Ann Varela: Who Traversed Statistics, Math, Physics and Astronomy?

An Interview with Ann Varela: Who Traversed Statistics, Math, Physics and Astronomy?

1) Ann, one mathematician, who was really a statistician, but delved into astronomy, while dabbling in physics and math was Pierre-Simon Laplace, who was born in France. What do we know about his early life?

Most accounts of Laplace’s early years state that he was from a lower to middle class family. However, those early records that would reveal the details were lost in a fire. Most of the research agrees that his father owned a farm and worked on the land. His mother apparently came from a farming family.

Laplace attended a Benedictine priory from age seven to sixteen, as his father wanted him to become a Roman Catholic priest. Affluent neighbors assisted financially with his early education. However, Laplace did not get his degree in divinity at Caen University. Instead, he did develop an extreme interest in mathematics, of which he was rather competent.

By the age of nineteen, Laplace suspended his college career and moved to Paris, France to begin work as a professor of mathematics at the École Militaire until 1776. During his tenure at École Militaire, Laplace became a prolific writer of mathematical papers. Maxima and minima of curves was the topic of his first paper. After that, he wrote about difference equations, integral calculus, mechanics, physical astronomy, and mathematical astronomy. By 1773, he produced thirteen papers to the Acadèmie des Sciences. Laplace was elected to the Acadèmie des Sciences and was charged with standardizing weights and measures, whereupon he worked on the metric system and encouraged a decimal base system. Specific distance units were chosen and represented fractions of the earth’s circumference. By 1799, the kilogram and meter were implemented as standard.

Laplace collaborated with Antoine Lavoisier, a French chemist whose life ended in 1794 by guillotine. Together, they demonstrated how decomposed compounds require an amount of heat equal to the heat required to create the formation of a compound from its elements. This clever experiment was fruitful, as it resulted in a published paper in 1880 called, “Memoir on Heat.” Kinetic theory of molecular motion was of great interest to these two scientists.

2) Difference equations and differential equations- why are these important and what was Laplace’s interest in these?

Around 1771, Laplace began working on differential equations and finite differences. The difference equation is a special type of recurrence relation. The difference equation includes the differences between consecutive values of a function of an integer variable. This means the equation defines a chain or sequence of values, once one or more initial terms are given. Each subsequent term of the chain is classified as a function of the preceding term. For example, the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, etc.) are generated by the difference equation Fn = Fn-1 + Fn-2. See Figure 1.

Figure 1

Differential equations relate functions with their derivatives. In other words, differential equations describe the ratio of change (slope) between the dependent variable (output value) and the independent variable (input value). Laplace studied the potential function and found how it satisfied a particular differential equation. The potential function is found in fluid dynamics, electromagnetism and other disciplines. Differential equations were critical to Laplace’s work with gravitational attraction.

3) Astronomical stability- what does this have to do with math?

Astronomical stability pertains to how each planet’s orbital motions effect each other. When considering these orbits in the short term, the planets appear relatively stable; however, their weak gravitational effects on one another can combine erratically. Scientists maintain our solar system is stable because it is unlikely that its planets will collide with each other or be ejected from the system in the very distant future.

Mathematicians and astronomers have searched for evidence for the stability of the planetary motions, and this mission led to many mathematical advances, along with verifications of stability of the solar system. In 1774, Laplace began researching cosmic mechanics and the stability of the solar system. By 1787, he established an explanation and investigation of the relationship between the lunar acceleration and certain changes in the irregularity of the Earth’s orbit. Laplace’s findings established the reasoning behind the stability of the whole solar system on the deduction that it consists of a collection of inflexible forms that move in a vacuum under a common gravitational pull.

4) The central limit theorem still stands pretty much today- Am I off on this—and can you explain the central limit theorem?

Given any set of independent and randomly generated variables, the mean (average) of the measurements (observations) scatter to form a symmetric mound shape (normal distribution). See Figure 2. The central limit theorem explains why the normal distribution occurs so often and why it is commonly used for an approximation of the mean of a collection of data. The central limit theorem shows how probabilistic and statistical approaches that work for normal distributions can be applicable to many problems involving other types of distributions. Laplace proved that the distribution of errors in large data samples from astral observations can be approximated by a normal distribution.

Figure 2

5) Apparently, Napoleon and Laplace knew each other but had a falling out. What happened?

