1、 density that satisfi es this usual integral relation, it must be a density with respect to (k1)-dimensional Lebesgue measure. Hence, technically, the density should be a function of k 1 of the k variables, with the k-th variable implicitly equal to one minus the sum of the others, so that all k var
2、iables sum to one. The choice of which k 1 variables to use in the density is arbitrary. For example, one way to write the density is as follows: f(q1,q2,.,qk1) = ?Pk i=1i ? Qk i=1(i) Qk1 i=1 qi1 i ? 1 Pk1 i=1 qi ?k1 . However, rather than needlessly complicate the presentation, we shall just write
3、the density as a function of the entire k-dimensional vector q. We also note that the constraint that P iqi= 1 forces the components of Q to be dependent. UWEETR-2010-00062 = 1,1,1 = .1,.1,.1 = 10,10,10 = 2,5,15 Figure 1: Density plots (blue = low, red = high) for the Dirichlet distribution over the
4、 probability simplex in R3for various values of the parameter . When = c,c,c for some c 0, the density is symmetric about the uniform pmf (which occurs in the middle of the simplex), and the special case = 1,1,1 shown in the top-left is the uniform distribution over the simplex. When 0 0 (k,)Gamma d
5、istribution with parameters k and D =AD=B means random variables A and B have the same distribution where (s) denotes the gamma function. The gamma function is a generalization of the factorial function: for s 0, (s + 1) = s(s), and for positive integers n, (n) = (n 1)! because (1) = 1. We denote th
6、e mean of a Dirichlet distribution as m = /0. Fig. 1 shows plots of the density of the Dirichlet distribution over the two-dimensional simplex in R3for a handful of values of the parameter vector . When = 1,1,1, the Dirichlet distribution reduces to the uniform distribution over the simplex (as a qu
7、ick exercise, check this using the density of the Dirichlet in (1).) When the components of are all greater than 1, the density is monomodal with its mode somewhere in the interior of the simplex, and when the components of are all less than 1, the density has sharp peaks almost at the vertices of t
8、he simplex. Note that the support of the Dirichlet is open and does not include the vertices or edge of the simplex, that is, no component of a pmf drawn from a Dirichlet will ever be zero. Fig. 2 shows plots of samples drawn IID from diff erent Dirichlet distributions. Table 2 summarizes some key p
9、roperties of the Dirichet distribution. When k = 2, the Dirichlet reduces to the Beta distribution. The Beta distribution Beta(,) is defi ned on (0,1) and has density f(x;,) = ( + ) ()()x 1(1 x)1. To make the connection clear, note that if X f3527d41efb383c74dad1180d061717Lff354488a165b9aefe3c2b530d
10、e8efe7Lff354b5b30475f8dfc43b3bd42edae89Lff35d095546f10305be18378769a4ce6Lff36579c44d3cbf14e09cf767490fa36BLff368e556bd8cda9b07081f9c2b13043嘶Lff3725ab01bc512ab9db285c043d5913Lff3781f282d3471c97ad70a3dce060f8琶Lff392250feb9524a34d8305a3e23f84cLff396317bb744d5c2f600549f58fdc8cLff396de568957aa8dbae180615
