Welcome to STAT 302,

Intro to Probability Instructor: Alexandre Bouchard

Winter/Spring 2018

www.stat.ubc.ca/~bouchard/courses/stat302-sp2017-18/

Plan for today:

• Logistics.

• Why you should care about probability.

• Some basic definitions and properties.

To get this information & more

• Main website : www.stat.ubc.ca/~bouchard/courses/stat302-sp2017-18/

• Piazza (contact link on webpage)

• Click on ‘Files’ to get lecture slides, assignments, etc.

Administrative details• Prerequisites: Math 200 or 226 (which may be

taken concurrently), or equivalent

• Exclusions: Stat 241/251, Math 302

• Textbook: OPTIONAL A first course in probability Sheldon Ross

• 9th or 8th edition

• Optional exercises on website (‘Syllabus’ tab)

• Software: probabilistic programming on the cloud (more details soon)

Assessment

• Assignments (4), 15%

• We encourage you to discuss your work with other students...

• However, you must write up your own solutions independently.

• No extension possible

Assessment

• Midterm, 25%

• Date: March 1 (check this week that this date is not a problem)

• Same place and time as usual lecture

• No make-up exam

Assessment• Webwork 5% - clickers 5%

• Get your clicker this week (will start collecting clicker scores on January 11)

• I will send an email when clicker registration website is setup (waiting for UBC IT)

• Final, 50%

• After lectures are over

• Date announced centrally by University

• You must pass the final to pass the course

To get help• Difficulty level deceptive: things get tricky, especially ~after

midterm. Stay on top and ask for help if confused.

• Piazza:

• Sign up by following the ‘Contacts’ tab of the course webpage

• We will only respond to question posted on Piazza (unless it is a personal matter)

• Come to office hours!

• Place, time: see ‘Contacts’ tab of course website (soon)

• Some additional ones for some of the assignments and exam

What STAT 302 is about:

• Probability spaces: arguably the best quantitative tool to model reality

• Properties of probability spaces

• Lots of examples

Why this topic is important

Machine Learning

Probability Optimization

Automated theorem proving Adaptive websites Affective computing Bioinformatics Brain–machine interfaces Cheminformatics Classifying DNA sequences Computational anatomy Computer Networks Computer vision, including object recognition Detecting credit-card fraud General game playing Information retrieval

Machine perception Medical diagnosis Economics Insurance Natural language processing Natural language understanding Optimization and metaheuristic Online advertising Recommender systems Robot locomotion Search engines Sentiment analysis (or opinion mining) Sequence mining

Internet fraud detection Linguistics Marketing Machine learning control Software engineering Speech and handwriting recognition Financial market analysis Structural health monitoring Syntactic pattern recognition Time series forecasting User behavior analytics Translation

Why this topic is important• Fundamental tool in statistics, computer science,

physics, econometrics, ... and increasingly, biology, linguistics, sociology, ...

• Creating models

• Inverting them (Bayesian statistics/conditioning)

• Computational power of randomness

• Also a branch of pure math in its own right

• Replacing logic as the philosophical foundations of science and cognition? (‘Dawning of the age of stochasticity’, D. Mumford)

Probability in action: Diverse examples

Engineering, technology, logistics

The Search for Malaysia Airlines Flight 370

Goal: finding the location of the crash

How to reconcile several sources of partial info: - Last known position- Fuel range- Last satellite ping

Question: how to prioritize search

http://tinyurl.com/lhzrufa

Ex. 1

Bayesian Search

1966: Palomares B-52 crash 1968: USS Scorpion disappearance

Ex. 1

How to trade-off exploration and exploitation?

Conditioning• Say you search in the square

of highest success probability

• You find nothing

• What should you do next?

• Note: even if the submarine is there, you might have missed it!

• Probability as a calculus of belief and uncertainty

Bayes theorem (Thomas Bayes),

1763

http://tinyurl.com/pcznhml

Ex. 1

Rational behavior and uncertainty

General question: how to act when- we are facing uncertainty- errors have different costs

Examples: - fraud detection- medical diagnosis- spam classifiers

Key tool: expected value

Ex. 3

Sciences

Ecology: Estimating animal population sizes

Example: finding the number of Sockeye salmon in the Pacific Ocean (!)

Very important problem for conservation, setting fishing quotas, etc.

Ex. 4

Insight: the capture-recapture trick

Population Capture and tag Recapture and count

Examples:

Ex. 4

Assessing significance• Histogram of # of births

organized by month:

• Question: is the # of births uniform across months?

• Note: even if the answer is yes, we would expect small differences across months.

• How small?

Ex. 5

Assessing significance

Ex. 5

• Tricky problem

• Mis-use of statistical tools has lead to the reproducibility crisis

Astrophysics: Estimating the age and fate of the Universe

• Goals: finding the Universe’s

• age

• density (=> fate)

• Data: Cosmic Microwave Background (CMB): remnants of Big Bang

• Detailed map from the Planck satellite

• Age, Physical constants => known distribution on CMP

• Invert using Bayes’ rule

Ex. 7

Phylogenetics: Reconstruction of ancient species

• Goals:

• better understand ancient species

• revive them?

• Fossil DNA degrades after few 1000s years

• Are dinosaurs’ genomes completely lost?

