Transcript lecture set 1

Introduction to
Predictive Learning
LECTURE SET 1
INTRODUCTION and OVERVIEW
Electrical and Computer Engineering
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OUTLINE of Set 1
1.1 Overview: what is this course about:
- subject matter
- philosophical connections
- prerequisites and HW1
- expected outcomes of this course
1.2 Historical Perspective
1.3 Motivation for Empirical Knowledge
1.4 General Experimental Procedure for
Estimating Models from Data
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1.1.1 Subject Matter
Uncertainty and Learning
• Decision making under uncertainty
• Biological learning (adaptation)
(examples and discussion)
• Induction, Statistics and Philosophy
Ex. 1: Many old men are bald
Ex. 2: Sun rises on the East every day
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(cont’d) Many old men are bald
•
Psychological Induction:
- inductive statement based on experience
- has certain predictive aspect
- no scientific explanation
•
Statistical View:
- the lack of hair = random variable
- estimate its distribution (depending on age) from
past observations (training sample)
•
Philosophy of Science Approach:
- find scientific theory to explain the lack of hair
- explanation itself is not sufficient
- true theory needs to make non-trivial predictions
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Conceptual Issues
• Any theory (or model) has two aspects:
1. explanation of past data (observations)
2. prediction of future (unobserved) data
• Achieving both goals perfectly not possible
• Important issues to be addressed:
- quality of explanation and prediction
- is good prediction possible at all ?
- if two models explain past data equally well, which one
is better?
- how to distinguish between true scientific and pseudoscientific theories?
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Beliefs vs True Theories
Men have lower life expectancy than women
• Because they choose to do so
• Because they make more money (on
average) and experience higher stress
managing it
• Because they engage in risky activities
• Because …..
Demarcation problem in philosophy
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1.1.2 Philosophical Connections
• Oxford English dictionary:
Induction is the process of inferring a
general law or principle from the
observations of particular instances.
• Clearly related to Predictive Learning.
• All science and (most of) human knowledge
involves induction
• How to form ‘good’ inductive theories?
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Challenge of Predictive Learning
• Explain the past and predict the future
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Background: philosophy
William of Ockham: entities should not
be multiplied beyond necessity
Epicurus of Samos: If more than one
theory is consistent with the
observations, keep all theories
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Background: philosophy
Thomas Bayes:
How to update/ revise beliefs in
light of new evidence
Karl Popper: Every true
(inductive) theory prohibits
certain events or occurences,
i.e. it should be falsifiable
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Background: philosophy
George W. Bush:
I am The Decider
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Background: philosophy
Bill Clinton:
I told the Truth
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1.1.3 Prerequisites and Hwk1
• Math: working knowledge of basic
Probability + Linear Algebra
• Statistical software (of your choice):
- MATLAB, also R-project, Mathematica etc.
Note: you will be using s/w implementations
of learning algorithm (not writing programs)
• Writing: no special requirements
• Philosophy: no special requirements
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Homework 1
• Purpose: testing background on probability
and computer skills
• Goal: estimate pdf of a random variable X
• Real Data: X=daily price changes of SP500
i.e. X (t ) 
Z (t )  Z (t  1)
 100% where Z(t) = closing price
Z (t  1)
• Typical + Useful Statistics
- Histogram (empirical pdf)
- mean, standard deviation
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(cont’d) Homework 1
Histogram = estimated pdf (from data)
• Example: histograms of 5 and 30 bins to model N(0,1)
also mean and standard deviation (estimated from data)
500
100
400
80
300
60
200
40
100
20
0
-3
-2
-1
0
1
2
3
4
0
-3
-2
-1
0
1
2
3
4
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(cont’d) Homework 1
NOTE: histogram ~ empirical pdf, i.e. y-axis scale is in %
(frequency).
Example: histogram of SP500 daily price changes in 1981:
1981
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
2.00%
1.80%
1.60%
1.40%
1.20%
1.00%
0.80%
0.60%
0.40%
0.20%
0.00%
-0.20%
-0.40%
-0.60%
-0.80%
-1.00%
-1.20%
-1.40%
-1.60%
-1.80%
-2.00%
0.00%
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1.1.4 Expected Outcomes (of this course)
Scientific/technical:
• Learning = generalization, concepts and issues
• Math theory: Statistical Learning Theory aka VC-theory
• Implications for Philosophy and Applications
Philosophical:
• Nature of human knowledge and intelligence
• Demarcation principle
• Human learning
Practical:
• Financial engineering
• Security
• Genomics
• Predicting successful marriage, climate modeling etc., etc.
What is this course NOT about
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Other Related Fields
•
•
•
•
•
The field of Pattern Recognition is concerned with the
automatic discovery of regularities in data.
Data Mining is the process of automatically discovering
useful information in large data repositories.
This book (on Statistical Learning) is about learning
from data.
The field of Machine Learning is concerned with the
question of how to construct computer programs that
automatically improve with experience.
Artificial Neural Networks perform useful computations
through the process of learning.

(1) unnecessary fragmentation  confusion
(2) all fields estimate useful models from data, i.e. extract
knowledge from data (the same as in classical statistics)
Real Issues: what is ‘useful’? What is ‘knowledge’?