During the French Revolution, Laplace was appointed to a general committee of weights and measures. He became acquainted with Lavoisier and Lagrange while developing a standard measurement system, which was later called the standard meter. I read several accounts concerning Laplace’s personality and most sources concurred that he was unabashedly outspoken and extremely confident in his theories and research. Laplace’s strong personality made it difficult for him to establish a good working relationship with governmental officials, so he was released from his post, but later reinstated by Napoleon as a senate member. After that, Laplace advanced to vice chancellor and then ultimately became the president of the senate.

Laplace now took on the task of calculating the motions of the planets, determining their statistics, resolving tidal problems, applications, astronomical tables, and historical information. He published five volumes of work, entitled Méchanique céleste, over a period of 26 years discussing the solar system. One complaint is that he did not always give credit to other scientists’ inspirations, collaborations, or contributions.

Jean-Baptiste Biot, one of Laplace’s contemporaries, helped him with revisions of the Méchanique céleste manuscript. Apparently, Biot wanted the manuscript to be supplemented with a generous amount of complex mathematical details. Laplace allowed Biot to make the necessary additions and other pertinent revisions. Laplace even went so far as to inform Napoleon and the French Academy of the manuscript to which Napoleon showed great interest, for he also was a mathematician. After the presentation, Laplace invited Biot to his study and showed him volumes of work dedicated to the same revisions that Biot had presented to Napoleon and the committee. This event seems like one rare instance that Laplace let one of his contemporaries get their deserved credit and bask in the lime light.

Laplace presented Napoleon with a copy of Méchanique céleste. It may be that Napoleon was not pleased with Laplace because Napoleon stated that Laplace failed to mention the Creator of the solar system in the manuscript. After all, Napoleon had signed an accord with the Pope restoring a degree of religious authority in France while Laplace carried on in his own style.

6) Now, Laplace may be most known for his “Principles of Probability”. Can you comment about each?

  1. Probability is the ratio of the “favored events” to the total possible events.
  2. The first principle assumes equal probabilities for all events. When this is not true, we must first determine the probabilities of each event. Then, the probability is the sum of the probabilities of all possible favored events.
  3. For independent events, the probability of the occurrence of all is the probability of each multiplied together.
  4. For events not independent, the probability of event B following event A (or event A causing B) is the probability of A multiplied by the probability that A and B both occur.
  5. The probability that A will occur, given that B has occurred, is the probability of A and B occurring divided by the probability of B.
  6. Three corollaries are given for the sixth principle, which amount to Bayesian probability. Where event Ai {A1, A2, …An} exhausts the list of possible causes for event B, Pr(B) = Pr(A1, A2, …, An). Then

For the first principle, the ratio of the “favored events” to the total possible events refers to the frequency of an event occurring (count) divided by the total number of events. This calculation represents the likelihood of an event happening. Say you have an old container of eight broken crayons containing the following colors: purple, orange, yellow, green, blue, red, red, and red. When calculating the probability of choosing a red crayon, you can determine that three out of eight crayons are red. Hence, 3/8 is the probability of choosing a red crayon from the box. You may also notice that choosing a red crayon is more likely to occur because there are more red crayons than any other crayon color in the box. The probability of choosing a particular color is not equally likely in this example.

The second principle brings up an important issue. It is necessary to know what the probability of each outcome is in order to determine if each outcome is equally likely to occur. Once it is determined what the probability of each outcome is, one may calculate the probability of various defined events. For example, if you roll a single standard die and define event A as the die lands on the number two, we may write the event as A = {Observe a 2.} Since the standard die has a two on only one of its six sides, the probability of the die landing on a two is one out of six, or 1/6. Each of the possible outcomes, one through six, is equally likely to occur. Given event B = {Observe a two or three.}, we would add the probability of landing on a two with the probability of landing on a 3. 1/6 + 1/6 = 2/6 which simplifies to 1/3.

The third and fifth principles are related to each other. They are both concerned with independent events. To clarify principle five, events are considered independent if the occurrence of one of the events does not alter the probability that the other event has occurred. This means the outcome of one event does not affect the likelihood of another event’s occurring. Consider the experiment of tossing a fair die, and let A = {Observe an odd number.} and B = {Observe a number less than or equal to four.} Are A and B independent events? We could start by listing the odd numbers: 1, 3, 5. This gives us a probability of 3/6 = 1/2 for event A. Next, we could list the numbers less than or equal to four: 1, 2, 3, 4. This gives us a probability of 4/6 = 2/3 for event B. The intersection (common measurements) of events A and B are the numbers 1 and 3, or 2/6 = 1/3. We can now use these probabilities in the conditional probability formula to see if the following statement is true: assuming B has occurred the conditional probability of A given B should be equal to the probability of A. The intersection of the events divided by the probability of B yields 1/3 divided by 2/3 = 1/2. Eureka! We have a match! Therefore, the events A and B are independent events.

