Molecular symmetry

From Wikipedia, the free encyclopedia

Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties, such as its dipole moment and its allowed spectroscopic transitions (based on selection rules such as the Laporte rule). Many university level textbooks on physical chemistry, quantum chemistry, and inorganic chemistry devote a chapter to symmetry.^{[1][2][3][4][5]}

While various frameworks for the study of molecular symmetry exist, group theory is the predominant one. This framework is also useful in studying the symmetry of molecular orbitals, with applications such as the Hückel method, ligand field theory, and the Woodward-Hoffmann rules. Another framework on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials.

Many techniques for the practical assessment of molecular symmetry exist, including X-ray crystallography and various forms of spectroscopy, for example infrared spectroscopy of metal carbonyls. Spectroscopic notation is based on symmetry considerations.

Symmetry concepts

The study of symmetry in molecules is an adaptation of mathematical group theory.

Elements

The symmetry of a molecule can be described by 5 types of symmetry elements.

• Symmetry axis: an axis around which a rotation by results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated C_n. Examples are the C₂ in water and the C₃ in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is assigned the z-axis in a Cartesian coordinate system.
• Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated σ. Water has two of them: one in the plane of the molecule itself and one perpendicular to it. A symmetry plane parallel with the principal axis is dubbed vertical (σ_v) and one perpendicular to it horizontal (σ_h). A third type of symmetry plane exists: If a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (σ_d). A symmetry plane can also be identified by its Cartesian orientation, e.g., (xz) or (yz).
• Center of symmetry or inversion center, abbreviated i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. There may or may not be an atom at the center. Examples are xenon tetrafluoride where the inversion center is at the Xe atom, and benzene (C₆H₆) where the inversion center is at the center of the ring.
• Rotation-reflection axis: an axis around which a rotation by , followed by a reflection in a plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated S_n. Examples are present in tetrahedral silicon tetrafluoride, with three S₄ axes, and the staggered conformation of ethane with one S₆ axis.
• Identity, abbreviated to E, from the German 'Einheit' meaning unity.^[6] This symmetry element simply consists of no change: every molecule has this element. While this element seems physically trivial, it must be included in the list of symmetry elements so that they form a mathematical group, whose definition requires inclusion of the identity element. It is so called because it is analogous to multiplying by one (unity).

Operations

XeF₄, with square planar geometry, has 1 C₄ axis and 4 C₂ axes orthogonal to C₄. These five axes plus the mirror plane perpendicular to the C₄ axis define the D_4h symmetry group of the molecule.

The 5 symmetry elements have associated with them 5 types of symmetry operations, which leave the molecule in a state indistinguishable from the starting state. They are sometimes distinguished from symmetry elements by a caret or circumflex. Thus, Ĉ_n is the rotation of a molecule around an axis and Ê is the identity operation. A symmetry element can have more than one symmetry operation associated with it. For example, the C₄ axis of the square xenon tetrafluoride (XeF₄) molecule is associated with two Ĉ₄ rotations (90°) in opposite directions and a Ĉ₂ rotation (180°). Since Ĉ₁ is equivalent to Ê, Ŝ₁ to σ and Ŝ₂ to î, all symmetry operations can be classified as either proper or improper rotations.

Molecular symmetry groups

Groups

The symmetry operations of a molecule (or other object) form a group, which is a mathematical structure usually denoted in the form (G,*) consisting of a set G and a binary combination operation say '*' satisfying certain properties listed below.

In a molecular symmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other. An example is the sequence of a C₄ rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy)C₄. By convention the order of operations is from right to left.

A molecular symmetry group obeys the defining properties of any group.

(1) closure property:
          For every pair of elements x and y in G, the product x*y is also in G.
          ( in symbols, for every two elements x, y∈G, x*y is also in G ).

This means that the group is closed so that combining two elements produces no new elements. Symmetry operations have this property because a sequence of two operations will produce a third state indistinguishable from the second and therefore from the first, so that the net effect on the molecule is still a symmetry operation.

(2) associative property:
          For every x and y and z in G, both (x*y)*z and x*(y*z) result with the same element in G.
          ( in symbols, (x*y)*z = x*(y*z ) for every x, y, and z ∈ G)
(3) existence of identity property:
          There must be an element ( say e ) in G such that product any element of G with e make no change to the element.
          ( in symbols, x*e=e*x= x for every x∈ G )
(4) existence of inverse property:
          For each element ( x ) in G, there must be an element y in G such that product of x and y is the identity element e.
          ( in symbols, for each x∈G there is a y ∈ G such that x*y=y*x= e for every x∈G )

The order of a group is the number of elements in the group. For groups of small orders, the group properties can be easily verified by considering its composition table, a table whose rows and columns correspond to elements of the group and whose entries correspond to their products.