11、bd59dc氶Lff397a05d37716082264266f54fc528f鐶Lff39ddb1288a1a201f637074f72eca33砶Lff39fc055c05bad1a2950231ebaa014d搶Lff3a3e230179c227aa416752257354ecLff3b2248c05a37c34afbaf2f9f2cac2epLff3c747f562a9c6657841554c259008cLff3deae504f0de517bed360b8c00f739鴶Lff3dfb601f7c45e78c36857473774be2Lff3e47d489df31b200a174c
12、a770baa47輶Lff3f9b548c13fad3148472fc6b460865Lff402307dcace35475d6e00ce7fae1d3Lff40a9e785e9a02506365719fab651c4踶Lff40c86aa9b91d356d4b2716a95ed6e7鸶Lff40fe2cbf48188ff0f33732b114a9a2谶Lff415fe3f95bd4d9c79e004e3e881717Lff36c677f6c214c37b58fef48ebd2596Lff35ae8a57b96c8a0e0566676557c76cLff36b4f10a7d41ab6051af
13、f8d43e83efLff38a5ae048c10f70123df3fac47e0aa褶Lff390eac406aaea9668c4132afaf30c4Lff3fe3291c27e44c9755a3bb6b2f2f0d鸶Lff356bf13550649ce501719a7e4e4f0cLff3534b3755b61b0f8ae04c912318fd2Lff3cb7fc2d30770d0980acab74b111d1丶Lff36bd345f223500be84a805f187a3c3Lff371f7f9569899eb06f09ca3ecc5dc1Lff3671d6ce544abfa080e1
14、483f24b40f椶Lff3de52cd2305c17b2cce4e75ecf2ab5Lff373f38faf97dc072ab7b383e780228收Lff38b4f3d9520f68253ab1131ad4c049弶Lff3904410e7f9a8ace2f33555c5cae16Lff3a72e283de768bac9416e4f8271927Lff3df1310c11c7231ec39a6629ea5672Lff35f9e5cd37da6f9e16ab2a089b0a5aLff3f6ad1b473b2baa45a1f9af7f083b6萶Lff3e73dc99ebbf81e1d24
15、a79d294b140Lff35a413a4a501d4875245edaa7082c2戶Lff3734db986f45a9262cd3281bcb2de7Lff3f1c86105b31242a0867500c99d3c8Lff389fcf3232e9b56bff691e322b0c3aLff3d2824a43df5b5c4fdb87d1ccc6a58Lff396213055280b69941e6abc6d25b81Lff3cf49713f2ec44494407d05e171b366Lff3d6c53caaccb2a68ccec5fe7069f46稶Lff39c1322a5f2ca85f0cd
16、d6eb6f69707Lff368dd11bd1eae0c699c28b535e3845匶Lff36942bd192450ebff1fc3544302432Lff36c724de5121e5c72793b45c49fa40Lff37137cd0249cc43e3bcf9cb7046dc4Lff3fee597ef012a826253d9aad64df7eLff41a65a9a09778a08cfae84c9d7506a戶Lff3c76f5838ffc7575f0aafdb2b3a488Lff39fc6f16a787edadd59b713f00a155Lff37d846c4be0070e4eb3f
17、3ba6420064锶Lff35a17185cdd38af90fd91af46569b7褶Lff3794453bbd79f74d72ddaa79829b51唶Lff35e5f7e911fa431766b58f69527932谶Lff3aed1765dcb16ecf2768eae1788886蜶Lff3d7e2e1abcbc0e1bf044bfc9af160cLff40efa70e018c2193c804392c1f955fLff3fe2d9aca6cf7c66a9cbc9ac1940deLff3fe7d571d65f90a77810a17b679e61Lff3a0643848f44875666
18、ede1821071756Lff3a488d9cf34d2469b550865faf9527Lff3f7a64a3ad49fa0739648ebb62249aLff36217ca06d39a346b89ea6588779ddLff3ee1a90ad759f6108d78f9366166ea琶Lff403b2764555186c4bfa08a979a41c0Lff37a89e629b26442994de976c5ab8c5Lff3db0c3d34990ca6cd7f33d7bcbfe7cLff3558f345bf76be589cac42503fd7afLff38b1a533c309ca3df9b
19、eef718c994bLff3d122f9d9ae02e8fbe824ffaa21894Lff3ccf07edff338fea93f11739abee05戶Lff3572965c1489f55cc8ec85aff2cc68崶Lff376b0fda788124cbbd054684a69072Lff378cbcdeef2504b51f1f9cadc17a73Lff3f142e94a7e99c69cba9244ff008deLff43451d0f7f4625a502be346c4f30b5Lff41e892316795c4302311d34af21a49Lff434acc4c08ad2934348f
20、cb0f528422倶Lff429767f47a78d654ce7f976ed2642eLff484303d6fc9391ea09b1f4ad5f5c8f萶Lff3e73dc99ebbf81e1d24a79d294b140Lff4519ea58a9a78aea8a2f9832d51bdaLff35a413a4a501d4875245edaa7082c2戶Lff3734db986f45a9262cd3281bcb2de7Lff433c216ee254609fba871324a2a5cdLff430b9bc537f8a739bf492badde3ff4Lff3f1c86105b31242a0867
