Exponential family

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"Natural parameters" linked here. For the use of this term in differential geometry, see the geometry of the differential curve.

In probability and statistics, exponential families are a set of probability distributions of a given form, which are specified below. This particular shape is chosen for mathematical convenience, based on some useful algebraic properties, as well as for the public, since exponential families have very natural distribution circuits to consider. The concept of exponential families is credited to E. J. G. Pitman, G. Darmois, and B. O. Koopman in 1935-36. The term exponential class is sometimes used instead of "exponential family".

The exponential distribution family provides a general framework for selecting possible distribution parameterization, in terms of natural parameters , and for determining useful sample statistics, called sufficient natural statistics of the family.

Video Exponential family

Definisi

Most of the commonly used distributions are in the exponential family, which are listed in the subsections below. The subdivision that follows it is an increasingly common sequence of exponential family mathematical definitions. Regular readers may want to limit attention to the first and simplest definitions, which relate to a single parameter family of discrete or continuous probability distributions.

Example of exponential family distribution

Exponential families include many of the most common distributions. Among many others, families include the following:

A number of common distributions are exponential families, but only when certain parameters are established and known. As an example:

• binomial (with a fixed number of experiments)
• multinomial (with a fixed number of experiments)
• binomial negative (with fixed number of failures)

Note that in each case, the parameters to be fixed define the limit on the size of the observed value.

Examples of general distributions that are not exponential families are t Students, mostly mixed distributions, and even distribution families are uniform when the limit is not fixed. See the section below in the example for further discussion.

Scalar parameters

Keluarga parameter eksponensial tunggal adalah seperangkat distribusi probabilitas yang fungsi kepadatan probabilitasnya (atau fungsi massa probabilitas, untuk kasus distribusi diskrit) dapat diekspresikan dalam bentuk

${\ displaystyle f_ {X} (x \ mid \ theta) = h (x) \ exp \ kiri (\ eta (\ theta) \ cdot T (x) -A (\ theta) \ right)}  ÂÂ$

where T ( x ), h ( x ), ? (? ), and A (? ) are known functions.

Suatu alternatif, bentuk ekuivalen yang sering diberikan adalah

${\ displaystyle f_ {X} (x \ mid \ theta) = h (x) g (\ theta) \ exp \ kiri (\ eta (\ theta ) \ cdot T (x) \ right)} Â Â$

atau dengan kata lain

${\ displaystyle f_ {X} (x \ mid \ theta) = \ exp \ kiri (\ eta (\ theta) \ cdot T (x) -A (\ theta) B (x) \ right)} Â Â$

Value ? are called family parameters.

Perhatikan bahwa x sering menjadi vektor pengukuran, dalam hal ini T ( x ) mungkin merupakan fungsi dari ruang kemungkinan nilai < i> x ke bilangan real. Secara umum, ? (? ) dan T ( x ) masing-masing dapat bernilai vektor sedemikian rupa sehingga ${\ displaystyle \ eta (\ theta) '\ cdot T (x)}  ÂÂ$ bernilai nyata.

If ? (? ) Ã, = ? , then the exponential family is said to be in the canonical form . By defining the changed parameter ? Ã, = Ã, ? (? ), it is always possible to convert an exponent family into a canonical form. Canonical form is not unique, because ? Can be multiplied by non-zero constants, provided that T ( x ) is multiplied by the inverse of the constant, or the constant > c can be added to ? and h ( x ) multiplied by ${\ displaystyle \ exp (-c \ cdot T (x))}$ to compensate.

Even when x is a scalar, and there is only one parameter, the function ? And T ( x ) can still be vectors, as described below.

Also note that the A (? ) function, or equivalently g (? ), is automatically determined after the function -other function has been selected, since it should take the form that causes the distribution to be normalized (sum it up or integrate it to one above the whole domain). Furthermore, these two functions can always be written as a function ? , even when ? (? ) is not a one-to-one function, ie two or more different values ​​of ? map to the same value ? (? ), and therefore ? (? ) can not be reversed. In such cases, all values? map to ? the same (? ) will also have the same values ​​for A (? ) and g (? ).

Factorization of the variables involved

where f and h are arbitrary functions x ; g and j are arbitrary functions ? ; and c is an arbitrary "constant" expression (i.e. an expression that does not involve x or ? ).

Ada pembatasan lebih lanjut tentang berapa banyak faktor seperti itu dapat terjadi. Misalnya, secondly express:

${\ displaystyle {[f (x) g (\ theta)]} ^ {h (x) j (\ theta)}, \ qquad {[f (x)]} ^ {h (x) j (\ theta)} [g (\ theta)] h (x) j (\ theta)},} Â Â$

adalah sama, yaitu produk dari dua faktor "diizinkan". Namun, ketika ditulis ulang ke dalam bentuk faktor,

${\ displaystyle {[f (x) g (\ theta)]} ^ {h (x) j (\ theta)} = {[f (x)]} ^ {h (x) j (\ theta)} [g (\ theta)] ^ {h (x) j (\ theta)} = e ^ {[h (x) \ ln f (x)] j (\ theta ) h (x) [j (\ theta) \ ln g (\ theta)]},}  ÂÂ$

it can be seen that it can not be expressed in the requested form. (However, this form is a member of the curved exponential family , which allows several factor factors in the exponent.)