Point group

The successive application (or composition) of one or more symmetry operations of a molecule has an effect equivalent to that of some single symmetry operation of the molecule. Moreover the set of all symmetry operations including this composition operation obeys all the properties of a group, given above. So (S,*) is a group where S is the set of all symmetry operations of some molecule, and * denotes the composition (repeated application) of symmetry operations. This group is called the point group of that molecule, because the set of symmetry operations leave at least one point fixed.
For some symmetries, an entire axis or an entire plane are fixed.

The symmetry of a crystal, however, is described by a space group of symmetry operations, which includes translations in space.

Examples

    (1)   The point group for the water molecule is C_2v, consisting of the symmetry operations E, C₂, σ_v and σ_v'. Its order is thus 4. Each operation is its own inverse. As an example of closure, a C₂ rotation followed by a σ_v reflection is seen to be a σ_v' symmetry operation: σ_v*C₂ = σ_v'. (Note that "Operation A followed by B to form C" is written BA = C).
    (2)   Another example is the ammonia molecule, which is pyramidal and contains a three-fold rotation axis as well as three mirror planes at an angle of 120° to each other. Each mirror plane contains an N-H bond and bisects the H-N-H bond angle opposite to that bond. Thus ammonia molecule belongs to the C_3v point group that has order 6: an identity element E, two rotation operations C₃ and C₃², and three mirror reflections σ_v, σ_v' and σ_v".

Common point groups

The following table contains a list of point groups labelled using the Schoenflies notation which is common in chemistry and molecular spectroscopy. The description of structure includes common shapes of molecules based on VSEPR theory.

Point group	Symmetry operations	Simple description of typical geometry	Example 1	Example 2	Example 3
C1	E	no symmetry, chiral	bromochlorofluoromethane	lysergic acid
Cs	E σh	mirror plane, no other symmetry	thionyl chloride	hypochlorous acid	chloroiodomethane
Ci	E i	inversion center	(S,R) 1,2-dibromo-1,2-dichloroethane (anti conformer)
C∞v	E 2C∞ ∞σv	linear	Hydrogen fluoride	nitrous oxide (dinitrogen monoxide)
D∞h	E 2C∞ ∞σi i 2S∞ ∞C2	linear with inversion center	oxygen	carbon dioxide
C2	E C2	"open book geometry," chiral	hydrogen peroxide
C3	E C3	propeller, chiral	triphenylphosphine
C2h	E C2 i σh	planar with inversion center	trans-1,2-dichloroethylene
C3h	E C3 C32 σh S3 S35	propeller	boric acid
C2v	E C2 σv(xz) σv'(yz)	angular (H2O) or see-saw (SF4)	water	sulfur tetrafluoride	sulfuryl fluoride
C3v	E 2C3 3σv	trigonal pyramidal	ammonia	phosphorus oxychloride
C4v	E 2C4 C2 2σv 2σd	square pyramidal	xenon oxytetrafluoride
D2	E C2(x) C2(y) C2(z)	twist, chiral	cyclohexane twist conformation
D3	E C3(z) 3C2	triple helix, chiral	Tris(ethylenediamine)cobalt(III) cation
D2h	E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)	planar with inversion center	ethylene	dinitrogen tetroxide	diborane
D3h	E 2C3 3C2 σh 2S3 3σv	trigonal planar or trigonal bipyramidal	boron trifluoride	phosphorus pentachloride
D4h	E 2C4 C2 2C2' 2C2 i 2S4 σh 2σv 2σd	square planar	xenon tetrafluoride	octachlorodimolybdate(II) anion
D5h	E 2C5 2C52 5C2 σh 2S5 2S53 5σv	pentagonal	ruthenocene	C70
D6h	E 2C6 2C3 C2 3C2' 3C2‘’ i 2S3 2S6 σh 3σd 3σv	hexagonal	benzene	bis(benzene)chromium
D2d	E 2S4 C2 2C2' 2σd	90° twist	allene	tetrasulfur tetranitride
D3d	E 2C3 3C2 i 2S6 3σd	60° twist	ethane (staggered rotamer)	cyclohexane chair conformation
D4d	E 2S8 2C4 2S83 C2 4C2' 4σd	45° twist	dimanganese decacarbonyl (staggered rotamer)
D5d	E 2C5 2C52 5C2 i 3S103 2S10 5σd	36° twist	ferrocene (staggered rotamer)
Td	E 8C3 3C2 6S4 6σd	tetrahedral	methane	phosphorus pentoxide	adamantane
Oh	E 8C3 6C2 6C4 3C2 i 6S4 8S6 3σh 6σd	octahedral or cubic	cubane	sulfur hexafluoride
Ih	E 12C5 12C52 20C3 15C2 i 12S10 12S103 20S6 15σ	icosahedral or dodecahedral	Buckminsterfullerene	dodecaborate anion	dodecahedrane

Representations

The symmetry operations can be represented in many ways. A convenient representation is by matrices. For any vector representing a point in Cartesian coordinates, left-multiplying it gives the new location of the point transformed by the symmetry operation. Composition of operations corresponds to matrix multiplication. In the C_2v example this is:

Although an infinite number of such representations exist, the irreducible representations (or "irreps") of the group are commonly used, as all other representations of the group can be described as a linear combination of the irreducible representations.

