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Maurice Tweedie

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Maurice Charles Kenneth Tweedie
External image
image icon Maurice Tweedie
Born(1919-09-30)30 September 1919
Reading, UK
Died14 March 1996(1996-03-14) (aged 76)
Liverpool, UK
EducationUniversity of Reading
Known forInverse Gaussian distribution
Tweedie distributions
Scientific career
InstitutionsVirginia Tech
University of Manchester
University of Liverpool
Academic advisorsPaul White
Boyd Harshbarger

Maurice Charles Kenneth Tweedie (30 September 1919 – 14 March 1996) was a British medical physicist and statistician from the University of Liverpool. He was known for research into the exponential family probability distributions.

Education and career

Tweedie read physics at the University of Reading and attained a BSc (general) and BSc (special) in physics in 1939 followed by a MSc in physics 1941. He found a career in radiation physics, but his primary interest was in mathematical statistics where his accomplishments far surpassed his academic postings.

Contributions

Tweedie distributions

Tweedie's contributions included pioneering work with the Inverse Gaussian distribution. Arguably his major achievement rests with the definition of a family of exponential dispersion models characterized by closure under additive and reproductive convolution as well as under transformations of scale that are now known as the Tweedie exponential dispersion models. As a consequence of these properties the Tweedie exponential dispersion models are characterized by a power law relationship between the variance and the mean which leads them to become the foci of convergence for a central limit like effect that acts on a wide variety of random data. The range of application of the Tweedie distributions is wide and includes:

Tweedie's formula

Tweedie is credited for a formula first published in Robbins (1956), which offers "a simple empirical Bayes approach to correcting selection bias". Let μ {\displaystyle \mu } be a latent variable we don't observe, but we know it has a certain prior distribution p ( μ ) {\displaystyle p(\mu )} . Let x = μ + ϵ {\displaystyle x=\mu +\epsilon } be an observable, where ϵ ∼ N ( 0 , Σ ) {\displaystyle \epsilon \sim N(0,\Sigma )} is a Gaussian noise variable (so p ( x | μ ) = N ( x | μ , Σ ) {\displaystyle p(x|\mu )=N(x|\mu ,\Sigma )} ) . Let ρ ( x ) = ∫ p ( x | μ ) p ( μ ) d μ {\displaystyle \rho (x)=\int p(x|\mu )p(\mu ){\text{d}}\mu } be the probability density of x {\displaystyle x} , then the posterior mean and variance of μ {\displaystyle \mu } given the observed x {\displaystyle x} are: E [ μ | x ] = x + Σ ∇ ρ ( x ) ρ ( x ) ; V a r [ μ | x ] = E [ μ μ T | x ] − E [ μ | x ] E [ μ | x ] T = Σ ( ∇ 2 ρ ( x ) ρ ( x ) − ∇ ρ ( x ) ∇ ρ ( x ) T ρ ( x ) 2 ) Σ + Σ {\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 } The posterior higher order moments of μ {\displaystyle \mu } are also obtainable as algebraic expressions of ∇ ρ , ρ , Σ {\displaystyle \nabla \rho ,\rho ,\Sigma } .

Proof for first part

Using ∇ N ( x | μ , Σ ) = N ( x | μ , Σ ) ∇ log ⁡ N ( x | μ , Σ ) = − Σ − 1 ( x − μ ) N ( x | μ , Σ ) {\displaystyle \nabla N(x|\mu ,\Sigma )=N(x|\mu ,\Sigma )\nabla \log N(x|\mu ,\Sigma )=-\Sigma ^{-1}(x-\mu )N(x|\mu ,\Sigma )} , we get Σ ∇ ρ ( x ) ρ ( x ) = Σ ∫ ∇ N ( x | μ , Σ ) p ( μ ) d μ ∫ N ( x | μ , Σ ) p ( μ ) d μ = − ∫ ( x − μ ) N ( x | μ , Σ ) p ( μ ) d μ ∫ N ( x | μ , Σ ) p ( μ ) d μ = − ∫ ( x − μ ) p ( μ | x ) d μ = − x + E [ μ | x ] , {\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,} where we have used Bayes' theorem to write p ( μ | x ) = p ( x | μ ) p ( μ ) p ( x ) = N ( x | μ , Σ ) p ( μ ) ∫ N ( x | μ , Σ ) p ( μ ) d μ . {\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 }}.}


Tweedie's formula is used in empirical Bayes method and diffusion models.