Untuk melihat mengapa spray bentuk

${\ displaystyle {[f (x)]} ^ {g (\ theta)}} Â Â$

It's a good idea, catches it

${\ displaystyle {[f (x)]} ^ {g (\ theta)} = e ^ {g (\ theta) \ ln f (x) }} Â Â$

dan karenanya faktor dalam exponen Demikian Pula,

${\ displaystyle {[f (x)]} ^ {h (x) g (\ theta)} = e ^ {h (x) g (\ theta) \ ln f (x)} = e ^ {[h (x) \ ln f (x)] g (\ theta)}} Â Â$

and again the factor in the exponent.

Perhatikan juga bahwa faktor yang terdiri dari jumlah di mana kedua jenis variabel yang terlibat (misalnya faktor bentuk ${\ displaystyle 1 f (x) g (\ theta)} Â$ ) tidak dapat diperhitungkan dalam mode ini (kecuali dalam beberapa kasus di mana terjadi secara langsung dalam eksponen); Inilah sebabnya, misalnya, distribuí Cauchy dan distribusi Student t bukan keluarga exponensial.

Parameter vektor

Definisi dalam hal satu parameter bilangan real dapat diperluas that satu parameter vektor nyata

${\ displaystyle {\ boldsymbol {\ theta}} = \ kiri (\ theta 1, \ theta2, \ ldots, \ theta s} \ right) ^ {T}.} Â Â$

Sebuah keluarga distribusi dikatakan milik keluarga vektor eksponensial jika fungsi kepadatan probabilitas (atau fungsi massa probabilitas, untuk distribusi diskrit) dapat ditulis sebagai

${\ displaystyle f_ {X} (x \ mid {\ boldsymbol {\ theta}}) = h (x) \ exp \ left (\ jumlah _ {i = 1} ^ {s} \ eta _ {i} ({\ boldsymbol {\ theta}}) T_ {i} (x) -A ({\ boldsymbol {\ theta}}) \ right)}    Atau dalam bentuk yang lebih ringkas,                            f                        X                         (        x         |                  )        =        h        (        x        )        exp                                    (                                    (                  )         ?                 T                (        x        )        -        A        (                  )                               )                               {\ displaystyle f_ {X} (x \ mid {\ boldsymbol {\ theta}}) = h (x) \ exp {\ Big (} {\ boldsymbol {\ eta}} ({\ boldsymbol {\ theta}}) \ cdot \ mathbf {T} (x) -A ({\ boldsymbol {\ theta}}) {\ Big)}}   Formula ini menulis jumlah sebagai produk titik dari fungsi bernilai-vektor                              (                  )             {\ displaystyle {\ boldsymbol {\ eta}} ({\ boldsymbol {\ theta}})}  dan                             T                (        x        )             {\ displaystyle \ mathbf {T} (x)}   .Suatu alternatif, bentuk ekuivalen yang sering terlihat adalah                            f                        X                         (        x         |                  )        =        h        (        x        )        g        (                  )        exp                                    (                                    (                  )         ?                 T                (        x        )                               )                               {\ displaystyle f_ {X} (x \ mid {\ boldsymbol {\ theta}}) = h (x) g ({\ boldsymbol {\ theta} }) \ exp {\ Big (} {\ boldsymbol {\ eta}} ({\ boldsymbol {\ theta}} \ cdot \ mathbf {T} (x) {\ Big)}}   Seperti dalam kasus skalar dihargai, keluarga exponensial dikatakan dalam bentuk shot jika                   ?        saya       :                        ?                      saya                         (                  )        =                ?                      saya                         .             {\ displaystyle \ forall i: \ quad \ eta _ {i} ({\ boldsymbol {\ theta}}) = \ theta i}   Sebuah keluarga vector exponential dikatakan melengkung nice dimension                                               1                                            ,                             ?                                  2                                            ,              ...              ,                             ?                                  d                                                       )                                T                             {\ displaystyle {\ boldsymbol {\ theta}} = \ kiri (\ theta 1, \ theta2, \ ldots, \ theta d) \ right) ^ {T}}   Kurang would give a dimensional vector                              (                  )        =                             (                                         ?                                  1                                            (                              )              ,                             ?                                  2                                            (                              )              ,              ...              ,                             ?                                  s                                            (                              )                         )                                T                        .             {\ displaystyle {\ boldsymbol {\ eta}} ({\ boldsymbol {\ theta}}) = \ kiri (\ eta1 ({\ boldsymbol {\ theta}}), \ eta2 ({\ boldsymbol {\ theta}}), \ ldots, \ eta ({\ boldsymbol {\ theta}}) \ right) ^ {T }.}   That is, if the dimension parameter vector is less than the number of functions of the parameter vector in the function representation of the above probability density. Note that the most common distributions in exponential families are not curved, and many algorithms designed to work with exponential family members implicitly or explicitly assume that the distribution is not curved.atau dengan kata lain                            f                        X                         (        x         |                  )        =        h        (        x        )        g        (                  )        exp                                    (                                            T                (        x        )                               )                               {\ displaystyle f_ {X} (x \ mid {\ boldsymbol {\ eta}}) = h (x) g ({\ boldsymbol {\ eta} }) \ exp {\ Big (} {\ boldsymbol {\ eta}} \ cdot \ mathbf {T} (x) {\ Big)}}   Perhatikan bahwa formulate it at terkadang dapat dilihat dengan                                                   T                                 T                (        x        )             {\ displaystyle {\ boldsymbol {\ eta}} ^ {T} \ mathbf {T} (x)}   di tempat                                      T                (        x        )             {\ displaystyle {\ boldsymbol {\ eta}} \ cdot \ mathbf {T} (x)}   . 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