two different measures of rational belief appropriate to different cases. Knowing the population we can express our incomplete knowledge of, or expectation of, the sample in terms of probability; knowing the sample we can express our incomplete knowledge of the population in terms of likelihood.

Fisher's invention of statistical likelihood was in reaction against an earlier form of reasoning called inverse probability. His use of the term "likelihood" fixed the meaning of the term within mathematical statistics.

A. W. F. Edwards (1972) established the axiomatic basis for use of the log-likelihood ratio as a measure of relative support for one hypothesis against another. The support function is then the natural logarithm of the likelihood function. Both terms are used in phylogenetics, but were not adopted in a general treatment of the topic of statistical evidence.

Interpretations under different foundations
Among statisticians, there is no consensus about what the foundation of statistics should be. There are four main paradigms that have been proposed for the foundation: frequentism, Bayesianism, likelihoodism, and AIC-based. For each of the proposed foundations, the interpretation of likelihood is different. The four interpretations are described in the subsections below.

Bayesian interpretation
In Bayesian inference, although one can speak about the likelihood of any proposition or random variable given another random variable: for example the likelihood of a parameter value or of a statistical model (see marginal likelihood), given specified data or other evidence,   the likelihood function remains the same entity, with the additional interpretations of (i) a conditional density of the data given the parameter (since the parameter is then a random variable) and (ii) a measure or amount of information brought by the data about the parameter value or even the model. Due to the introduction of a probability structure on the parameter space or on the collection of models, it is possible that a parameter value or a statistical model have a large likelihood value for given data, and yet have a low probability, or vice versa. This is often the case in medical contexts. Following Bayes' Rule, the likelihood when seen as a conditional density can be multiplied by the prior probability density of the parameter and then normalized, to give a posterior probability density. More generally, the likelihood of an unknown quantity $X$ given another unknown quantity $Y$  is proportional to the probability of $Y$  given $X$ .

Likelihoodist interpretation
In frequentist statistics, the likelihood function is itself a statistic that summarizes a single sample from a population, whose calculated value depends on a choice of several parameters θ1 ... θp, where p is the count of parameters in some already-selected statistical model. The value of the likelihood serves as a figure of merit for the choice used for the parameters, and the parameter set with maximum likelihood is the best choice, given the data available.

The specific calculation of the likelihood is the probability that the observed sample would be assigned, assuming that the model chosen and the values of the several parameters θ give an accurate approximation of the frequency distribution of the population that the observed sample was drawn from. Heuristically, it makes sense that a good choice of parameters is those which render the sample actually observed the maximum possible post-hoc probability of having happened. Wilks' theorem quantifies the heuristic rule by showing that the difference in the logarithm of the likelihood generated by the estimate's parameter values and the logarithm of the likelihood generated by population's "true" (but unknown) parameter values is asymptotically χ2 distributed.

Each independent sample's maximum likelihood estimate is a separate estimate of the "true" parameter set describing the population sampled. Successive estimates from many independent samples will cluster together with the population's "true" set of parameter values hidden somewhere in their midst. The difference in the logarithms of the maximum likelihood and adjacent parameter sets' likelihoods may be used to draw a confidence region on a plot whose co-ordinates are the parameters θ1 ... θp. The region surrounds the maximum-likelihood estimate, and all points (parameter sets) within that region differ at most in log-likelihood by some fixed value. The χ2 distribution given by Wilks' theorem converts the region's log-likelihood differences into the "confidence" that the population's "true" parameter set lies inside. The art of choosing the fixed log-likelihood difference is to make the confidence acceptably high while keeping the region acceptably small (narrow range of estimates).

As more data are observed, instead of being used to make independent estimates, they can be combined with the previous samples to make a single combined sample, and that large sample may be used for a new maximum likelihood estimate. As the size of the combined sample increases, the size of the likelihood region with the same confidence shrinks. Eventually, either the size of the confidence region is very nearly a single point, or the entire population has been sampled; in both cases, the estimated parameter set is essentially the same as the population parameter set.

AIC-based interpretation
Under the AIC paradigm, likelihood is interpreted within the context of information theory.

title: Likelihood function


text:
Frederick T. Frelinghuysen

Frederick Theodore Frelinghuysen (August 4, 1817 – May 20, 1885) was an American lawyer and politician from New Jersey who served as a U.S. Senator and later as United States Secretary of State under President Chester A. Arthur.