Character tables

For each point group, a character table summarizes information on its symmetry operations and on its irreducible representations. As there are always equal numbers of irreducible representations and classes of symmetry operations, the tables are square.
The table itself consists of characters that represent how a particular irreducible representation transforms when a particular symmetry operation is applied. Any symmetry operation in a molecule's point group acting on the molecule itself will leave it unchanged. But, for acting on a general entity, such as a vector or an orbital, this need not be the case. The vector could change sign or direction, and the orbital could change type. For simple point groups, the values are either 1 or −1: 1 means that the sign or phase (of the vector or orbital) is unchanged by the symmetry operation (symmetric) and −1 denotes a sign change (asymmetric).

The representations are labeled according to a set of conventions:

• A, when rotation around the principal axis is symmetrical
• B, when rotation around the principal axis is asymmetrical
• E and T are doubly and triply degenerate representations, respectively
• when the point group has an inversion center, the subscript g (German: gerade or even) signals no change in sign, and the subscript u (ungerade or uneven) a change in sign, with respect to inversion.
• with point groups C_∞v and D_∞h the symbols are borrowed from angular momentum description: Σ, Π, Δ.

The tables also capture information about how the Cartesian basis vectors, rotations about them, and quadratic functions of them transform by the symmetry operations of the group, by noting which irreducible representation transforms in the same way. These indications are conventionally on the righthand side of the tables. This information is useful because chemically important orbitals (in particular p and d orbitals) have the same symmetries as these entities.

The character table for the C_2v symmetry point group is given below:

C2v	E	C2	σv(xz)	σv'(yz)
A1	1	1	1	1	z	x2, y2, z2
A2	1	1	−1	−1	Rz	xy
B1	1	−1	1	−1	x, Ry	xz
B2	1	−1	−1	1	y, Rx	yz

Consider the example of water (H₂O), which has the C_2v symmetry described above. The 2p_x orbital of oxygen has B₁ symmetry as in the fourth row of the character table above, with x in the sixth column). It is oriented perpendicular to the plane of the molecule and switches sign with a C₂ and a σ_v'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, −1, 1, −1}, corresponding to the B₁ irreducible representation. Likewise, the 2p_z orbital is seen to have the symmetry of the A₁ irreducible representation, 2p_y B₂, and the 3d_xy orbital A₂. These assignments and others are noted in the rightmost two columns of the table.

Historical background

Hans Bethe used characters of point group operations in his study of ligand field theory in 1929, and Eugene Wigner used group theory to explain the selection rules of atomic spectroscopy.^[7] The first character tables were compiled by László Tisza (1933), in connection to vibrational spectra. Robert Mulliken was the first to publish character tables in English (1933), and E. Bright Wilson used them in 1934 to predict the symmetry of vibrational normal modes.^[8] The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy.^[9]

Nonrigid molecules

The symmetry groups described above are useful for describing rigid molecules which undergo only small oscillations about a single equilibrium geometry, so that the symmetry operations all correspond to simple geometrical operations. However Longuet-Higgins has proposed a more general type of symmetry groups suitable for non-rigid molecules with multiple equivalent geometries.^[10][11]
These groups are known as permutation-inversion groups, because a symmetry operation may be an energetically feasible permutation of equivalent nuclei, or an inversion with respect to the centre of mass, or a combination of the two.

For example, ethane (C₂H₆) has three equivalent staggered conformations. Conversion of one conformation to another occurs at ordinary temperature by internal rotation of one methyl group relative to the other. This is not a rotation of the entire molecule about the C₃ axis, but can be described as a permutation of the three identical hydrogens of one methyl group. Although each conformation has D_3d symmetry as in the table above, description of the internal rotation and associated quantum states and energy levels requires the more complete permutation-inversion group.

Similarly, ammonia (NH₃) has two equivalent pyramidal (C_3v) conformations which are interconverted by the process known as nitrogen inversion. This is not an inversion in the sense used for symmetry operations of rigid molecules, since NH₃ has no inversion center. Rather it is a reflection of all atoms about the centre of mass (close to the nitrogen), which happens to be energetically feasible for this molecule. Again the permutation-inversion group is used to describe the interaction of the two geometries.

A second and similar approach to the symmetry of nonrigid molecules is due to Altmann.^[12][13] In this approach the symmetry groups are known as Schrödinger supergroups and consist of two types of operations (and their combinations): (1) the geometric symmetry operations (rotations, reflections, inversions) of rigid molecules, and (2) isodynamic operations which take a nonrigid molecule into an energetically equivalent form by a physically reasonable process such as rotation about a single bond (as in ethane) or a molecular inversion (as in ammonia).^[13]

Uncertainty

From Wikipedia, the free encyclopedia

We are frequently presented with situations wherein a decision must be made when we are uncertain of exactly how to proceed.

