1607:
1218:
39:
962:
1602:{\displaystyle \Sigma \,{\frac {\nabla \rho (x)}{\rho (x)}}={\frac {\Sigma \int \nabla N(x|\mu ,\Sigma )\,p(\mu ){\text{d}}\,\mu }{\int N(x|\mu ,\Sigma )\,p(\mu )\,{\text{d}}\mu }}=-{\frac {\int (x-\mu )N(x|\mu ,\Sigma )\,p(\mu )\,{\text{d}}\mu }{\int N(x|\mu ,\Sigma )\,p(\mu )\,{\text{d}}\mu }}=-\int (x-\mu )p(\mu |x)\,{\text{d}}\mu =-x+E,}
636:
1803:
1213:
1612:
957:{\displaystyle E=x+\Sigma {\frac {\nabla \rho (x)}{\rho (x)}};\quad Var=E-EE^{T}=\Sigma \left({\frac {\nabla ^{2}\rho (x)}{\rho (x)}}-{\frac {\nabla \rho (x)\nabla \rho (x)^{T}}{\rho (x)^{2}}}\right)\Sigma +\Sigma }
1030:
571:
2414:
Song, Yang; Sohl-Dickstein, Jascha; Kingma, Diederik P.; Kumar, Abhishek; Ermon, Stefano; Poole, Ben (2020). "Score-Based
Generative Modeling through Stochastic Differential Equations".
496:
1017:
430:
389:
2466:
1950:
1868:
2471:
357:
982:
611:
328:
631:
591:
2446:
310:
Tweedie is credited for a formula first published in
Robbins (1956), which offers "a simple empirical Bayes approach to correcting selection bias". Let
1840:. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. Calcutta: Indian Statistical Institute. pp. 579–604.
2441:
1798:{\displaystyle p(\mu |x)={\frac {p(x|\mu )p(\mu )}{p(x)}}={\frac {N(x|\mu ,\Sigma )\,p(\mu )}{\int N(x|\mu ,\Sigma )\,p(\mu )\,{\text{d}}\mu }}.}
1208:{\displaystyle \nabla \rho (x)=\nabla N(x|\mu ,\Sigma )=N(x|\mu ,\Sigma )\nabla \log N(x|\mu ,\Sigma )=-\Sigma ^{-1}(x-\mu )N(x|\mu ,\Sigma )}
2335:
247:
like effect that acts on a wide variety of random data. The range of application of the
Tweedie distributions is wide and includes:
1836:
Tweedie, M.C.K. (1984). "An index which distinguishes between some important exponential families". In Ghosh, J.K.; Roy, J (eds.).
501:
2476:
2073:
Kendal, WS; Jørgensen, B (2011). "Taylor's power law and fluctuation scaling explained by a central-limit-like convergence".
2023:
2115:
Kendal WS. 2002. A frequency distribution for the number of hematogenous organ metastases. Invasion
Metastasis 1: 126–135.
212:
105:
216:
2451:
1988:
Kendal, WS (2004). "Taylor's ecological power law as a consequence of scale invariant exponential dispersion models".
1859:
299:
435:
987:
394:
167:
130:
195:
160:
134:
1809:
244:
235:. As a consequence of these properties the Tweedie exponential dispersion models are characterized by a
220:
179:
109:
95:
42:
362:
2461:
2456:
2288:
2230:
2082:
2039:
232:
293:
240:
2276:
2415:
2378:
164:
2127:"An exponential dispersion model for the distribution of human single nucleotide polymorphisms"
2396:
2331:
2258:
2199:
2148:
2098:
2055:
191:
159:(30 September 1919 – 14 March 1996) was a British medical physicist and statistician from the
1969:
Jørgensen, B; Martinez, JR; Tsao, M (1994). "Asymptotic behaviour of the variance function".
333:
2386:
2370:
2323:
2296:
2279:(2015). "Self-organized criticality attributed to a central limit-like convergence effect".