Frelinghuysen was born in Millstone, New Jersey, and was adopted by his uncle Theodore Frelinghuysen after his father's death. He graduated from Rutgers College and studied law under his uncle. Frelinghuysen was involved in various political roles, including serving as a delegate to the 1860 Republican National Convention and as Attorney General of New Jersey. He was also appointed to fill a vacancy in the United States Senate.

In 1870, Frelinghuysen was nominated as U.S. ambassador to the United Kingdom but declined the position. He served again as a U.S. Senator from 1871 to 1877, and in 1881, was appointed U.S. Secretary of State by President Chester A. Arthur, serving until 1885. During his tenure, he withdrew the U.S. from the War of the Pacific and negotiated a treaty change with Hawaii, allowing for a naval base at Pearl Harbor.

After his term as Secretary of State, Frelinghuysen returned to Newark, New Jersey, where he died three months later. He married Matilda Elizabeth Griswold in 1842, with whom he had six children. Frelinghuysen University in Washington D.C. was named in his honor in 1917.

Early life and education
Frelinghuysen was born in Millstone, New Jersey, to Frederick Frelinghuysen (1788–1820) and Mary Dumont. His father died when he was just three years old, and he was adopted by his uncle, Theodore Frelinghuysen (1787–1862).

His grandfather, Frederick Frelinghuysen (1753–1804), was an eminent lawyer, one of the framers of the first New Jersey Constitution, a soldier in the American Revolutionary War, a member (1778–1779 and 1782–1783) of the Continental Congress from New Jersey, and from 1793 to 1796 a member of the United States Senate.

His uncle, Theodore Frelinghuysen, was Attorney General of New Jersey from 1817 to 1829, was a U.S. Senator from New Jersey from 1829 to 1835, was the Whig candidate for Vice President of the United States on the Henry Clay ticket in the 1844 Presidential election, and was Chancellor of New York University from 1839 until 1850 and president of Rutgers College from 1850 to 1862.

Frelinghuysen graduated from Rutgers College in 1836, and studied law in Newark with his uncle, to whose practice he succeeded in 1839, after he was admitted to the bar.

Career
Following his admission to the bar, he became attorney for the Central Railroad of New Jersey, the Morris Canal and Banking Company and other corporations.

Political career
According to The New York Times, Frelinghuysen was a member of the Whig Party until joining the Republican Party upon its inception. He was also crucial in establishing the Republican Party in New Jersey.

During the American Civil War, Frelinghuysen was active in public office rather than joining the Union Army. He was a delegate in 1861 to the Peace Congress, and appointed Attorney General of New Jersey by Governor Charles S. Olden that year to serve in the post until 1867. Frelinghuysen was encouraged by some to run for governor in 1862, though declined.

Frelinghuysen was a delegate to the 1860 Republican National Convention from New Jersey and from 1861 to 1867 was Attorney General of New Jersey. He was a delegate to the Peace conference of 1861 in Washington, and in 1866 was appointed by the Governor of New Jersey, as a Republican, to fill a vacancy in the United States Senate. In the winter of 1867, he was elected to fill the unexpired term, but a Democratic majority in the New Jersey Legislature prevented his re-election in 1869.

In 1870, he was nominated by President Ulysses S. Grant, and confirmed by the Senate, as United States Ambassador to the United Kingdom to succeed John Lothrop Motley, but declined the mission. From 1871 to 1877 he was again a member of the United States Senate, in which he was prominent in debate and in committee work, and was chairman of the U.S. Senate Committee on Foreign Affairs during the Alabama Claims negotiations.

He was a strong opponent of the Reconstruction measures of President Andrew Johnson, for whose conviction he voted for in Johnson's impeachment trial. Frelinghuysen supported the Radical Republicans' program for Reconstruction that emphasized a harsh treatment of former Confederates. He later allied with the GOP Stalwart faction whose members tended to utilize corruption/patronage effectively, though was considered to have a clean record.

He was a member of the joint committee which drew up and reported (1877) the Electoral Commission Bill, and subsequently served as a member of the Electoral Commission that decided the 1876 Presidential election. As a Republican, he voted with the eight-member majority on all counts.

U.S. Secretary of State
On December 12, 1881, he was appointed United States Secretary of State by President Chester A. Arthur to succeed James G. Blaine, and served until the inauguration of President Grover Cleveland in 1885.