Uncertainty is the situation which involves imperfect and / or unknown information. In other words it is a term used in subtly different ways in a number of fields, including insurance, philosophy, physics, statistics, economics, finance, psychology, sociology, engineering, metrology, and information science. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable and/or stochastic environments, as well as due to ignorance and/or indolence.^[1]

Concepts

Although the terms are used in various ways among the general public, many specialists in decision theory, statistics and other quantitative fields have defined uncertainty, risk, and their measurement as:

1. Uncertainty: The lack of certainty. A state of having limited knowledge where it is impossible to exactly describe the existing state, a future outcome, or more than one possible outcome.
2. Measurement of Uncertainty: A set of possible states or outcomes where probabilities are assigned to each possible state or outcome – this also includes the application of a probability density function to continuous variables.
3. Risk: A state of uncertainty where some possible outcomes have an undesired effect or significant loss.
4. Measurement of Risk: A set of measured uncertainties where some possible outcomes are losses, and the magnitudes of those losses – this also includes loss functions over continuous variables.^[2]

Knightian uncertainty. In his seminal work Risk, Uncertainty, and Profit (1921), University of Chicago economist Frank Knight established the important distinction between risk and uncertainty:^[3]

“

Uncertainty must be taken in a sense radically distinct from the familiar notion of risk, from which it has never been properly separated.... The essential fact is that 'risk' means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomena depending on which of the two is really present and operating.... It will appear that a measurable uncertainty, or 'risk' proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all.

”

“

You cannot be certain about uncertainty.

”

There are other taxonomies of uncertainties and decisions that include a broader sense of uncertainty and how it should be approached from an ethics perspective:^[4]

A taxonomy of uncertainty

There are some things that you know to be true, and others that you know to be false; yet, despite this extensive knowledge that you have, there remain many things whose truth or falsity is not known to you. We say that you are uncertain about them. You are uncertain, to varying degrees, about everything in the future; much of the past is hidden from you; and there is a lot of the present about which you do not have full information. Uncertainty is everywhere and you cannot escape from it.

“

”

Dennis Lindley, Understanding Uncertainty (2006)

For example, if it is unknown whether or not it will rain tomorrow, then there is a state of uncertainty. If probabilities are applied to the possible outcomes using weather forecasts or even just a calibrated probability assessment, the uncertainty has been quantified. Suppose it is quantified as a 90% chance of sunshine. If there is a major, costly, outdoor event planned for tomorrow then there is a risk since there is a 10% chance of rain, and rain would be undesirable.

Furthermore, if this is a business event and $100,000 would be lost if it rains, then the risk has been quantified(a 10% chance of losing $100,000). These situations can be made even more realistic by quantifying light rain vs. heavy rain, the cost of delays vs. outright cancellation, etc.

Some may represent the risk in this example as the "expected opportunity loss" (EOL) or the chance of the loss multiplied by the amount of the loss (10% × $100,000 = $10,000). That is useful if the organizer of the event is "risk neutral", which most people are not. Most would be willing to pay a premium to avoid the loss. An insurance company, for example, would compute an EOL as a minimum for any insurance coverage, then add onto that other operating costs and profit. Since many people are willing to buy insurance for many reasons, then clearly the EOL alone is not the perceived value of avoiding the risk.

Quantitative uses of the terms uncertainty and risk are fairly consistent from fields such as probability theory, actuarial science, and information theory. Some also create new terms without substantially changing the definitions of uncertainty or risk. For example, surprisal is a variation on uncertainty sometimes used in information theory. But outside of the more mathematical uses of the term, usage may vary widely. In cognitive psychology, uncertainty can be real, or just a matter of perception, such as expectations, threats, etc.

Vagueness or ambiguity are sometimes described as "second order uncertainty", where there is uncertainty even about the definitions of uncertain states or outcomes. The difference here is that this uncertainty is about the human definitions and concepts, not an objective fact of nature. It has been argued that ambiguity, however, is always avoidable while uncertainty (of the "first order" kind) is not necessarily avoidable.

Uncertainty may be purely a consequence of a lack of knowledge of obtainable facts. That is, there may be uncertainty about whether a new rocket design will work, but this uncertainty can be removed with further analysis and experimentation. At the subatomic level, however, uncertainty may be a fundamental and unavoidable property of the universe. In quantum mechanics, the Heisenberg Uncertainty Principle puts limits on how much an observer can ever know about the position and velocity of a particle. This may not just be ignorance of potentially obtainable facts but that there is no fact to be found. There is some controversy in physics as to whether such uncertainty is an irreducible property of nature or if there are "hidden variables" that would describe the state of a particle even more exactly than Heisenberg's uncertainty principle allows.