Let us now discuss principle three since we have established that the events are independent. We can find the probability of the occurrence of all of the events by multiplying all of the probabilities together. What this is actually describing is the intersection of the events defined. If all events occur, that represents both events occurring at the same time. Using the events described above, we have the probability of A, P(A) = 1/2 and the probability of B, P(B) = 2/3. Therefore, (1/2)(2/3) = 2/6 = 1/3. We know this is correct because the numbers 1 and 3 (two out of six total) were, in fact, present in both event A and event B.

Principle four refers to dependent events. When two events depend on each other, the probability of the compound event is the product of the probability of the first event and the probability that, this event having occurred, the other will occur. To illustrate this principle we could imagine three boxes A, B, and C. Two of the boxes contain chocolate candies and one of the boxes contains almonds. The probability of drawing a chocolate candy from box C is 2/3, since two of the three boxes contain only chocolate candies. However, once a chocolate candy has been drawn from box C, the uncertainty as to which of the boxes contains almonds applies only to boxes A and B, and the probability of drawing a chocolate from box B is 1/2. The product of 2/3 and 1/2, is 2/6 or 1/3, is then the probability of drawing two chocolate candies at the same time from boxes B and C. In this scenario, it is possible to see the influence of past events on the probability of future events.

Principle six describes the probability of any one of various events occurring is the ratio where the numerator is the probability of any of these events resulting from a cause and the denominator being the sum of all the individual event’s probabilities. This principle reveals the reason why we attribute routine events to a specific cause. The use of probability to make inferences is a type of statistical approach known as Bayesian statistical methods, named after the 18th century English philosopher Thomas Bayes.

To begin, Ai {A1, A2, …An} represents a list of mutually exclusive (non-intersecting) and continuous events, such that the sum of the probabilities of each event is equal to one. The next corollary, Pr(B) = Pr(A1, A2, …, An) shows that there is the same likelihood as there is probability that the event will take place, supposing the event to be continuous. Finally, we arrive at Bayes’s rule that states that the probability of the occurrence of any one of these causes is equal to the probability of the event resulting from this cause divided by the sum of the similar probabilities relative to all the causes. It is imperative to note that if the causes are not equally likely, one must substitute the product of this probability by the possibility of the cause itself for the probability of the event resulting from each cause.

Let us refer to Figure 3 for an example of Bayes’s rule.

Figure 3

To find the probability that a green ball chosen at random came from Box 2, we start by writing Bayes’s rule:

P(B2|G) = [P(B2)P(G|B2)] / [P(B2)P(G|B2) + P(B2c)P(G|B2c)]

Next, we must realize that P(B2c) is the same as P(B1) and P(G|B2c) is the same as P(G|B1) by the definition of complements.

Now we are ready to substitute the known values into the formula to find the solution.

P(B2|G) = [(0.5)(0.3)] / [(0.5)(0.3) + (0.5)(0.5)]

P(B2|G) = 0.15 / (0.15 + 0.25) = 0.15 / 0.4 = 0.375 = 37.5%

Therefore, the probability that the green ball came from Box 2 is 37.5%. It is now quite easy to determine the probability that the green ball came from Box 1 by again using the rule of complements. Since there are only two boxes, we can subtract the probability of the green ball coming from Box 2 from 1: 1 – 0.375 = 0.675 = 67.5%. The probability that the green ball came from Box 1 is 67.5%. Now if you just refer back to the boxes, you can see that our conclusion makes sense since Box 1 had more green balls in it to begin with.Pr ( A i B ) = Pr ( A i ) Pr ( B A i ) ∑ j Pr ( A j ) Pr ( B A j ) . {displaystyle Pr(A_{i}mid B)=Pr(A_{i}){frac {Pr(Bmid A_{i})}{sum _{j}Pr(A_{j})Pr(Bmid A_{j})}}.}

7) What have I neglected to ask about this fine mathematician, statistician, and scholar?

Laplace spent a great deal of his time studying heat, magnetism, and electricity, but he also had innovations in pure and applied mathematics including his method for approximating integrals and a solution of the linear partial differential equation of the second order. He created Laplace’s equation, and founded the Laplace transform, which appears in many areas of mathematical physics. The Laplacian differential operator, which is widely used in mathematics, is also named after him.

One amusing quotation of Laplace’s is, “What we know is not much. What we do not know is immense.”

Apparently, Laplace’s brain was removed after his death and kept by his personal physician, François Magendie. Sometime later, it went on display in a traveling anatomical museum in Britain.