Upon taking the post, Frelinghuysen was tasked with resolving a number of consequences resulted by the actions of his predecessor Blaine. Taking a pacifistic and patient approach, he shared the vision held by William H. Seward of the United States dominating the global market in setting an example for other nations to follow, he withdrew the U.S. from the War of the Pacific between Chile and Peru in which his predecessor unsuccessfully backed the Peruvians.

Frelinghuysen's other actions included canceling a scheduled Pan-American conference against President Arthur's wishes that Blaine had originally planned. In addition, he negotiated a treaty change with Hawaii that allowed for a naval base for the U.S. in Pearl Harbor, which was later known for being bombed by Japan in World War II.

In contrast to his predecessors in the position of U.S. Secretary of State, Frelinghuysen proved unable to urge Great Britain to modify the Clayton-Bulwer Treaty terms in a re-negotiation attempt, and instead pushed through a treaty with Nicaragua that would permit the construction of a canal in the country under joint ownership. However, it was withdrawn later during the presidency of Grover Cleveland by the U.S. Senate, which did not ratify it. Other efforts rejected by Congress included proposals to negotiate reciprocity Spain, Mexico, and Santo Dominigo, in addition to opening an international consortium between the Congo, U.S. and other countries.

Frelinghuysen served in the post until the end of President Arthur's term, effectively resigning in early March 1885.

After his term as Secretary of State Frelinghuysen returned to his home in Newark where he died less than three months after retiring.

Legacy
In 1917, Frelinghuysen University in Washington D.C. was named in honor of his service to African American causes.

Marriage and children
On January 25, 1842, Frelinghuysen married Matilda Elizabeth Griswold (1817–1889). She was the daughter of George Griswold, a merchant in New York City who "made an immense fortune in the time of the clipper trade with China." Together, they were the parents of three daughters and three sons, including:

 * Matilda Griswold Frelinghuysen (1846–1926), who married Henry Winthrop Gray (1840–1906), a prominent merchant in 1889.
 * Charlotte Louisa "Lucy" Frelinghuysen (1847–1930),
 * Frederick Frelinghuysen (1848–1924), who married Estelle B. Kinney, daughter of Thomas T. Kinney, in 1902.
 * George Griswold Frelinghuysen (1851–1936), who married Sara Linen Ballantine, granddaughter of Peter Ballantine, in 1881.
 * Sarah Helen Frelinghuysen (1856–1939), who first married Judge John J. Davis (1851–1902), the grandson of Massachusetts governor John Davis. After his death, she married Brig. Gen. Charles Laurie McCawley (1865–1935), the son of Charles G. McCawley, the 8th Commandant of the Marine Corps, in 1906.
 * Theodore Frelinghuysen (1860–1928), who married Alice Dudley Coats (1861–1889) in 1885. After her death, he married Elizabeth Mary "Lily" (née Thompson) Cannon, a daughter of William G. Thompson and descendant of Elijah Brush, both Mayors of Detroit, and the widow of Henry Lee Grand Cannon.

Frelinghuysen died at Newark on May 20, 1885, aged 67. He was buried at Mount Pleasant Cemetery, Newark. His widow died a few years later in February 1889.

Descendants
Through his eldest son Frederick, he was the grandfather of George Griswold Frelinghuysen II, who married Anne de Smolianinof; Estelle C. "Suzy" Frelinghuysen, who married fellow painter George Lovett Kingsland Morris; Frederick Frelinghuysen; Thomas Frelinghuysen; and Theodore Frelinghuysen.

Through his daughter Sarah and granddaughter Mathilda Elizabeth Frelinghuysen (née Davis) Lodge (1876–1960), who married George Cabot Lodge, he was the great-grandfather of Henry Cabot Lodge Jr. (1902–1985), the diplomat and United States Senator from Massachusetts, and John Davis Lodge (1903–1985), also a diplomat, U.S. Representative, and Governor of Connecticut.

title: Frederick T. Frelinghuysen


text:
Wikipedia talk:How do I do this

I have a problem. I have a user Id (Deb) and a password, and I can login. But when I try to access "Preferences", I am told I am not logged in. Then I log in again, and I get a message telling me I am logged in. Then I try "Preferences" again, and get the same message telling me I am not logged in. My contributions are not recognised. What am I doing wrong, please? Deb (and I have the same problem, too!--firepink)

 * Have you tried checking the box next to "Save the password as a cookie" on the login screen? AxelBoldt

Thank you, Axel - this resolved the problem. I wish I knew why. Deb

Sorry for this question but as a newcomer I haven't been able to solve it on my own. How do I link up the article I have contributed to others within Wikipedia that might be related? Ansdrew

If you want to create a link in your article to another Wikipedia article, you first need to find that article's exact title (with search), and then you put a text like World War II in your article to create the link. If you want the text of the link to be different from the title of the linked article, you can use the syntax Second World War. You can even create links to articles that don't exist yet, hopefully prompting others to fill the gap.