References

  1. ^ Tweedie, M.C.K. (1984). "An index which distinguishes between some important exponential families". In Ghosh, J.K.; Roy, J (eds.). Statistics: Applications and New Directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. Calcutta: Indian Statistical Institute. pp. 579–604. MR 0786162.
  2. Smith, C.A.B. (1997). "Obituary: Maurice Charles Kenneth Tweedie, 1919–96". Journal of the Royal Statistical Society, Series A. 160 (1): 151–154. doi:10.1111/1467-985X.00052.
  3. Tweedie, MCK (1957). "Statistical properties of inverse Gaussian distributions. I." Ann Math Stat. 28 (2): 362–377. doi:10.1214/aoms/1177706964.
  4. Tweedie, MCK (1957). "Statistical properties of inverse Gaussian distributions. II". Ann Math Stat. 28: 695–705.
  5. Jørgensen, B (1987). "Exponential dispersion models". Journal of the Royal Statistical Society, Series B. 49 (2): 127–162.
  6. Jørgensen, B; Martinez, JR; Tsao, M (1994). "Asymptotic behaviour of the variance function". Scand J Stat. 21: 223–243.
  7. Kendal, WS (2004). "Taylor's ecological power law as a consequence of scale invariant exponential dispersion models". Ecol Complex. 1 (3): 193–209. doi:10.1016/j.ecocom.2004.05.001.
  8. ^ Kendal, WS; Jørgensen, BR (2011). "Tweedie convergence: a mathematical basis for Taylor's power law, 1/f noise and multifractality". Phys. Rev. E. 84 (6): 066120. Bibcode:2011PhRvE..84f6120K. doi:10.1103/physreve.84.066120. PMID 22304168.
  9. Kendal, WS; Jørgensen, B (2011). "Taylor's power law and fluctuation scaling explained by a central-limit-like convergence". Phys. Rev. E. 83 (6): 066115. Bibcode:2011PhRvE..83f6115K. doi:10.1103/physreve.83.066115. PMID 21797449.
  10. Kendal WS. 2002. A frequency distribution for the number of hematogenous organ metastases. Invasion Metastasis 1: 126–135.
  11. Kendal, WS (2003). "An exponential dispersion model for the distribution of human single nucleotide polymorphisms". Mol Biol Evol. 20 (4): 579–590. doi:10.1093/molbev/msg057. PMID 12679541.
  12. Kendal, WS (2004). "A scale invariant clustering of genes on human chromosome 7". BMC Evol Biol. 4: 3. doi:10.1186/1471-2148-4-3. PMC 373443. PMID 15040817.
  13. Kendal, WS (2001). "A stochastic model for the self-similar heterogeneity of regional organ blood flow". Proc Natl Acad Sci U S A. 98 (3): 837–841. Bibcode:2001PNAS...98..837K. doi:10.1073/pnas.98.3.837. PMC 14670. PMID 11158557.
  14. Kendal, W. (2015). "Self-organized criticality attributed to a central limit-like convergence effect". Physica A. 421: 141–150. Bibcode:2015PhyA..421..141K. doi:10.1016/j.physa.2014.11.035.
  15. Robbins, Herbert E. (1992), Kotz, Samuel; Johnson, Norman L. (eds.), "An Empirical Bayes Approach to Statistics", Breakthroughs in Statistics, Springer Series in Statistics, New York, NY: Springer New York, pp. 388–394, doi:10.1007/978-1-4612-0919-5_26, ISBN 978-0-387-94037-3, retrieved 21 September 2023
  16. Efron, B (2011). "Tweedie's Formula and Selection Bias". Journal of the American Statistical Association. 106 (496): 1602–1614. doi:10.1198/jasa.2011.tm11181. JSTOR 23239562. PMC 3325056. PMID 22505788.
  17. Song, Yang; Sohl-Dickstein, Jascha; Kingma, Diederik P.; Kumar, Abhishek; Ermon, Stefano; Poole, Ben (2020). "Score-Based Generative Modeling through Stochastic Differential Equations". arXiv:2011.13456 .
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