Logistics

1
Logistics

• Class size? Who is new? Who is listening?
• Everyone on Athena mailing list
  concepts-and-theories? If not write to me.
• Everyone on stellar yet? If not, write to Melissa
  Yeh (mjyeh_at_mit.edu).
• Interest in having a printed course pack, even if
  a few readings get changed?

2
Plan for tonight

• Why be Bayesian?
• Informal introduction to learning as
  probabilistic inference
• Formal introduction to probabilistic inference
• A little bit of mathematical psychology
• An introduction to Bayes nets

3
Plan for tonight

• Why be Bayesian?
• Informal introduction to learning as
  probabilistic inference
• Formal introduction to probabilistic inference
• A little bit of mathematical psychology
• An introduction to Bayes nets

4
Virtues of Bayesian framework

• Generates principled models with strong
  explanatory and descriptive power.

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Virtues of Bayesian framework

• Generates principled models with strong
  explanatory and descriptive power.
• Unifies models of cognition across tasks and
  domains.
• Categorization
• Concept learning
• Word learning
• Inductive reasoning
• Causal inference
• Conceptual change

• Biology
• Physics
• Psychology
• Language
• . . .

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Virtues of Bayesian framework

• Generates principled models with strong
  explanatory and descriptive power.
• Unifies models of cognition across tasks and
  domains.
• Explains which processing models work, and why.
• Associative learning
• Connectionist networks
• Similarity to examples
• Toolkit of simple heuristics

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Virtues of Bayesian framework

• Generates principled models with strong
  explanatory and descriptive power.
• Unifies models of cognition across tasks and
  domains.
• Explains which processing models work, and why.
• Allows us to move beyond classic dichotomies.
• Symbols (rules, logic, hierarchies, relations)
  versus Statistics
• Domain-general versus Domain-specific
• Nature versus Nurture

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Virtues of Bayesian framework

• Generates principled models with strong
  explanatory and descriptive power.
• Unifies models of cognition across tasks and
  domains.
• Explains which processing models work, and why.
• Allows us to move beyond classic dichotomies.
• A framework for understanding theory-based
  cognition
• How are theories used to learn about the
  structure of the world?
• How are theories acquired?

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Rational statistical inference(Bayes, Laplace)

• Fundamental question
• How do we update beliefs in light of data?
• Fundamental (and only) assumption
• Represent degrees of belief as probabilities.
• The answer
• Mathematics of probability theory.

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What does probability mean?

• Frequentists Probability as expected frequency
• P(A) 1 A will always occur.
• P(A) 0 A will never occur.
• 0.5 lt P(A) lt 1 A will occur more often than not.
  
• Subjectivists Probability as degree of belief
• P(A) 1 believe A is true.
• P(A) 0 believe A is false.
• 0.5 lt P(A) lt 1 believe A is more likely to be
  true than false.

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What does probability mean?

• Frequentists Probability as expected frequency
• P(heads) 0.5 If we flip 100 times, we
  expect to see about 50 heads.
• Subjectivists Probability as degree of belief
• P(heads) 0.5 On the next flip, its an
  even bet whether it comes up heads or tails.
• P(rain tomorrow) 0.8
• P(Saddam Hussein is dead) 0.1
• . . .

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Is subjective probability cognitively viable?

• Evolutionary psychologists (Gigerenzer, Cosmides,
  Tooby, Pinker) argue it is not.

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• To understand the design of statistical
  inference mechanisms, then, one needs to examine
  what form inductive-reasoning problems -- and the
  information relevant to solving them -- regularly
  took in ancestral environments. Asking for
  the probability of a single event seems
  unexceptionable in the modern world, where we are
  bombarded with numerically expressed statistical
  information, such as weather forecasts telling us
  there is a 60 chance of rain today. In
  ancestral environments, the only external
  database available from which to reason
  inductively was one's own observations and,
  possibly, those communicated by the handful of
  other individuals with whom one lived.
• The probability of a single event cannot be
  observed by an individual, however. Single
  events either happen or they dont -- either it
  will rain today or it will not. Natural
  selection cannot build cognitive mechanisms
  designed to reason about, or receive as input,
  information in a format that did not regularly
  exist.