If you want to create links from other Wikipedia articles to your new article, you first surf around a bit to find the most appropriate Wikipedia articles (searching for your new article's title might be useful), and then edit them, adding a link pointing to your new article with the same syntax as above. AxelBoldt

- ''Q. So, how do I change the entry heading for a topic? "TheLaw" is a topic and should be "Law".

A. Registered users can move a page: this moves the page content and edit history to a new title, and creates a redirecting page at the old title.'' This does not answer the question. It answers "Who may move a page?" and "What does moving a page do?" It does not answer "How do I move a page?" --FOo

Feel free to rewrite! (I find the "Move this page" link in the nav-bar is a sort of a clue :-) ) -- Tarquin
 * I know the feeling FoO. I have never seen this move page feature either. --rmhermen
 * Exactly. I'm guessing either it's not accessible to all registered users, or ... ? --FOo
 * (Later): I get it. It's only available in the "Standard" skin, whereas I was set to the "Nostalgia" skin. This means that Wikipedia "skins" are not skins in the usual sense, since they change the features available rather than just making them look different!

title: Wikipedia talk:How do I do this


text:
Kannada script

The Kannada script (IAST: Kannaḍa lipi; obsolete: Kanarese or Canarese script in English) is an abugida of the Brahmic family, used to write Kannada, one of the Dravidian languages of South India especially in the state of Karnataka. It is one of the official scripts of the Indian Republic. Kannada script is also widely used for writing Sanskrit texts in Karnataka. Several minor languages, such as Tulu, Konkani, Kodava, Sanketi and Beary, also use alphabets based on the Kannada script. The Kannada and Telugu scripts share very high mutual intellegibility with each other, and are often considered to be regional variants of single script. Other scripts similar to Kannada script are Sinhala script (which included some elements from the Kadamba script ), and Old Peguan script (used in Burma).

The Kannada script (ಅಕ್ಷರಮಾಲೆ akṣaramāle or ವರ್ಣಮಾಲೆ varṇamāle) is a phonemic abugida of forty-nine letters. The character set is almost identical to that of other Brahmic scripts. Consonantal letters imply an inherent vowel. Letters representing consonants are combined to form digraphs (ಒತ್ತಕ್ಷರ ottakṣara) when there is no intervening vowel. Otherwise, each letter corresponds to a syllable.

The letters are classified into three categories: ಸ್ವರ svara (vowels), ವ್ಯಂಜನ vyañjana (consonants), and ಯೋಗವಾಹಕ yōgavāhaka (semiconsonants).

The Kannada words for a letter of the script are ಅಕ್ಷರ akshara, ಅಕ್ಕರ akkara, and ವರ್ಣ varṇa. Each letter has its own form (ಆಕಾರ ākāra) and sound (ಶಬ್ದ śabda), providing the visible and audible representations, respectively. Kannada is written from left to right.

History
The Brahmi script evolved into the Kadamba script by the 5th century, which in turn developed into the Old Kannada script and in the 7th century Telugu script emerged from the same Old Kannada script. The Kannada and Telugu scripts then separated by around 1300 C.E.

Over the centuries some changes have been made to the Kannada script. These changes consist of:
 * 1) Modification of existing glyphs: In the early Kannada script, no orthographic distinction was made between the short mid  ಎ, ಒ and long mid  ಏ, ಓ. However, distinct signs were employed to denote the special consonants viz. the trill  ಱ the retroflex lateral  ಳ and the retroflex rhotic  ೞ, by the 5th century.
 * 2) Introduction of new characters: Kannada script includes characters like  ಶ,  ಷ, ಋ,  ೠ,  ಌ,  ೡ,  ಐ,  ಔ,  ಅಂ,  ಅಃ, and mahāprāṇa characters like  ಖ,  ಘ,  ಛ,  ಝ,  ಥ,  ಧ,  ಠ,  ಢ,  ಫ,  ಭ. The introduction was done so that Sanskrit (and loanwords into the Kannada language from the donor language Sanskrit) could be written using the Kannada script. These changes have facilitated the use of the Kannada script for writing many of the literary Indic languages, including Sanskrit.