Measurements

The most commonly used procedure for calculating measurement uncertainty is described in the "Guide to the Expression of Uncertainty in Measurement" (GUM) published by ISO. A derived work is for example the National Institute for Standards and Technology (NIST) Technical Note 1297, "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results", and the Eurachem/Citac publication "Quantifying Uncertainty in Analytical Measurement". The uncertainty of the result of a measurement generally consists of several components. The components are regarded as random variables, and may be grouped into two categories according to the method used to estimate their numerical values:

• Type A, those evaluated by statistical methods
• Type B, those evaluated by other means, e.g., by assigning a probability distribution

By propagating the variances of the components through a function relating the components to the measurement result, the combined measurement uncertainty is given as the square root of the resulting variance. The simplest form is the standard deviation of a repeated observation.
In metrology, physics, and engineering, the uncertainty or margin of error of a measurement, when explicitly stated, is given by a range of values likely to enclose the true value. This may be denoted by error bars on a graph, or by the following notations:

• measured value ± uncertainty
• measured value +uncertainty
  −uncertainty
• measured value(uncertainty)

In the last notation, parentheses are the concise notation for the ± notation. For example applying 10 ¹⁄₂ meters in a scientific or engineering application, it could be written 7001105000000000000♠10.5 m or 7001105000000000000♠10.50 m, by convention meaning accurate to within one tenth of a meter, or one hundredth. The precision is symmetric around the last digit. In this case its half a tenth up and half a tenth down, so 10.5 means between 10.45 and 10.55. Thus it is understood that 10.5 means 7001105000000000000♠10.5±0.05, and 10.50 means 7001105000000000000♠10.50±0.005, also written 7001105000000000000♠10.5(0.5) and 7001105000000000000♠10.50(5). But if the accuracy is within two tenths, the uncertainty is ± one tenth, and it is required to be explicit: 7001105000000000000♠10.5±0.1 and 7001105000000000000♠10.50±0.01 or 7001105000000000000♠10.5(1)and 7001105000000000000♠10.50(1). The numbers in parenthesis apply to the numeral left of themselves, and are not part of that number, but part of a notation of uncertainty. They apply to the least significant digits. For instance, 7000100794000000000♠1.00794(7) stands for 7000100794000000000♠1.00794±0.00007, while 7000100794000000000♠1.00794(72) stands for 7000100794000000000♠1.00794±0.00072.^[5] This concise notation is used for example by IUPAC in stating the atomic mass of elements.

The middle notation is used when the error is not symmetrical about the value – for example 7000340000000000000♠3.4+0.3
−0.2.
This can occur when using a logarithmic scale, for example.

Often, the uncertainty of a measurement is found by repeating the measurement enough times to get a good estimate of the standard deviation of the values. Then, any single value has an uncertainty equal to the standard deviation. However, if the values are averaged, then the mean measurement value has a much smaller uncertainty, equal to the standard error of the mean, which is the standard deviation divided by the square root of the number of measurements. This procedure neglects systematic errors, however.

When the uncertainty represents the standard error of the measurement, then about 68.3% of the time, the true value of the measured quantity falls within the stated uncertainty range. For example, it is likely that for 31.7% of the atomic mass values given on the list of elements by atomic mass, the true value lies outside of the stated range. If the width of the interval is doubled, then probably only 4.6% of the true values lie outside the doubled interval, and if the width is tripled, probably only 0.3% lie outside. These values follow from the properties of the normal distribution, and they apply only if the measurement process produces normally distributed errors. In that case, the quoted standard errors are easily converted to 68.3% ("one sigma"), 95.4% ("two sigma"), or 99.7% ("three sigma") confidence intervals.^{[citation needed]}

In this context, uncertainty depends on both the accuracy and precision of the measurement instrument. The lower the accuracy and precision of an instrument, the larger the measurement uncertainty is. Notice that precision is often determined as the standard deviation of the repeated measures of a given value, namely using the same method described above to assess measurement uncertainty. However, this method is correct only when the instrument is accurate. When it is inaccurate, the uncertainty is larger than the standard deviation of the repeated measures, and it appears evident that the uncertainty does not depend only on instrumental precision.

Uncertainty and the media

Uncertainty in science, and science in general, is often interpreted much differently in the public sphere than in the scientific community.^[6] This is due in part to the diversity of the public audience, and the tendency for scientists to misunderstand lay audiences and therefore not communicate ideas clearly and effectively.^[6] One example is explained by the information deficit model. Also, in the public realm, there are often many scientific voices giving input on a single topic.^[6] For example, depending on how an issue is reported in the public sphere, discrepancies between outcomes of multiple scientific studies due to methodological differences could be interpreted by the public as a lack of consensus in a situation where a consensus does in fact exist.^[6] This interpretation may have even been intentionally promoted, as scientific uncertainty may be managed to reach certain goals. For example, global warming contrarian activists took the advice of Frank Luntz to frame global warming as an issue of scientific uncertainty, which was a precursor to the conflict frame used by journalists when reporting the issue.^[7]

“Indeterminacy can be loosely said to apply to situations in which not all the parameters of the system and their interactions are fully known, whereas ignorance refers to situations in which it is not known what is not known”.^[8] These unknowns, indeterminacy and ignorance, that exist in science are often “transformed” into uncertainty when reported to the public in order to make issues more manageable, since scientific indeterminacy and ignorance are difficult concepts for scientists to convey without losing credibility.^[6] Conversely, uncertainty is often interpreted by the public as ignorance.^[9] The transformation of indeterminacy and ignorance into uncertainty may be related to the public’s misinterpretation of uncertainty as ignorance.