21、500c99d3c8Lff41eb82cab65bd74bed4f1f55e7ad9cLff389fcf3232e9b56bff691e322b0c3aLff3d2824a43df5b5c4fdb87d1ccc6a58Lfba703d516fbfc8338d30522e671283c制L7ba710c2311fa7fbf964f239a0ea50b0阶L7ba7587ec45e5cafd3434e57abcd04bcL7ba76e5b26d985b05ca814798580ffc5挶L7ba7ab720547873e408868d3f1633ec1L7ba7b020f8af6cd8466c3f
22、151e1b34c4耶*L7ba7d1a2b589ce090e756a6a1a7d9992L7ba845af86d5c5d81729fede86642ee9L7ba8f984385abd86e4c9d62675a01ba1L7ba9091637f8c32fb62f88c4a985aa59L7ba90bdb7c8f59655266a64721f611e5L7ba9378a4a1e2bc330cb194a0e62dc79L7ba995417ac7119713d6751db1df13d8L7ba9dc4e2675c138943600c778b7010a匶L7baa2f4392d87a8e604583
23、e516b920aeL7baa752f759f833b114281fd5a8a282bL7baa83bf1cf1451b4c2410ab9865459dL7baaa619ed26d0318a37a500bc7cadd8L7baaabab8a9ba4ebe03378d5d1c78533L7baae76edc02ed772b17f8dd09c4e9f2礶7L7baaf7ea44134a69bea3a966320f5c2bL7bab0a3357599b4816790a9c9091ffd4L7bab1475055740459c673f7ac62c0916L7bab3bbb5bf4f61a1683223
24、8ad6539acQL7babc613099ec4920f3d9203660c04d0L7bac1306c3f122eb82d8aef806f39b12L7bac17ed06285b24ce4d2b212fed281cL7bac6ab093b1e941bc5a9c38694fa2d0缶L7bac8e0d31fed8b125ca422e8fa03e43L7bacf57553d7f91a8188b518333b3141L7badfed0d24d0a4f7c11153afbd7c6a5制L7baf1442c5099a10dc12b4b8ad3a9c9d丶L7baf1c8ab1b2ae9f386b47
25、d263584880L7baf3cf0429a8a97ce86051be80a762cL7baf9d55effbc007756cb7759eb899c4谶L7bafc2b9461c08a806e330b75ca80cccL7bafd17d60f6af131a2d3e940d1705e6L7bb03fe137339ecb0e547528346efdbeL7bb04ea55b77ba01a7088ef5ef802184L7bb08175976e1ade0b5a6c571bb26d6eL7bb09c98a71768ff3fca12c0f1dfc8daL7bb0d7d1750fc65ae4967368
26、6332fcf8L7bb0df22daca8426beb84deebf4f8b18L7bb0f4402cae19a438e4d21340bfca0dSL7bb1122998d67aa2788b0c3803d67028昶L7bb114d0ec3477355eb2a867a58e5e9cL7bb17b9e26b0147963f66ca77b07d4e8L7bb19b05a39996004786fcf19df420b5oL7bb1b777bc8347541ed9d9cabd0656a8L7bb1c862057cf4dca91827d799869937L7bb2038dab7e517c13403014
27、31003a8d簶AL7bb233318eb2b645c06879a4ad60723a嬶L7bb23e2caf6baff71592b41cd57757cc戶L7bb17877f7be40cb704f0e787f7b7854舶L7bab804af33a37441ed5760f15b73e7fL7baf8d9e777e8af8617598a0e7e0df15收L7badd81d8e1a3ad5a87c3d113d7620c3L7bad66425897d0ac863a4292acbfdcedL7ba9b534e9ef12c0a78cf1ab7eae0c59L7bb1f0001a693ed465988
28、e4cd46fc08bL7bb247cf02e8c6307b50a9c63e59025f鐶L7bb0716985a9768dd6746e061077db04圶L7ba77212782e0763c09c5d0a94f4c8a2逶L7baadce68cfab1b9c8c88f53df8037b7L7ba83e01aa00f11503c44be59e82d1aeL7babcd05b1beeb1dab83a19338c9233dL7baf077c7af717e5120a366696dc1d4cL7bab0d3e97abe6658cd04bf9308e6df4L7ba7cf2c4609a3c270f66
29、799e0126e05L7baff790bd25e1dea0680658a243728bL7bb049839065ddb02eac3f8e6f35836aL7bafed558e6e9e7892c792771cd6144c鰶L7bab6103c1fdc47ffc7e5647a1e27669L7baa22080d65cb370978468570e7179bL7bac2c50b3a717f5d5d78e1534dcb9de然L7bad56b32737065e88f8fff773bb74c6L7ba99e652d2f5f1ce8ba7cf8eff34a3eL7bb052fe6555895f6fad60
30、1b8508f47fL7bb0189103abed61397afb5447b2a8ae圶L7ba83c9f48db03b31178f02b5faa2b9bL7bac0f2fbf0a4226067de306ffcfa7c5耶L7baa9619bafb137119c35fd7ca16d9d3L7ba9efc91fead76446dbae005a7a4e71L7bacfcb4264a5ba3618ccc076cb72ffdL7bad5f690eb79ff14a42a1da7388697aL7ba954e1985727f1de512eb6dc4f291a圶L7bb0a457c4ffb5d6be2159