Vowels
There are thirteen vowel letters (ಸ್ವರ svara) in modern Kannada. When a vowel follows a consonant, it is written with a diacritic rather than as a separate letter. There are also three obsolete vowels.

Written Kannada is composed of akshara or kagunita, corresponding to syllables. The letters for consonants combine with diacritics for vowels. The consonant letter without any diacritic, such as ಕ ka, has the inherent vowel a ಅ. This is called ದೀರ್ಘ dīrgha. A consonant without a vowel is marked with a 'killer' stroke, such as ಕ್ k. This is known as ಹ್ರಸ್ವ hrasva.

Yōgavāha
There are two yōgavāha (part-vowel, part consonant) letters, known as ardhavisarga, used in modern Kannada and two others used in Sanskrit transcription.

Consonant letters
Two categories of consonant letters (ವ್ಯಂಜನ vyan̄jana) are defined in Kannada: the structured consonants and the unstructured consonants.

Structured consonants
The structured consonants are classified according to their place of articulation, that is, where the tongue touches the palate.

Unstructured consonants
The unstructured consonants are consonants that do not fall into any of the above structures:

Obsolete Kannada letters
Kannada literary works employed the letters ಱ (transliterated  or 'rh') and ೞ (transliterated , 'lh' or 'zh'), whose manner of articulation most plausibly could be akin to those in present-day Malayalam and Tamil. The letters dropped out of use in the 12th and 18th centuries, respectively. Later Kannada works replaced 'rh' and 'lh' with ರ (ra) and ಳ (la) respectively.

It is still used to write the Badaga language and a vowel + virama + ḻ is used to transcribe its retroflex vowels.

Another letter (or unclassified vyanjana (consonant)) that has become extinct is 'nh' or 'inn'. Likewise, this has its equivalent in Telugu, where it is called Nakaara pollu. The usage of this consonant was observed until the 1980s in Kannada works from the mostly coastal areas of Karnataka (especially the Dakshina Kannada district). Now, hardly any mainstream works use this consonant. This letter has been replaced by ನ್ (consonant n).

Consonant conjuncts
The Kannada script is rich in conjunct consonant clusters, with most consonants having a standard subjoined form and few true ligature clusters. A table of consonant conjuncts follows although the forms of individual conjuncts may differ according to the font.

Of special note is the sequence concerning the letter ರ (ra). Unlike other letters, the conjunct form is written second even if it is pronounced first in the sequence.

For example, the /rnaː/ in the word Karnāṭaka (ಕರ್ನಾಟಕ) is written ರ್ನಾ rather than ರ‍್ನಾ.

The nasal consonants ಙ (ṅa), ಞ (ña), ಣ (ṇa), ನ (na), and ಮ (ma) are usually written as an anusvara ಂ when preceding another consonant rather than a consonant conjunct.

For example, the /ŋg/ in the word Beṅgaḷūru (ಬೆಂಗಳೂರು) is usually written ಂಗ rather than ಙ್ಗ (ಬೆಙ್ಗಳೂರು).

Full list of consonant + vowel combinations
The formations shown boldface above are seldom used.

Numerals
The decimal numerals in the script are:

Transliteration
Several transliteration schemes/tools are used to type Kannada characters using a standard keyboard. These include Baraha (based on ITRANS), Pada Software and several internet tools like Google transliteration, Quillpad (predictive transliterator). Nudi, the Government of Karnataka's standard for Kannada Input, is a phonetic layout loosely based on transliteration.

In popular culture
Due to its resemblance to an eye and an eyebrow, the Kannada letter ಠ ṭha is used in the "look of disapproval" (displayed as "ಠ_ಠ"), a popular emoticon used to convey disapproval or contempt. Similarly, the akshara ರೃ rr̥a has been used in emoticons to represent a monocle, while ಥ tha has been used to represent a tearing eye.

Unicode
Kannada script was added to the Unicode Standard in October 1991 with the release of version 1.0.