Journalists often either inflate uncertainty (making the science seem more uncertain than it really is) or downplay uncertainty (making the science seem more certain than it really is).^[10] One way that journalists inflate uncertainty is by describing new research that contradicts past research without providing context for the change^[10] Other times, journalists give scientists with minority views equal weight as scientists with majority views, without adequately describing or explaining the state of scientific consensus on the issue.^[10] In the same vein, journalists often give non-scientists the same amount of attention and importance as scientists.^[10]

Journalists may downplay uncertainty by eliminating “scientists’ carefully chosen tentative wording, and by losing these caveats the information is skewed and presented as more certain and conclusive than it really is”.^[10] Also, stories with a single source or without any context of previous research mean that the subject at hand is presented as more definitive and certain than it is in reality.^[10] There is often a “product over process” approach to science journalism that aids, too, in the downplaying of uncertainty.^[10] Finally, and most notably for this investigation, when science is framed by journalists as a triumphant quest, uncertainty is erroneously framed as “reducible and resolvable”.^[10]

Some media routines and organizational factors affect the overstatement of uncertainty; other media routines and organizational factors help inflate the certainty of an issue. Because the general public (in the United States) generally trusts scientists, when science stories are covered without alarm-raising cues from special interest organizations (religious groups, environmental organization, political factions, etc.) they are often covered in a business related sense, in an economic-development frame or a social progress frame.^[11] The nature of these frames is to downplay or eliminate uncertainty, so when economic and scientific promise are focused on early in the issue cycle, as has happened with coverage of plant biotechnology and nanotechnology in the United

States, the matter in question seems more definitive and certain.^[11]

Sometimes, too, stockholders, owners, or advertising will pressure a media organization to promote the business aspects of a scientific issue, and therefore any uncertainty claims that may compromise the business interests are downplayed or eliminated.^[10]

Applications

• Investing in financial markets such as the stock market.
• Uncertainty or error is used in science and engineering notation. Numerical values should only be expressed to those digits that are physically meaningful, which are referred to as significant figures. Uncertainty is involved in every measurement, such as measuring a distance, a temperature, etc., the degree depending upon the instrument or technique used to make the measurement. Similarly, uncertainty is propagated through calculations so that the calculated value has some degree of uncertainty depending upon the uncertainties of the measured values and the equation used in the calculation.^[12]
• Uncertainty is designed into games, most notably in gambling, where chance is central to play.
• In scientific modelling, in which the prediction of future events should be understood to have a range of expected values.
• In physics, the Heisenberg uncertainty principle forms the basis of modern quantum mechanics.
• In weather forecasting it is now commonplace to include data on the degree of uncertainty in a weather forecast.
• Uncertainty is often an important factor in economics. According to economist Frank Knight, it is different from risk, where there is a specific probability assigned to each outcome (as when flipping a fair coin). Uncertainty involves a situation that has unknown probabilities, while the estimated probabilities of possible outcomes need not add to unity.
• In entrepreneurship: New products, services, firms and even markets are often created in the absence of probability estimates. According to entrepreneurship research, expert entrepreneurs predominantly use experience based heuristics called effectuation (as opposed to causality) to overcome uncertainty.
• In metrology, measurement uncertainty is a central concept quantifying the dispersion one may reasonably attribute to a measurement result. Such an uncertainty can also be referred to as a measurement error. In daily life, measurement uncertainty is often implicit ("He is 6 feet tall" give or take a few inches), while for any serious use an explicit statement of the measurement uncertainty is necessary. The expected measurement uncertainty of many measuring instruments (scales, oscilloscopes, force gages, rulers, thermometers, etc.) is often stated in the manufacturer's specification.
• Mobile phone radiation and health
• Uncertainty has been a common theme in art, both as a thematic device (see, for example, the indecision of Hamlet), and as a quandary for the artist (such as Martin Creed's difficulty with deciding what artworks to make).
• Uncertainty assessment is significantly important mining ventures where decisions are made based on uncertain models/outcomes.^[13] Predictions of oil and gas production from subsurface reservoirs are always uncertain.^[14][15]
• Uncertainty is almost unavoidable in Environmental Impact Assessments, EIAs.^[16] These assessments typically involve situations in which the full set of possible options and impacts for a particular project might not be known or there is no consensus as to which option to choose or which impact to consider. Uncertainty also occurs in EIAs when there is no certainty about the magnitude of impacts, there is no evidence on the possible interactions among the impacts, and the assumptions made for the assessment are not easily verifiable. Uncertainty is also reflected in situations in which: (i) there is no agreement as to the criteria to use to evaluate the importance of the impacts; (ii) the effectiveness of measures to manage impacts is uncertain; or (iii) when it is extremely difficult to detect early changes in the environment in order to minimize them over time. These uncertainties pose tremendous challenges in successfully managing impacts produced by, for instance, a development project. This is especially challenging since the uncertainties in EIAs, such as those referred to above, are usually obscured in the EIA report. Gustavsson ^[17] claimed that such uncertainties are often hidden because of either a desire for rapid approval of the EIA or to avoid controversy among practitioners, the public, and project developers that could compromise project realization. Cardenas ^[18] provides a comprehensive review on the issues of uncertainty in environmental impact assessments.