31、00a757aff7L7bb21da0e2f83a352d5d05bd670fb015鼶L7bad768fd72704960eb87509d12b3e09餶L7ba8da601e48f461c71ef583dfe59dd4踶L7bb0680f3fb229ed9b54b3329d7430afL7bb04bda481ac3c519b6648a789d3003礶L7bb0635dd86acc762255ff212fa3fd45餶L7badaa77c9fa54db1af5e5345bd4a922倶L7bae39a1f30941099384421a1fadddd3L7baf2bd0f88f91c27ce
32、02da1f69af3ab崶L7bab0196d19fa74fdd8f0dc412dd90c5L7ba98440b10c1bb3d9395ffa89801f06砶L7ba93c316cf2ff8c03cae87adbc74dab笶L7babfda280b80a79da6360be591fd72e霶L7baa508c2f7f98507910daae3969fa2bL7bb03cea657b668b79c26f5f7feb2fdc紶L7babc613cacdbf628d2762930a50359b8ec5a4a10ac63ee7f2c25a25b69e960L98ec64ea906e8da6174
33、b2d9190fe4984L98ece248e635784bbefe2994e3f60842昶L98ed05f1898ff8d19423952172b679b3L98ed95791059474b34c36e9b97f237abL98edbc06c90011efac7827151c0583e6L98ee52bff5626875bf34b92bc1384fcc_L98ee697f2fb0031f24baaf2afcff15b8L98eef3043e238011c0a33bf28cdf3246縶L98ef207c60000346f301b91b6625847eL98efd982470c880c20b
34、38bd59b4c63a4L98efffaf563b66a422d6e706b5e2cf14L98f01522ea1220d0475cf48f2f780cd36L98f13699cc1a863c6951f99631764827鸶L98f2121f3fd98fb2e0d989e2617df344L98f2139560188c06e9478cb1080685f8昶L98f27758f0ecb811dac2ffe608d7344cKL98f27aeb80615ba72212ff014bb8a24f倶L98f28927c3aa20c344250447fa5287480L98f2cee02dfe4691
35、666b0a726833f9a6嘶rL98f2f24f15b2d28921e35aeebb18117cL98f3105784f2105dce58290ce38f0287L98f31af5fcddd5ebfcbd537f3e4f1509L98f32068f3376c323cc20c2dbbf91967L98f386a66daa61d1ade974c3a22948d3L98f3b3df5bae4a66a284ce13255b44c9L98f3d9cd1a2ce7cdd056d0986c3384f0)L98f3df98ff0be78d7f34915e2338b039L98ed4a03ede06f98
36、702533fdadd5e772餶L98ed7a981f69d6c0032717abde94daeeL98f1ce6b2be20df987c75addb9f35e0f伶L98f39df6ffd184fc7c419439cad8ca0aL98ef5d19319389c005b9a482724bd617L98ef0e49ee2b048c787446008a6f2eb0L98f1948bac1ecace9ac88bc138a881cc礶L98f31222c68de2d9e42ce9ae68d38aea鬶L98f2a89b324c8680691842727ee5529cL98ef5381f0d993c
37、dd4096be6601d2e59L98ed151e758d4f4274b05e3b04cba92bL98f2019b7eadf97208e6025f3154175c踶L98ed4756d1e36f381616d49939f98c95L98f1862e6e8e67fd7a2dd0e124c6e7d3L98f30207b8c1248e221b93cb44458d90L98f2ed47f0600561b3331c4813a8e2c5L98f18f7814142f33aaec1f86f4082badL98ed50e50864e39e2ce50fc4823baf50L98ef61dc2affc9971
38、66ce0a25537d3bfL98f2c4b63c5f404ab423f8ff932cff12挶L98ee942a88cf28648f0e898f9cdffc9eL98f121f70ba9cc43d9818779efe39d83L98f1c2780545ade65fdb95128cfde674L98f31fac164c987daad368ae57296d09訶L98f0c5026e8bfee2ffa5d4eaec3d4164L98ef1b39c9c90285a2572d4409191335崶L98ecfed3ae522876480af5dbc5cb80daL98ede50d93d7d7b6f
39、bf6274a056d92d1L98ec7452f518cf0cabde1198e38fa6efL98ee4cabae156842aa24d95696ab8032L98eff6966fac3dcd0f493c05e66dcf3dL98eee8ef76247a2dfacdbbf28b611e7bL98efa81b59cddc171a03f740d4e7f0a3稶L98ecacb4cd4d43a3cb2a8b2a59b37491鴶L98f1d58e1e2ac9fc0bde11c66bbe5216L98ef1cb8e1b81dc68b75b01b858e407aL98ecb1af821557229c
40、612f506ae1f56bL98f27e90cfa80fa9a40fe3ab50edf494L98f3c3e4e7437614e5441566354f4302弶L98ece6b7d3f0c1fdaf7a3b0f75f44f9eL98f303db8f36122490c871f68c2a919c吶L98ef8f786bcca1ca0a71f56174fd216d唶L98ef8f786bcca1ca0a71f56174fd216dL98f09a147ff7f0165c53aa2c4663bbadL98f823e0aef2f1274799ff9f69c65678L98f73bee27d57ef90920c2d848572d68L98ef5381f0d993cdd4096be6601d2e59L98ed151e758d4f4274b05e3
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