Ex. 7

Phylogenetic tree• Idea: use the genomes

from the descendants of dinosaurs (modern birds)

• We know how DNA change over time (probabilistically)

• Marginalization of unknowns (as in family tree example)

• Additional challenge: structure of tree is unknown

Ex. 7

Outline of the course

• Discrete probability models

• Conditioning and Bayes

• Expectation

• Continuous probability models

• Asymptotics

Random variables

• Fundamental object of study

• Examples of a random variable X

• The height of a UBC student picked at random

• Gambling example: ‘Rademacher coin’

• Tails: +1

• Heads; -1

Surprising challenge• Sums of random variables

• Omnipresent

• Taking the sum of number is easy, so taking the sum of random variables should also be easy, right?

• Not quite... consider for example the problem of computing the probability that the sum of 100 Rademacher coins is greater that 50.

• Would have been hard in the pre-computer era

• Generalized versions of this problem still hard with computer

Asymptotics to the rescue• Another surprise: sums of

random variables can be approximated by something simple when large number of terms involved

• No matter what each X is!!! (almost)

• Also explains why we will spent disproportionate amount of time on some specific types of random variables (normal/gaussian, Poisson, ...)

300 coins

Outline of the course

• Discrete probability models

• Conditioning and Bayes

• Expectation

• Continuous probability models

• Asymptotics

Discrete:

Probability:

Model:

when we can enumerate the scenarios

extension of the notion of proportion

a simplification of reality amenable to mathematical investigation

We will make these concepts more precise today:

Definitions and basic properties

Example: a bag of distinct objects of the same size

Probability when outcomes are equally

likely:

Ex. 9

• Proportion of red shapes?

• Probability of drawing a red shape?

• Outcome: an individual object in the bagex.: s =

# of outcomes of interest# of outcomes

Notation

Probability when outcomes are equally

likely:

Def 1

• Proportion of red shapes?

• Probability of drawing a red shape?

• Outcome: an individual object in the bag

# of outcomes of interest# of outcomes

This is a set of outcomesNickname: event

Typical notation: red, E, blue, F, ..E = {a, c, e}

a

bc

d

e

Sample spaceTypical notation: S

Notation: P, Q, ..P is a function:

- input: an event- output: a number in [0, 1]

P : 2S → [0, 1]Example: P(E) = 3/5 P(E) = |E| / |S|

Basic properties• We know P(E)

• Example: P(square) = 2/5

• What is P(Ec) ?

• Example: P(not square)

• Ec means:

• the complement of E

• the outcomes not in E

• = S \ E (minus, for sets)

P(Ec) = 1 - P(E)

Prop. 1

Basic properties• We know P(E), P(F)

• Example: P(blue) = 2/5 P(star) = 1/5

• What is P(E ⋃ F) ?

• Example: P(blue or star)

P(E ⋃ F) = P(E) + P(F)

Prop. 2

if E and F are disjoint, i.e. E ⋂ F = ∅

• Try now on:

• Example: P(red) = 3/5 P(square) = 2/5

Odds

• Equivalent, but less used terminology

• The odds of red shapes is #(red) / #(not red) = 3/2

• odds(E) = P(E) / P(Ec) = P(E) / (1 - P(E) )

Def. 2

Summary so far• Outcome: a scenario, s =

• Sample space S: all the possible scenarios

• Event: a set of outcomes, e.g. E = {s ∈ S : s is red}

• Probability: in the discrete, equally weighted case, P(E) = |E| / |S|

• Properties:

• P(E ⋃ F) = P(E) + P(F) when E & F are disjoint (non-overlapping)

• P(Ec) = 1 - P(E)

Building models

Two coins

• Flip 2 coins. What is the probability that the 2 coins both show heads?

A. 1/2

B. 1/3

C. 1/4

D. none of above

Ex. 10

Probability that 2 coins show heads

• 1/2? Either they do, or they don’t.

• 1/3? Either both heads, both tails, or heads-tails.

• 1/4? Imagine one coin is painted red, one is painted blue. There are then 4 possibilities:

- red: heads, blue: heads

- red: heads, blue: tails

- red: tails, blue: heads

- red: tails, blue: tails

- These correspond to 3 different models

- None is ‘true’

- But the third is more useful (accurate at doing predictions)

Models are approximations

• Reality: a complex dynamical system

Stroboscopic image of a coin flip by Andrew Davidhazy

• Model: a ‘bag’ with 4 ‘objects’ in it (H,H)

(T,H)

(T,T)

(H,T)

Which model to use?

Probability theory alone does cannot answer this question.

Reality Model (simplification)

Prediction / answers

trial-and-error, scientific method (test predictions)

probability theory

But...

• Probability theory still useful:

• Given a model, it makes certain predictions

• We can then test those predictions

• Example: law of large numbers

• Relates probability and frequency in repeated experiments

Reality check

• Dataset of 10,000 actual tosses of two coins available on the web (!)

0 2000 6000 100000.0

0.2

0.4

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0.8

1.0

0 2000 6000 100000.0

0.2

0.4

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0.8

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0 2000 6000 10000

0.0

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1.0

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1000

0.30.247

http://tinyurl.com/ljtc63g coin toss

rela

tive

freq

uenc

y

All models are wrong• Given the large number of throws, 0.247

may seem a bit far from 0.25 (we will formalize this idea later)

• is there something fishy?

• after reading the fine prints of the data, realized all throws started with same face in hand of thrower

• see P. Diaconis’ paper, http://tinyurl.com/yked5fk

• Essentially, all models are wrong, but some are useful. -- G. Box