(Brase, Cosmides and Tooby, 1998)
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Is subjective probability cognitively viable?

• Evolutionary psychologists (Gigerenzer, Cosmides,
  Tooby, Pinker) argue it is not.
• Reasons to think it is
• Intuitions are old and potentially universal
  (Aristotle, the Talmud).
• Represented in semantics (and syntax?) of natural
  language.
• Extremely useful .

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Why be subjectivist?

• Often need to make inferences about singular
  events
• e.g., How likely is it to rain tomorrow?
• Cox Axioms
• A formal model of common sense
• Dutch Book Survival of the Fittest
• If your beliefs do not accord with the laws of
  probability, then you can always be out-gambled
  by someone whose beliefs do so accord.
• Provides a theory of learning
• A common currency for combining prior knowledge
  and the lessons of experience.

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Cox Axioms (via Jaynes)

• Degrees of belief are represented by real
  numbers.
• Qualitative correspondence with common sense,
  e.g.
• Consistency
• If a conclusion can be reasoned in more than one
  way, then every possible way must lead to the
  same result.
• All available evidence should be taken into
  account when inferring a degree of belief.
• Equivalent states of knowledge should be
  represented with equivalent degrees of belief.
• Accepting these axioms implies Bel can be
  represented as a probability measure.

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Plan for tonight

• Why be Bayesian?
• Informal introduction to learning as
  probabilistic inference
• Formal introduction to probabilistic inference
• A little bit of mathematical psychology
• An introduction to Bayes nets

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Example flipping coins

• Flip a coin 10 times and see 5 heads, 5 tails.
• P(heads) on next flip? 50
• Why? 50 5 / (55) 5/10.
• Future will be like the past.
• Suppose we had seen 4 heads and 6 tails.
• P(heads) on next flip? Closer to 50 than to 40.
• Why? Prior knowledge.

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Example flipping coins

• Represent prior knowledge as fictional
  observations F.
• E.g., F 1000 heads, 1000 tails strong
  expectation that any new coin will be fair.
• After seeing 4 heads, 6 tails, P(heads) on next
  flip 1004 / (10041006) 49.95
• E.g., F 3 heads, 3 tails weak expectation
  that any new coin will be fair.
• After seeing 4 heads, 6 tails, P(heads) on next
  flip 7 / (79) 43.75. Prior knowledge too
  weak.

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Example flipping thumbtacks

• Represent prior knowledge as fictional
  observations F.
• E.g., F 4 heads, 3 tails weak expectation
  that tacks are slightly biased towards heads.
• After seeing 2 heads, 0 tails, P(heads) on next
  flip 6 / (63) 67.
• Some prior knowledge is always necessary to avoid
  jumping to hasty conclusions.
• Suppose F After seeing 2 heads, 0 tails,
  P(heads) on next flip 2 / (20) 100.

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Origin of prior knowledge

• Tempting answer prior experience
• Suppose you have previously seen 2000 coin flips
  1000 heads, 1000 tails.
• By assuming all coins (and flips) are alike,
  these observations of other coins are as good as
  actual observations of the present coin.

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Problems with simple empiricism

• Havent really seen 2000 coin flips, or any
  thumbtack flips.
• Prior knowledge is stronger than raw experience
  justifies.
• Havent seen exactly equal number of heads and
  tails.
• Prior knowledge is smoother than raw experience
  justifies.
• Should be a difference between observing 2000
  flips of a single coin versus observing 10 flips
  each for 200 coins, or 1 flip each for 2000
  coins.
• Prior knowledge is more structured than raw
  experience.

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A simple theory

• Coins are manufactured by a standardized
  procedure that is effective but not perfect.
• Justifies generalizing from previous coins to the
  present coin.
• Justifies smoother and stronger prior than raw
  experience alone.
• Explains why seeing 10 flips each for 200 coins
  is more valuable than seeing 2000 flips of one
  coin.
• Tacks are asymmetric, and manufactured to less
  exacting standards.