The Unicode block for Kannada is U+0C80–U+0CFF:

title: Kannada script


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Talk:Mahogany

Early mix of comments (previously untitled section)
Could someone clean that is not my imagination, so when you want come with me, i ready this? There were separate entries for neem and mahogany, but they were basically the same. So i made a redirect from one to another. But i guess i did wrong: mahogany is family, neem is one of species of that family. So redirect should be removed, and something useful added instead in mahogany szopen

OK, this is off the top of my head, so don't shoot me if I've got it wrong, but isn't "mahogany" also used as a term for the timber of a whole stack of trees, half of them unrelated to one another? It's incorrect usage, no doubt - just as the timber trade in Oz calls all sorts of things "ash" which are indisputably not ash at all but eucalypt - but we should mention that usage of "mahogany" somewhere in the article, I think. Probably the same applies to teak. Tannin 14:22 4 Jun 2003 (UTC)

That is quite correct. The same goes for cherry, ebony, ironwood, oak, rosewood, teak ,etc. Mahogany is used for any number of woods (incorrectly, in varying degrees). Exceptions are about half the woods listed on the wikipedia page such as those of Cedrela, Toona, etc which I never heard of being traded as mahogany. I guess this shows that wikipedia is doomed. People hate facts, they prefer to perpetuate myths. PvR 16 sep 2005

They may not be traded in commerce as mahogany, but they are called mahoganies by botanists, as they are related plants in the family Meliaceae. The timber trade is not the best source of scientific naming, full as it is of unscrupulous traders trying to sell inferior quality woods as 'the real thing' to unsuspecting customers by attaching false labels and knowing that the customers do not have the wherewithal to check the true identity of the wood they are being sold. - MPF 09:11, 17 September 2005 (UTC)

Well, it all depends on your frame of reference. If you are going by only a single listing ever then anything is possible. You can even generate a usage by entering it on the web somewhere. I thought the point of wikipedia was to give helpful information, rather than mislead. If you are going by a single case here, and a single case there, you can write the weirdest things and "substantiate" these with references.

In the case of mahogany the general pattern is quite clear. If people are speaking about mahogany (other than as a color) they are speaking about wood. There are dozens of species the wood of which may be referred to as mahogany. Look it up. You try and turn up links about the trees in question, rather than the wood, and you will have a hard time of it. Nobody is interested in the trees.

If a taxonomist is speaking about a particular species he will invariably use the scientific name. Besides that, he will list local names. Look it up. If a botanist is running a ecological study he likely will use the local name. Look it up. PvR 18 Sept. 2005

 * I don't understand what you are talking about. This is a general article about usage of the term "mahogany" as it applies to the trees and the wood.  Hence, both are specified.  As for usage by systematists or ecologists - both use both common and scientific names.  In my experience systematists use common names far more than ecologists.
 * As for "helpful" vs. "misleading" information - the point of Wikipedia is not to restrict information to single conventions of usage - the point is to provide breadth and context. I am well aware that the species-specific stuff needs a lot of work, but it really doesn't do much good if you simply make it your mission to make the articles conform to a single aspect of usage.  I quite frankly would rather not have any articles under common names, because they are horribly confusing and vague, but policy leans toward common names.  So, given the multiplicity of common names, and given the vague way that the term "mahogany" is used - for the wood, for various meliaceous species, for Swietenia spp...we need to list all.  Personally I use the word "mahogany" for three species of Swietenia.  I would leave out Khaya and all the rest.  But this is not about my usage, and it should not be about any single person's usage.  It's about the breadth of usage.  The purpose is to provide that breadth.  So there is no harm and a lot of good in letting readers know that when someone calls a tree mahogany they might be speaking of a number of different species.  Guettarda 18:30, 18 September 2005 (UTC)

 * Frame of reference is everything, of course. If 80% of all people use "mahogany" as being a color; 20% of all people as being a timber (Swietenia or a reasonable substitute); 0,05 % of all people as being a tree of the genus Toona, then it is indeed misleading to list these three options as equal usage.