2248:
2238:
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2179:
2138:
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1997:
1909:
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284:
228:
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330:
be a latent variable we don't observe, but we know it has a certain prior distribution
2194:
2167:
239:
relationship between the variance and the mean which leads them to become the foci of
2435:
2253:
2218:
1929:
Tweedie, MCK (1957). "Statistical properties of inverse
Gaussian distributions. II".
270:
126:
84:
2322:, Springer Series in Statistics, New York, NY: Springer New York, pp. 388–394,
2219:"A stochastic model for the self-similar heterogeneity of regional organ blood flow"
2327:
2001:
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66:
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260:
2143:
2126:
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236:
2400:
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2152:
2102:
2059:
215:. Arguably his major achievement rests with the definition of a family of
2243:
2184:
2382:
2314:
Robbins, Herbert E. (1992), Kotz, Samuel; Johnson, Norman L. (eds.),
2420:
2024:"Tweedie convergence: a mathematical basis for Taylor's power law,
198:
where his accomplishments far surpassed his academic postings.
1899:"Statistical properties of inverse Gaussian distributions. I."
187:
183:
2168:"A scale invariant clustering of genes on human chromosome 7"
186:(general) and BSc (special) in physics in 1939 followed by a
566:{\displaystyle \rho (x)=\int p(x|\mu )p(\mu ){\text{d}}\mu }
211:
Tweedie's contributions included pioneering work with the
1948:
Jørgensen, B (1987). "Exponential dispersion models".
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579:
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365:
336:
316:
1864:"Obituary: Maurice Charles Kenneth Tweedie, 1919–96"
140:
122:
101:
91:
73:
52:
31:
20:
1951:Journal of the Royal Statistical Society, Series B
1869:Journal of the Royal Statistical Society, Series A
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1601:
1207:
1011:
976:
956:
625:
605:
585:
565:
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424:
383:
351:
322:
984:are also obtainable as algebraic expressions of
2363:Journal of the American Statistical Association
2017:
2015:
2013:
2011:
8:
2316:"An Empirical Bayes Approach to Statistics"
1838:Statistics: Applications and New Directions
1609:where we have used Bayes' theorem to write
491:{\displaystyle p(x|\mu )=N(x|\mu ,\Sigma )}
1012:{\displaystyle \nabla \rho ,\rho ,\Sigma }
593:, then the posterior mean and variance of
425:{\displaystyle \epsilon \sim N(0,\Sigma )}
17:
2467:Academics of the University of Manchester
2419:
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1220:
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532:
503:
471:
448:
437:
396:
364:
335:
315:
2472:Academics of the University of Liverpool
1825:
2359:"Tweedie's Formula and Selection Bias"
964:The posterior higher order moments of
190:in physics 1941. He found a career in
28:
1831:
1829:
233:Tweedie exponential dispersion models
163:. He was known for research into the
7:
2447:Alumni of the University of Reading
2022:Kendal, WS; Jørgensen, BR (2011).
1761:
1717:
1480:
1427:
1349:
1296:
1273:
1267:
1229:
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1139:
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1034:
1006:
991:
951:
945:
894:
879:
835:
823:
675:
669:
482:
416:
290:regional blood flow heterogeneity,
194:, but his primary interest was in
14:
432:is a Gaussian noise variable (so
384:{\displaystyle x=\mu +\epsilon }
223:under additive and reproductive
37:
710:
157:Maurice Charles Kenneth Tweedie
22:Maurice Charles Kenneth Tweedie
2442:People from Reading, Berkshire
1777:
1771:
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1352:
1339:
1332:
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1299:
1286:
1279:
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743:
734:
727:
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650:
643:
573:be the probability density of
552:
546:
540:
533:
526:
514:
508:
485:
472:
465:
456:
449:
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419:
407:
346:
340:
1:
1808:Tweedie's formula is used in
217:exponential dispersion models
213:Inverse Gaussian distribution
106:Inverse Gaussian distribution
2328:10.1007/978-1-4612-0919-5_26
2002:10.1016/j.ecocom.2004.05.001
178:Tweedie read physics at the
2320:Breakthroughs in Statistics
2301:10.1016/j.physa.2014.11.035
2493:
2095:10.1103/physreve.83.066115
2052:10.1103/physreve.84.066120
2028:noise and multifractality"
300:self-organized criticality
231:that are now known as the
2375:10.1198/jasa.2011.tm11181
168:probability distributions
150:
115:
36:
27:
2223:Proc Natl Acad Sci U S A
391:be an observable, where
229:transformations of scale
131:University of Manchester
1915:10.1214/aoms/1177706964
1883:10.1111/1467-985X.00052
352:{\displaystyle p(\mu )}
305:
196:mathematical statistics
161:University of Liverpool
135:University of Liverpool
1810:empirical Bayes method
1799:
1603:
1209:
1013:
978:
958:
627:
607:
587:
567:
492:
426:
385:
353:
324:
283:genomic structure and
2477:British statisticians
2244:10.1073/pnas.98.3.837
2185:10.1186/1471-2148-4-3
2144:10.1093/molbev/msg057
1897:Tweedie, MCK (1957).