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Limitations

• Can all domain knowledge be represented so
  simply, in terms of an equivalent number of
  fictional observations?
• Suppose you flip a coin 25 times and get all
  heads. Something funny is going on .
• But with F 1000 heads, 1000 tails, P(heads) on
  next flip 1025 / (10251000) 50.6. Looks
  like nothing unusual.

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Plan for tonight

• Why be Bayesian?
• Informal introduction to learning as
  probabilistic inference
• Formal introduction to probabilistic inference
• A little bit of mathematical psychology
• An introduction to Bayes nets

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Basics

• Propositions A, B, C, . . . .
• Negation
• Logical operators and, or
• Obey classical logic, e.g.,

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Basics

• Conservation of belief
• Joint probability
• For independent propositions
• More generally

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Basics

• Example
• A Heads on flip 2
• B Tails on flip 2

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Basics

• All probabilities should be conditioned on
  background knowledge K e.g.,
• All the same rules hold conditioned on any K
  e.g.,
• Often background knowledge will be implicit,
  brought in as needed.

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Bayesian inference

• Definition of conditional probability
• Bayes theorem

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Bayesian inference

• Definition of conditional probability
• Bayes rule
• Posterior probability
• Prior probability
• Likelihood

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Bayesian inference

• Bayes rule
• What makes a good scientific argument? P(HD) is
  high if
• Hypothesis is plausible P(H) is high
• Hypothesis strongly predicts the observed data
• P(DH) is high
• Data are surprising P(D) is low

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Bayesian inference

• Deriving a more useful version

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Bayesian inference

• Deriving a more useful version

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Bayesian inference

• Deriving a more useful version

Conditionalization
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Bayesian inference

• Deriving a more useful version

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Bayesian inference

• Deriving a more useful version

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Bayesian inference

• Deriving a more useful version

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Bayesian inference

• Deriving a more useful version

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Random variables

• Random variable X denotes a set of mutually
  exclusive exhaustive propositions (states of the
  world)
• Bayes theorem for random variables

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Random variables

• Random variable X denotes a set of mutually
  exclusive exhaustive propositions (states of the
  world)
• Bayes rule for more than two hypotheses

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Sherlock Holmes

• How often have I said to you that when you have
  eliminated the impossible whatever remains,
  however improbable, must be the truth? (The Sign
  of the Four)

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Sherlock Holmes

• How often have I said to you that when you have
  eliminated the impossible whatever remains,
  however improbable, must be the truth? (The Sign
  of the Four)

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Sherlock Holmes

• How often have I said to you that when you have
  eliminated the impossible whatever remains,
  however improbable, must be the truth? (The Sign
  of the Four)

0
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Sherlock Holmes

• How often have I said to you that when you have
  eliminated the impossible whatever remains,
  however improbable, must be the truth? (The Sign
  of the Four)

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Plan for tonight

• Why be Bayesian?
• Informal introduction to learning as
  probabilistic inference
• Formal introduction to probabilistic inference
• A little bit of mathematical psychology
• An introduction to Bayes nets

47
Representativeness in reasoning

• Which sequence is more likely to be produced by
  flipping a fair coin?
• HHTHT
• HHHHH

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A reasoning fallacy

• Kahneman Tversky people judge the probability
  of an outcome based on the extent to which it is
  representative of the generating process.