 * BTW: you will note that I did write it up as being in use for a multitude of woods/trees, but this was edited out by MDF in favor of his personal hobby horse. If you want to list usage in detail, then that is fine by me but it would need to be done with some care and with context intact. I just read that "wiki" means a fast look-up: in that case what is wanted is the main use, not the clutter of an occasional usage that happens to have caught the fancy of somebody. PvR, 19 Sep 2005

 * No, wiki-wiki means "fast" but Wikipedia is meant to be an encyclopaedia - the "wiki" part has to do with the software, which lets anyone edit. By the way, would you be willing to register a user name?  It would be a lot easier and would let these discussions progress more easily.  It would also be nice if you could cut other users a little more slack - people here - especially MDF - are generally not here to push an agenda, but to work on trying to build a free encyclopaedia.  Try to be constructive in your comments and your edit summaries.  Your contributions are appreciated, and would be a lot more if you could be a tad more polite - especially to as hard-working a volunteer as MDF.  Guettarda 16:41, 19 September 2005 (UTC)

 * Well, I will think about registering a user name, but it looks to be having more disadvantages than advantages. As for MDF, I don't know much about him, except that he does not check his facts before he acts, cannot spell botanical names, is awkward with botanical names in general and is about the worst case of "a-species-name-should-reflect-the genus-name"-agenda pushing that I have seen.  Just about everybody I know will agree that this is a very bad agenda to be pushing.  In dealing with reality a little pragmatism is called for. An encyclopedia that is badly at odds with the facts is going nowhere. PvR, 19 Sep 2005

 * I don't understand what you mean by "a-species-name-should-reflect-the genus-name"-agenda. Can you explain?  As the advantages/disadvantages of a user name - advantages include being able to maintain a "watch list" of pages of interest and the fact that it keeps your IP at least a little more hidden.  I don't see the disadvantages - it isn't that much harder to block an IP or series of IPs than it is to block a user name (please don't interpret that as a threat or anything, it's just an example of some people's reservations).  MDF has thousands of good edits, by the way, and has done an awful lot to improve plant articles, and to expand the coverage.  Spelling mistakes are easy to fix - the hard work is putting together the articles in the first place and protecting them from vandalism.  Guettarda 20:06, 19 September 2005 (UTC)

 * Well, the "a-species-name-should-reflect-the genus-name"-agenda is fairly old item. After binary names for species became widely accepted (some two hundred years ago) it struck some people that common names should follow the same pattern. A genus should have a common name (say "Oak" for Quercus) and all the species of Quercus should have a two-part name ending in "oak".

 * This works moderately well where there is a stable taxonomy and a genus with clear delimitations, but on the whole it does not work well at all. All temperate species of Quercus can be called "oaks", but it is hardly true that all "oaks" belong to Quercus. The USDA tried to apply this for the US trees with moderate success, even if it looks a little artificial sometimes: "douglas-fir" as one word, to force a common name for a genus (Pseudotsuga) without using "fir" (Abies) and "red cedar" to avoid a combination with "cedar". For more complex situations this works poorly.

 * Obviously, MDF is a hard case in this respect. Just look at Torreya taxifolia, which now has the common name "Stinking Torreya" that occurs on Wikipedia only, displacing the well-established "stinking cedar" that is in all the books. Obviously MDF goes beyond an eccentric choice among existing names and he does not hesitate to fabricate names to his liking. He is not dealing with reality anymore. This is going beyond pushing an agenda, crossing over into zealous bigotry.

 * I am sure vandalism is a big problem, the more so as I see apparently well-established Wikipedia users and adminstrators engaging in it, wholeheartedly, oblivious to even the hardest facts. Whatever happened to the idea that an encyclopedia should describe reality? - PvR, 20 Sept. 2005

The "a-species-name-should-reflect-the genus-name" idea is new to me, but I have never had any patience with common names, and I think that the idea of "official common names" (like the bird people do) is silly. In my opinion, any documentable common name should be in the article, but common names are really just trivia.

More to the point, you accuse MDF of "vandalism" - that's a pretty serious accusation, and you shouldn't use it trivially. He one common name with another - if you thought he made a mistake you should have said so, not called it vandalism. I have no idea what his source is, or what his motivation was for making the change. The point is, that's what Talk pages exist for - so that you can talk, not insult people with edit summaries. Guettarda 17:41, 20 September 2005 (UTC)

 * Well, I will give Talk pages a try, although I have noticed that in general people are not very inclined to change their opinion, or even willing to check facts. My admittedly few attempts at using Talk pages bear out that this general pattern applies here too. Certainly MDF does not use Talk pages before he tears into things. Yes, "vandalism" ("the malicious and deliberate defacement or destruction of somebody else's [or public] property") is a strong term; maybe too strong, maybe "zealous bigotry" is more accurate?