Bayesian probability

From Wikipedia, the free encyclopedia

Bayesian probability is one interpretation of the concept of probability. In contrast to interpreting probability as frequency or propensity of some phenomenon, Bayesian probability is a quantity that we assign to represent a state of knowledge,^[1] or a state of belief.^[2] In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability.

The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses, i.e., the propositions whose truth or falsity is uncertain.
Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then updated in the light of new, relevant data (evidence).^[3] The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation.

The term "Bayesian" derives from the 18th century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of Bayesian inference.^[4] Mathematician Pierre-Simon Laplace pioneered and popularised what is now called Bayesian probability.^[5]

Broadly speaking, there are two views on Bayesian probability that interpret the probability concept in different ways. According to the objectivist view, the rules of Bayesian statistics can be justified by requirements of rationality and consistency and interpreted as an extension of logic.^[1][6] According to the subjectivist view, probability quantifies a "personal belief".^[2]

Bayesian methodology

Bayesian methods are characterized by the following concepts and procedures:

• The use of random variables, or, more generally, unknown quantities,^[7] to model all sources of uncertainty in statistical models. This also includes uncertainty resulting from lack of information (see also the aleatoric and epistemic uncertainty).
• The need to determine the prior probability distribution taking into account the available (prior) information.
• The sequential use of the Bayes' formula: when more data becomes available, calculate the posterior distribution using the Bayes' formula; subsequently, the posterior distribution becomes the next prior.
• For the frequentist a hypothesis is a proposition (which must be either true or false), so that the frequentist probability of a hypothesis is either one or zero. In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the truth value is uncertain.

Objective and subjective Bayesian probabilities

Broadly speaking, there are two views on Bayesian probability that interpret the 'probability' concept in different ways. For objectivists, probability objectively measures the plausibility of propositions, i.e. the probability of a proposition corresponds to a reasonable belief everyone (even a "robot") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency.^[1][6] For subjectivists, probability corresponds to a 'personal belief'.^[2] For subjectivists, rationality and coherence constrain the probabilities a subject may have, but allow for substantial variation within those constraints. The objective and subjective variants of Bayesian probability differ mainly in their interpretation and construction of the prior probability.

History

The term Bayesian refers to Thomas Bayes (1702–1761), who proved a special case of what is now called Bayes' theorem in a paper titled "An Essay towards solving a Problem in the Doctrine of Chances".^[8] In that special case, the prior and posterior distributions were Beta distributions and the data came from Bernoulli trials. It was Pierre-Simon Laplace (1749–1827) who introduced a general version of the theorem and used it to approach problems in celestial mechanics, medical statistics, reliability, and jurisprudence.^[9] Early Bayesian inference, which used uniform priors following Laplace's principle of insufficient reason, was called "inverse probability" (because it infers backwards from observations to parameters, or from effects to causes).^[10] After the 1920s, "inverse probability" was largely supplanted by a collection of methods that came to be called frequentist statistics.^[10]
In the 20th century, the ideas of Laplace were further developed in two different directions, giving rise to objective and subjective currents in Bayesian practice. Harold Jeffreys' Theory of Probability (first published in 1939) played an important role in the revival of the Bayesian view of probability, followed by works by Abraham Wald (1950) and Leonard J. Savage (1954). The adjective Bayesian itself dates to the 1950s; the derived Bayesianism, neo-Bayesianism is of 1960s coinage.^[11] In the objectivist stream, the statistical analysis depends on only the model assumed and the data analysed.^[12] No subjective decisions need to be involved. In contrast, "subjectivist" statisticians deny the possibility of fully objective analysis for the general case.

In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of Markov chain Monte Carlo methods, which removed many of the computational problems, and an increasing interest in nonstandard, complex applications.^[13] Despite the growth of Bayesian research, most undergraduate teaching is still based on frequentist statistics.^{[14][citation needed]} Nonetheless, Bayesian methods are widely accepted and used, such as in the field of machine learning.^[15]

Justification of Bayesian probabilities

The use of Bayesian probabilities as the basis of Bayesian inference has been supported by several arguments, such as the Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem.

