Laplace, Pierre Simon de (1749-1827)

1. Laplace, Pierre Simon de (1749-1827)

2. Agenda • Biography • History of The Central limit Theorem (CLT) • Derivation of the CLT • First Version of the CLT • CLT for the Binomial Distribution • Laplace and Bayesian ideas • 4th and 5th Principles • 6th Principle • 7th Principle • References

3. Biography • Pierre Simon Laplace was born in Normandy on March 23, 1749, and died at Paris on March 5, 1827 • French scientist, mathematician and astronomer; established mathematically the stability of the Solar system and its origin - without a divine intervention • Professor of mathematics in the École militaire of Paris at the age of 19.

4. Biography (Cont’d) • Under Napoleon, Laplace was a member, then chancellor, of the Senate, and received the Legion of Honour in 1805. Appointed Minister of the Interior, in 1799. Removed from office by Napoleon after six weeks only!! • Named a Marquis in 1817 after the restoration of the Bourbons • Main publications: • Mécanique céleste (1771, 1787) • Théorie analytique des probabilités 1812 – first edition dedicated to Napoleon

5. History Of Central Limit Theorem From De Moivre to Laplace • De Moivre investigated the limits of the binomial distribution as the number of trials increases without bound and found that the function exp(-x2) came up in connection with this problem. • The formulation of the normal distribution, (1/√2)exp(-x2/2), came with Thomas Simpson.

6. History Of CLT (Cont’d) • This idea was was expanded upon by the German mathematician Carl Friedrich Gauss who then developed the principle of least squares. • Independently, the French mathematicians Pierre Simon de Laplace and Legendre also developed these ideas.It was with Laplace's work that the first inklings of the Central Limit Theorem appeared. • In France, the normal distribution is known as Laplacian Distribution; while in Germany it is known as Gaussian.

7. Derivation of the CLT • Initial Work: Laplace was calculating the probability distribution of the sum of meteor inclination angles. He assumed that all the angles were r.v’s following a triangular distribution between 0 and 90 degrees • Problems: • The deviation between the arithmetic mean which was inflicted with observational errors, and the theoretical value • The exact calculation was not achievable due to the considerable amount of celestrial bodies • Solution: Find an approximation !!

8. First Version of the CLT • Laplace introduced the m.g.f, which is known as Laplace Transform of f • He then introduced the Characteristic function: • If is a sample of i.i.d. obs., • Assume that we have a discrete r.v. x, that takes on the values –m, -m+1,…,0,…,m-1,m with prob. p-m ,…,pm. • Let Sn be the sum of the n possible errors.

10. CLT for the Binomial Distribution

11. Final Note on The Proof of The CLT • It was Lyapunov's analysis that led to the modern characteristic function approach to the Central Limit Theorem. • Where

12. Laplace & Bayesian ideasOverview “Philosophical essay on probabilities” by Laplace • General principles on probability • Expectation • Analytical methods • Applications

13. 4th & 5th principles: conditional & marginal give the joint “Here the question posed by some philosophers concerning the influence of the past on the probability of the future, presents itself”

14. CAUSE EVENT 6th principle: “discrete” Bayes theorem “Fundamental principle of that branch of the analysis of chance that consists of reasoning a posteriori from events to causes”

15. CAUSE EVENT FUTURE EVENT 7th principle: probability of future based on observations “(…) the correct way of relating past events to the probability of causes and of future events (…)”

16. References • Laplace, Pierre Simon De. “Philosophical essay on probabilities. • Weatherburn, C.E. “A First Course in Mathematical Statistics”.