1800:
1604:
1210:
1014:
979:
959:
628:
608:
588:
568:
493:
427:
386:
354:
325:
207:Tweedie distributions
180:University of Reading
110:Tweedie distributions
96:University of Reading
1613:
1219:
1031:
1023:Proof for first part
988:
977:{\displaystyle \mu }
968:
637:
617:
606:{\displaystyle \mu }
597:
577:
502:
436:
395:
363:
334:
323:{\displaystyle \mu }
314:
276:hematogenous cancer
271:random matrix theory
257:fluctuation scaling,
174:Education and career
2293:2015PhyA..421..141K
2235:2001PNAS...98..837K
2217:Kendal, WS (2001).
2166:Kendal, WS (2004).
2125:Kendal, WS (2003).
2087:2011PhRvE..83f6115K
2044:2011PhRvE..84f6120K
613:given the observed
2452:Medical physicists
2369:(496): 1602–1614.
1795:
1599:
1205:
1009:
974:
954:
623:
603:
583:
563:
488:
422:
381:
349:
320:
165:exponential family
2357:Efron, B (2011).
2337:978-0-387-94037-3
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1784:
1689:
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1503:
1450:
1378:
1372:
1318:
1259:
938:
871:
705:
626:{\displaystyle x}
586:{\displaystyle x}
558:
306:Tweedie's formula
227:as well as under
219:characterized by
192:radiation physics
154:
153:
141:Academic advisors
117:Scientific career
63:30 September 1919
48:
47:
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1814:diffusion models
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329:
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146:Boyd Harshbarger
80:
62:
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41:
40:
29:
18:
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2110:
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2067:
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2020:
2009:
1987:
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749:
691:
674:
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574:
500:
499:
434:
433:
393:
392:
361:
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332:
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312:
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294:multifractality
209:
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182:and attained a
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145:
133:
129:
108:
87:
82:
78:
69:
64:
58:
56:
43:Maurice Tweedie
38:
23:
12:
11:
5:
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2433:
2428:
2427:
2406:
2349:
2336:
2306:
2268:
2229:(3): 837–841.
2209:
2158:
2137:(4): 579–590.
2117:
2108:
2065:
2007:
1996:(3): 193–209.
1980:
1961:
1940:
1921:
1908:(2): 362–377.
1889:
1876:(1): 151–154.
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85:Liverpool, UK
77:14 March 1996
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2032:Phys. Rev. E
2031:
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1990:Ecol Complex
1989:
1983:
1974:
1971:Scand J Stat
1970:
1964:
1955:
1949:
1943:
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1924:
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1901:
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252:Taylor's law
210:
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123:Institutions
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2457:1919 births
2287:: 141–150.
241:convergence
225:convolution
67:Reading, UK
2436:Categories
2421:2011.13456
2277:Kendal, W.
1977:: 223–243.
1937:: 695–705.
1820:References
278:metastasis
144:Paul White
59:1919-09-30
2281:Physica A
1787:μ
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285:evolution
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92:Education
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