Axiomatic approach

Richard T. Cox showed that^[6] Bayesian updating follows from several axioms, including two functional equations and a hypothesis of differentiability. The assumption of differentiability or even continuity is controversial; Halpern found a counterexample based on his observation that the Boolean algebra of statements may be finite.^[16] Other axiomatizations have been suggested by various authors to make the theory more rigorous.^[7]

Dutch book approach

The Dutch book argument was proposed by de Finetti, and is based on betting. A Dutch book is made when a clever gambler places a set of bets that guarantee a profit, no matter what the outcome of the bets. If a bookmaker follows the rules of the Bayesian calculus in the construction of his odds, a Dutch book cannot be made.

However, Ian Hacking noted that traditional Dutch book arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books. For example, Hacking writes^[17] "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour."

In fact, there are non-Bayesian updating rules that also avoid Dutch books (as discussed in the literature on "probability kinematics" following the publication of Richard C. Jeffreys' rule, which is itself regarded as Bayesian ^[18]). The additional hypotheses sufficient to (uniquely) specify Bayesian updating are substantial, complicated, and unsatisfactory.^[19]

Decision theory approach

A decision-theoretic justification of the use of Bayesian inference (and hence of Bayesian probabilities) was given by Abraham Wald, who proved that every admissible statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures.^[20] Conversely, every Bayesian procedure is admissible.^[21]

Personal probabilities and objective methods for constructing priors

Following the work on expected utility theory of Ramsey and von Neumann, decision-theorists have accounted for rational behavior using a probability distribution for the agent. Johann Pfanzagl completed the Theory of Games and Economic Behavior by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and Oskar Morgenstern: their original theory supposed that all the agents had the same probability distribution, as a convenience.^[22] Pfanzagl's axiomatization was endorsed by Oskar Morgenstern: "Von Neumann and I have anticipated" the question whether probabilities "might, perhaps more typically, be subjective and have stated specifically that in the latter case axioms could be found from which could derive the desired numerical utility together with a number for the probabilities (cf. p. 19 of The Theory of Games and Economic Behavior). We did not carry this out; it was demonstrated by Pfanzagl ... with all the necessary rigor".^[23]

Ramsey and Savage noted that the individual agent's probability distribution could be objectively studied in experiments. The role of judgment and disagreement in science has been recognized since Aristotle and even more clearly with Francis Bacon. The objectivity of science lies not in the psychology of individual scientists, but in the process of science and especially in statistical methods, as noted by C. S. Peirce.^[24] Recall that the objective methods for falsifying propositions about personal probabilities have been used for a half century, as noted previously. Procedures for testing hypotheses about probabilities (using finite samples) are due to Ramsey (1931) and de Finetti (1931, 1937, 1964, 1970). Both Bruno de Finetti and Frank P. Ramsey acknowledge^{[citation needed]} their debts to pragmatic philosophy, particularly (for Ramsey) to Charles S. Peirce.

The "Ramsey test" for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century.^[25] This work demonstrates that Bayesian-probability propositions can be falsified, and so meet an empirical criterion of Charles S. Peirce, whose work inspired Ramsey. (This falsifiability-criterion was popularized by Karl Popper.^[26][27])
Modern work on the experimental evaluation of personal probabilities uses the randomization, blinding, and Boolean-decision procedures of the Peirce-Jastrow experiment.^[28] Since individuals act according to different probability judgments, these agents' probabilities are "personal" (but amenable to objective study).

Personal probabilities are problematic for science and for some applications where decision-makers lack the knowledge or time to specify an informed probability-distribution (on which they are prepared to act). To meet the needs of science and of human limitations, Bayesian statisticians have developed "objective" methods for specifying prior probabilities.

Indeed, some Bayesians have argued the prior state of knowledge defines the (unique) prior probability-distribution for "regular" statistical problems; cf. well-posed problems. Finding the right method for constructing such "objective" priors (for appropriate classes of regular problems) has been the quest of statistical theorists from Laplace to John Maynard Keynes, Harold Jeffreys, and Edwin Thompson Jaynes: These theorists and their successors have suggested several methods for constructing "objective" priors:

• Maximum entropy
• Transformation group analysis
• Reference analysis

Each of these methods contributes useful priors for "regular" one-parameter problems, and each prior can handle some challenging statistical models (with "irregularity" or several parameters). Each of these methods has been useful in Bayesian practice. Indeed, methods for constructing "objective" (alternatively, "default" or "ignorance") priors have been developed by avowed subjective (or "personal") Bayesians like James Berger (Duke University) and José-Miguel Bernardo (Universitat de València), simply because such priors are needed for Bayesian practice, particularly in science.^[29] The quest for "the universal method for constructing priors" continues to attract statistical theorists.^[29]

Thus, the Bayesian statistician needs either to use informed priors (using relevant expertise or previous data) or to choose among the competing methods for constructing "objective" priors.