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OUTLINE of Set 1
1.1 Overview: what is this course about
1.2 Historical Perspective
Handling uncertainty and risk:
- probabilistic
vs - risk minimization
1.3 Motivation for Empirical Knowledge
1.4 General Experimental Procedure for
Estimating Models from Data
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1.2.1Handling Uncertainty and Risk
• Ancient times
• Probability for quantifying uncertainty
- degree-of-belief
- frequentist (Cardano-1525, Pascale, Fermat)
• Newton and causal determinism
• Probability theory and statistics (20th century)
• Modern science (A. Einstein)
Goal of science: estimating a true model or
system identification
Scientific knowledge ~
deterministic+causal(explicable, assured)
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Gods, Prophets and Shamans
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Handling Uncertainty and Risk(2)
• Making decisions under uncertainty
= risk management
• Probabilistic approach:
- estimate probabilities (of future events)
- assign costs and minimize expected risk
• Risk minimization approach:
- apply decisions to known past events
- select one minimizing expected risk
• Common in all living things: learning,
generalization
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Cultural and Psychological Aspects
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•
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All men by nature desire knowledge
Man has an intense desire for assured
knowledge
Assured Knowledge ~ belief in
- religion
- reason (causal determinism)
- science / pseudoscience
- data-analytic models (~ Big Data)
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Human Generalization
•
Example 1: continue given sequence
6, 10, 14, 18,….
•
Example 2:
Sceitnitss osbevred: it is nt inptrant how
lteters are msspled isnide the word. It is
ipmoratnt that the fisrt and lsat letetrs
do not chngae, tehn the txet is itneprted
corrcetly
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OUTLINE of Set 1
1.1 Overview: what is this course about
1.2 Historical Perspective
1.3 Motivation for Empirical Knowledge
- scientific knowledge
- growth of empirical knowledge
- the nature of human knowledge
1.4 General Experimental Procedure for
Estimating Models from Data
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1.3.1 Scientific Knowledge
•
Combines ideas/models and facts/data
• First-principle knowledge:
hypothesis  experiment  theory
~ deterministic, simple causal models
• Modern data-driven discovery:
Computer program + DATA  knowledge
~ statistical, complex systems
• Two different philosophies
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Scientific Knowledge
•
•
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Classical Knowledge (last 3-4 centuries):
- objective
- recurrent events (repeatable by others)
- quantifiable (described by math models)
Knowledge ~ causal, deterministic, logical
Humans cannot reason well about
- noisy/random data
- multivariate high-dimensional data
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Knowledge Discovery in Digital Age
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•
•
Most information in the form of digital data
Can we get assured knowledge from data?
Big Data ~ technological nirvana
data + connectivity  more knowledge
Wired Magazine, 16/07: We can stop looking for (scientific)
models. We can analyze the data without hypotheses
about what it might show. We can throw the numbers into
the biggest computing clusters the world has ever seen and
let statistical algorithms find patterns where science cannot.
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•
Reality: many questionable studies
Duke biologists discovered an unusual link between the
popular singer and a new species of fern, i.e.
- bisexual reproductive stage of the ferns;
- the team found the sequence GAGA when analyzing the
fern’s DNA base pairs
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Scientific Data Mining: Kepler’s Laws
• How planets move among the stars?
- Ptolemaic system (geocentric)
- Copernican system (heliocentric)
• Tycho Brahe (16 century)
- measure positions of the planets in the sky
- use experimental data to support one’s
view (hypothesis)
• Johannes Kepler:
- used volumes of Tycho’s data to discover
three remarkably simple laws
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Kepler’s Laws
(1) The orbit is an ellipse with sun at its focus
(2) The line joining a planet to the sun sweeps equal
areas during the same time
(3) The ratio P2/D3 is constant, where P is the orbit
period and D is the orbit size.
NO computers, statistics, machine learning or Big Data
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Kepler’s Laws vs. ‘Lady Gaga’ knowledge
• Both search for assured knowledge
• Kepler’s Laws
- well-defined hypothesis stated a priori
- prediction capability
- human intelligence
- no marketing
• Lady Gaga knowledge ~ belief
- no hypothesis stated a priori
- no prediction capability
- computer intelligence (software program)
- very marketable (to wide audience)
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1.3.2Growth of empirical knowledge
•
•
•
•
Huge growth of the amount of data in
20th century (computers and sensors)
Complex systems (engineering, life
sciences and social)
Classical first-principle science is
inadequate for empirical knowledge
Need for Data-Analytic Modeling:
How to estimate good predictive
models from noisy data
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1.3.3 Nature of human knowledge
•
Three types of knowledge
- scientific (first-principle, deterministic)
- empirical
- metaphysical (beliefs)
•
Boundaries are poorly understood
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More on Empirical Knowledge
Demarcation:
•
Empirical Knowledge vs Beliefs
Examples
•
Empirical vs First Principle Knowledge
Examples, Discussion
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Representation of Empirical Knowledge
•
Facts, observations ~ data points (x, y)
Knowledge: predictive relationship ~ function f(x)
Examples:
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Example: Polynomial Curve Fitting
• The Problem: estimate the best polynomial model
- true (target) function ~ second-order (in blue)
- But the best model is linear
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Modeling Assumptions
•
•
•
•
•
Future data is similar to the past
Explanation is different from prediction
Prediction implies many future (test) inputs)
Multiplicity of good solutions
Estimation of inductive models that can
predict (generalize) is difficult (~ill-posed)
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1.4 General Experimental Procedure
1. Understand Application Goals/ Requirements
2. Hypothesis Formulation (Problem Formalization)
3. Data Generation/Collection/ Experiment Design
4. Data Cleaning and Preprocessing
5. Model Estimation (learning)
6. Model Interpretion/ Assessment and Drawing
Conclusions
Note:
- each step is complex and usually involves several
iterations
- final model depends on all previous steps
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Honest Disclosure of Results
• Recall Tycho Brahe (16th century)
• Modern drug studies
Review of studies submitted to FDA
• Of 74 studies reviewed, 38 were
judged to be positive by the FDA.
All but one were published.
• Most of the studies found to
have negative or questionable
results were not published,
researchers found.
Source: The New England Journal of
Medicine, WSJ Jan 17, 2008)
Publication bias:
widespread in modern research
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