Math 2a: Lecture 1. Agenda. Course organization. History of the subject. Why is the subject relevant today? First examples

Transcription

1 Math 2a: Lecture 1 Agenda Course organization History of the subject Why is the subject relevant today? First examples

2 More about the Course This is an introductory course in statistics and probability theory. Different from a mathematical theory course: This course is well described by a problem solving and practical approach, with a strong emphasis on conceptual understanding.

3 What is Statistics? Statistics: The art of making numerical conjectures about puzzling questions. Three examples: 1. Who is going to win the next election? Will the Governor be recalled? If yes, which candidate will win? 2. What are the effects of a new medical treatment? 3. What is the average income in the U.S.? To answer these questions, we need a mathematical tool: probability theory. In short, probability is the language of uncertainty.

4 Organization of the Course First 5 weeks: introduction to probability theory, particularly those aspects most relevant to statistics. Last 5 weeks: Introduction to fundamental ideas and techniques of statistical inference.

5 This Lecture History Why is it relevant today? First examples and calculations

6 History Interesting Informative Development of probability theory is connected with a specific time period with a new conception of reality Game of chance are very ancient Surprisingly, the mathematical language is very recent

7 The 17th Century The Baroque century People realize they are not the center of the universe More room for uncertainty Example: Pascal s bet To believe or not to believe?

8 Who? Why? First names: the French connection Pascal Fermat De Moivre Developed mathematics Games of chance Risk management Actuarial sciences

9 (a) Fermat (b) Pascal (c) De Moivre

10 Beginning of the 20th Century The true logic of this world is in the calculus of probabilities. Maxwell Explosion of this discipline Central figures Kolmogorov De Finetti Von Mises Eruption of uncertainty in hard sciences Probability becomes a natural language for describing real world phenomena Popper: Scientific method Explains why the field developed

11 A. N. Kolmogorov (USSR, )

12 W. Heisenberg (Germany, )

13 Interpretation of Probabilities Very difficult to try defining probability 17th Century: Probability of an event = # favorable outcomes/ # possible outcomes Interpretation of probabilities is a mess The concept of randomness is delicate Coin flipping? Computer generated random numbers? Sex of an offspring?

14 Which type of mathematics? We use probability to do the mathematics of random phenomena. What is random? A coin toss? Randomness is often a good way to describe a phenomena that is too complex to describe exactly. What we think as random/deterministic has often more to do with the nature of our knowledge than with the essence of the phenomena The models that have been used for biological phenomena are mainly stochastic models (differently from physics that got to stochastic modeling only very late)

15 Pragmatic Approach Model reality uncertainty We want to explore the consequences of those mathematical tools

16 What is the role of models? What we see is the solution to a computational problem, our brains compute the most likely causes from the photon absorptions within our eyes. H. Helmholtz All models are wrong. Some are useful. (Cox) We use models to learn about nature and to mimic nature abilities (vision is one example)

17 Genetics I shall never believe that God plays dice with the world Albert Einstein Randomness: how we can describe reality Mendel theory of genetics Laws of inheritance of physical traits Mechanism of heredity is based on gene pairs Gene pairs control biological characteristics in several ways. One way is dominance. Illustrates the power of simple probability models Today s applications: finding causing-disease genes

18 J. G. Mendel (Austria, )

19 Laws of Genetic Inheritance Mendel, second half of the 19th century. A a A a Father Mother Genotype Physical trait Here Allele with 2 possible values A, a. A: Dominant a: Recessive e.g. Eye color A: Black a: Blue

20 Example d: dominant allele r: recessive allele d/d d/r r/d dominant trait r/r recessive trait

21 Example Eye Color Emmanuel blue/black Chiara blue/black Chance of having an offspring with blue eyes p = 1/2 1/2 = 1/4.

22 Statistics Again, late development, late 20 s, early 30 s. Central figure: R. A. Fisher.

23 R. A. Fisher (UK, )

24 A first example This is adapted from Fisher s famous lady tasting tea. A friend of yours claims he/she has extra-sensorial powers. For example, this person claims he/she can guess the color of a card with the card facing down. How would you check whether this may be true?

25 A second example: Salk vaccine field trial (1954) First polio epidemic hit the U.S. in During the next 40 years, polio claimed many hundreds of thousands of lives (especially children). In the 50 s, Jonas Salk developed a promising vaccine. Is this vaccine effective?

26 Salk vaccine field trial, II Wish to know the effect of a treatment (like a vaccine) on a response (like getting the disease) to find out, compare a treatment group with a control group The control group must be comparable to the treatment group. Why? Difference likely to be due to the treatment Randomly assign subjects to the two groups to make sure that both groups are alike Control group receives placebo (if possible) Compare responses between the two groups Draw conclusions

27 Salk vaccine field trial, III # of subjects Disease infection rate Treatment 200,000 28% (56,000) Control 200,000 71% (142,000) Devil s advocate claims the Salk vaccine had no effect. That is, the difference between the polio rates for the treatment and control groups is just due to chance. How likely is that? On this hypothesis 198,000 children are fated to contract polio assignment to treatment and control has nothing to do with it (random) polio has the same chance to turn up in both goups

28 The observed difference is due to the variability in the grouping. What we need to understand, is the chance of coming up with an assignment such that 142,000 subjects out of 198,000 or more end up in the same group. Abstraction An urn contains 400,000 balls of which 198,000 are black (polio cases) and 202,000 are white (healthy cases) You randomly select 200,000 balls. What is the chance that your selection contains fewer than 56,000 or more than 198,000 black balls? Chance is below double-digit precision of a computer.