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Probability & Statistics Primer: Key Concepts & Distributions

Dive into the basics of probability and statistics with this lecture covering sample spaces, random variables, and various distributions. Understand joint, conditional, and marginal distributions, as well as the concept of statistical independence.

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Probability & Statistics Primer: Key Concepts & Distributions

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  1. Introduction to probability and statistics Alireza FotuhiSiahpirani & Brittany Baur sroy@biostat.wisc.edu Computational Network BiologyBiostatistics & Medical Informatics 826https://compnetbiocourse.discovery.wisc.edu Sep 13th 2016 Some of the material covered in this lecture is adapted from BMI 576

  2. Goals for today • Probability primer • Introduction to linear regression

  3. A few key concepts • Sample spaces • Random variables • Discrete and continuous continuous distributions • Joint, conditional and marginal distributions • Statistical independence

  4. Definition of probability • Intuitively, we use “probability” to refer to our degree of confidence in an event of an uncertain nature. • Always a number in the interval [0,1] 0 means “never occurs” 1 means “always occurs”

  5. Sample space • Sample space: a set of possible outcomes for some experiment • Examples • Flight to Chicago: {on time, late} • Lottery: {ticket 1 wins, ticket 2 wins,…,ticket n wins} • Weather tomorrow: {rain, not rain} or {sun, rain, snow} or {sun, clouds, rain, snow, sleet} • Roll of a die: {1,2,3,4,5,6} • Coin toss: {Heads, Tail}

  6. Random variables • Random variable: A variable that represents the outcome of a uncertain experiment • A random variable can be • Discrete/Categorical: Outcomes take a fixed set of values • Roll of die, flight to Chicago, weather tomorrow • Continuous: Outcomes take continuous values • Height, weight

  7. Notation • Uppercase letters and words denote random variables • X, Y • Lowercase letters and words denote values • x, y • Probability that X takes value x • We will also use the shorthand form • For Boolean random variables, we will use the shorthand P(x) for P(X=x) P(fever) forP(Fever = true) P(¬fever) forP(Fever = false)

  8. 0.3 0.2 0.1 sun rain sleet snow clouds Discrete probability distributions • A probability distribution is a mathematical function that specifies the probability of each possible outcome of a random variable • We denote this as P(X) for random variable X • It specifies the probability of each possible value of X, x • Requirements:

  9. Joint probability distributions • Joint probability distribution: the function given by P(X = x, Y = y) • Read as “X equals xandY equals y” • Example probability that it’s sunny and my flight is on time

  10. Marginal probability distributions • The marginal distribution of X is defined by “the distribution of X ignoring other variables” • This definition generalizes to more than two variables, e.g.

  11. Marginal distribution example joint distribution marginal distribution for X

  12. Conditional distributions • The conditional distribution of Xgiven Y is defined as: • Or in short • The distribution of X given that we know the value of Y • Intuitively, how much does knowing Y tell us about X?

  13. Conditional distribution example conditional distribution for X givenY=on-time joint distribution

  14. Independence • Two random variables, X and Y, are independent if • Another way to think about this is knowing X does not tell us anything about Y

  15. Independence example #1 joint distribution marginal distributions Are X and Y independent here? NO.

  16. Independence example #2 joint distribution marginal distributions Are X and Y independent here? YES.

  17. Conditional independence • Two random variables X and Y are conditionally independent given Z if  • “once you know the value of Z, knowing Y doesn’t tell you anything about X” • Alternatively

  18. Conditional independence example Are Fever and Headache independent? NO.

  19. Conditional independence example Are Fever and Headache conditionally independent given Flu: YES.

  20. Chain rule of probability • For two variables • For three variables etc. • To see that this is true, note that

  21. Example discrete distributions • Binomial distribution • Multinomial distribution

  22. The binomial distribution • Two outcomes per trial of an experiment • Distribution over the number of successes in a fixed number n of independent trials (with same probability of success p in each) • e.g. the probability of x heads in ncoin flips p=0.5 p=0.1 P(X=x) P(X=x) x x

  23. The multinomial distribution • A generalization of the binomial distribution to more than two outcomes • Provides a distribution of the number of times any of the outcomes happen. • For example consider rolling of a die n= 100 times. Each time we can have one of k = 6 outcomes, • is the variable representing the number of times the die landed on the ith face, • is the probability of the die landing on theith face

  24. Continuous random variables • When our outcome is a continuous number we need a continuous random variable • Examples: Weight, Height • We specify a density function for random variable X as • Probabilities are specified over an interval • Probability of taking on a single value is 0.

  25. Continuous random variables contd • To define a probability distribution for a continuous variable, we need to integrate f(x)

  26. Example continuous distributions • Uniform distribution • Gaussian distribution • Exponential distribution

  27. Uniform distribution • A variable X is said to have a uniform distribution, between , where, , if a b x Adapted from Wikipedia

  28. Gaussian Distribution • The univariate Gaussian distribution is defined by two parameters, Mean: and Standard deviation: From Wikipedia: Normal distribution, https://en.wikipedia.org/wiki/Normal_distribution

  29. Exponential Distribution • Exponential distribution for a random variable X is defined as From Wikipedia: Exponential distribution, https://en.wikipedia.org/wiki/Exponential_distribution

  30. Goals for today • Probability primer • Introduction to linear regression

  31. Linear regression y x Slope Intercept Linear regression assumes that output (y) is a linear function of the input (x) Given: Data= Estimate:

  32. Residual Sum of Squares (RSS) Residual y x To find the , we need to minimize the Residual Sum of Squares

  33. Minimizing RSS y Residual x

  34. Linear regression with p inputs • Y: output • Inputs: intercept Parameters/coefficients Given: Data= Estimate:

  35. Ordinary least squares for estimating • Pick the that minimizes the residual sum of squares RSS

  36. How to minimize RSS? • Easier to think in matrix form This is the square of y-Xb in matrix world

  37. Simple matrix calculus 1. 2. 3. The Matrix Cookbookhttps://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf

  38. Estimating by minimizing RSS Works well when (XTX)-1 is invertible. But often this is not true. Need to regularize or add a prior

  39. References • Chapter 2, Probabilistic Graphical Models. Principles and Techniques. Friedman & Koller. • Slides adapted from Prof. Mark Craven’s Introduction to Bioinformatics lectures. • All of Statistics, Larry Wasserman. • Chapter 3, The Elements of Statistical Learning, Hastie, Tibshirani, Friedman.

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