 * As to "the hard work is putting together the articles in the first place" that is of course quite true, although it is even harder to acquire the expertise needed to put together a good article. Wikipedia's approach to have everybody add bits and pieces will work well up to a point. Once somebody has "put together a good article" it risks getting skewed by having details added by people who feel detail is better than precision (or that detail is precision). That is even when the bits added are not wrong to begin with. PvR 21 Sep 2005.

Anon's reverts
ano po b ung scientific name ng mahogany sa pinas
 * The name "mahogany" is used botanically for Entandophragma spp. If you don't have access to biological abstract, try Google Scholar.
 * "and/or" is redundant - or is non-exclusive in English (as opposed to in Boolean logic).

Guettarda 17:34, 17 September 2005 (UTC)

It is in use, somewhat. You will note that I included this usage but that this was edited out by MDF. The nice thing about English is that just about anything is possible, except possibly locally. - contributed by

Please don't break up my edits like that. Thanks. As for English, no, you can't just do what you want - the language has rules of usage and style. It is useful to follow them. Guettarda 18:30, 18 September 2005 (UTC)

OK. As for "and/or" it is in the dictionary I use, as perfectly regular. PvR

Musical instruments
In the last sentence of the article, how do the wood's "dark properties" have to do with the tone produced by instruments in which it is used? This doesn't seem to be a meaningful statement. Lou Sander 13:08, 27 November 2006 (UTC)

Correct me if I'm wrong but...in general, the darker and denser the wood the "darker" the tone...meaning more low-emd (bass and mid) versus high end (treble)..this is why many guitars will have a mahogany body and a maple neck...the lighter maple providing more clarity and high-end, I believe.

Mahogany
What is Mahogany?

what waht waht waht what what what hwa trhwerhwhahwahawa ?????????? it makes no sence

Commercial sources
It would appear the Uses section of this article is the product of commercial ghost-writing and Denon may be the source. Google searches for specific sentences return sites affiliated with the company or promoting its products; headphones in particular.

"Mahogany is a very popular material for drum making, because of its great integrity and capability to produce a very dark, warm tone compared to other more common wood types like maple or birch."

"The famous Beatles sound of the 60s was made with Ludwig Drums in mahogany shells."

OopsNotSignedIn 02:23, 3 September 2007 (UTC)

??????????????
ok, i do NOT get what you guys are talking about.I'm wondering, how long does mahogany take to grow? mahogany wood is beautiful, it's so sad that everyones killing the species.

Fat puut Zayne Martin (talk) 21:05, 19 December 2021 (UTC)

Haymaker
I moved this sentence here for clarification:

 * Mahogany is a Japanese analogue to the English "haymaker".

As far as I know, a haymaker is just a kind of wild, swinging, or windup punch. This sentence appeared towards the end of the discussion of musical instruments. No idea what it means here. - Smerdis of Tlön (talk) 21:09, 18 December 2007 (UTC)

In what parts of the world does Mahogany grow?
I've found some mystery seed here in South India. Someone suggested that it may be mahogany. In what parts of the world does Mahogany grow? —Preceding unsigned comment added by 207.46.55.28 (talk) 11:28, 30 June 2009 (UTC)

Misspelling or variation?
There is a topic http://en.wikipedia.org/wiki/Lauan that should possibly be the destination for the unlinked word Luauan (just before Meranti) as it seems to be merely a variation of the spelling. Else, if the spelling were corrected, it would link automagically.

Are the seeds 'samara'?
Mahogany seeds look like samaras. Can anyone confirm they are samaras? 38.109.87.242 (talk) 18:18, 21 May 2015 (UTC)

Yes they are smaras. Plant surfer 19:22, 21 May 2015 (UTC)

Mahogany <-> Mahagoni
How come the "A" and "O" changed position compared to most other languages (including the Latin species names)? 178.2.94.217 (talk) 13:15, 2 August 2017 (UTC)

External links modified
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Duplicate article
https://en.wikipedia.org/wiki/Swietenia_macrophylla That's another article about the same tree — Preceding unsigned comment added by 213.57.109.194 (talk) 08:06, 4 September 2020 (UTC)

Mahogany vs Mahagoni
Is mahogany the same as mahagoni https://de.wikipedia.org/wiki/Swietenia_mahagoni ? Both are species of the genus swietenia... 201.218.117.14 (talk) 23:27, 5 December 2022 (UTC)

title: Talk:Mahogany