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Predictive versus inductive reasoning
Hypothesis
H
Data
D
50
Predictive versus inductive reasoning
Prediction given ?
H
D
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Predictive versus inductive reasoning
Prediction given ?
Induction ? given
H
D
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Bayes Rule in odds form

• P(H1D) P(DH1) P(H1)
• P(H2D) P(DH2) P(H2)
• D data
• H1, H2 models
• P(H1D) posterior probability that model 1
  generated the data.
• P(DH1) likelihood of data given model 1
• P(H1) prior probability that model 1 generated
  the data

x
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Bayesian analysis of coin flipping

• D HHTHT
• H1, H2 fair coin, trick all heads coin.
• P(DH1) 1/32 P(H1) 999/1000
• P(DH2) 0 P(H2) 1/1000
• P(H1D) / P(H2D) infinity

P(H1D) P(DH1) P(H1) P(H2D)
P(DH2) P(H2)
x
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Bayesian analysis of coin flipping

• D HHHHH
• H1, H2 fair coin, trick all heads coin.
• P(DH1) 1/32 P(H1) 999/1000
• P(DH2) 1 P(H2) 1/1000
• P(H1D) / P(H2D) 999/32 301

P(H1D) P(DH1) P(H1) P(H2D)
P(DH2) P(H2)
x
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Bayesian analysis of coin flipping

• D HHHHHHHHHH
• H1, H2 fair coin, trick all heads coin.
• P(DH1) 1/1024 P(H1) 999/1000
• P(DH2) 1 P(H2) 1/1000
• P(H1D) / P(H2D) 999/1024 11

P(H1D) P(DH1) P(H1) P(H2D)
P(DH2) P(H2)
x
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The role of theories

• The fact that HHTHT looks representative of a
  fair coin and HHHHH does not reflects our
  implicit theories of how the world works.
• Easy to imagine how a trick all-heads coin could
  work high prior probability.
• Hard to imagine how a trick HHTHT coin could
  work low prior probability.

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Plan for tonight

• Why be Bayesian?
• Informal introduction to learning as
  probabilistic inference
• Formal introduction to probabilistic inference
• A little bit of mathematical psychology
• An introduction to Bayes nets

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Scaling up

• Three binary variables Cavity, Toothache, Catch
  (whether dentists probe catches in your tooth).

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Scaling up

• Three binary variables Cavity, Toothache, Catch
  (whether dentists probe catches in your tooth).
• With n pieces of evidence, we need 2n1
  conditional probabilities.
• Here n2. Realistically, many more X-ray, diet,
  oral hygiene, personality, . . . .

60
Conditional independence

• All three variables are dependent, but Toothache
  and Catch are independent given the presence or
  absence of Cavity.
• Both Toothache and Catch are caused by Cavity,
  but via independent causal mechanisms.
• In probabilistic terms
• With n pieces of evidence, x1, , xn, we need 2 n
  conditional probabilities

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A simple Bayes net

• Graphical representation of relations between a
  set of random variables
• Causal interpretation independent local
  mechanisms
• Probabilistic interpretation factorizing complex
  terms

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A more complex system
Battery
Radio
Ignition
Gas
Starts
On time to work

• Joint distribution sufficient for any inference

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A more complex system
Battery
Radio
Ignition
Gas
Starts
On time to work

• Joint distribution sufficient for any inference

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A more complex system
Battery
Radio
Ignition
Gas
Starts
On time to work

• Joint distribution sufficient for any inference
• General inference algorithm local message passing

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Explaining away

• Assume grass will be wet if and only if it rained
  last night, or if the sprinklers were left on

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Explaining away
Compute probability it rained last night, given
that the grass is wet
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Explaining away
Compute probability it rained last night, given
that the grass is wet
68
Explaining away
Compute probability it rained last night, given
that the grass is wet
69
Explaining away
Compute probability it rained last night, given
that the grass is wet
70
Explaining away
Compute probability it rained last night, given
that the grass is wet
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Explaining away
Compute probability it rained last night, given
that the grass is wet and sprinklers were left
on
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Explaining away
Compute probability it rained last night, given
that the grass is wet and sprinklers were left
on
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Explaining away
Discounting to prior probability.
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Contrast w/ spreading activation
Rain
Sprinkler
Grass Wet

• Observing rain, Wet becomes more active.
• Observing grass wet, Rain and Sprinkler become
  more active.
• Observing grass wet and sprinkler, Rain cannot
  become less active. No explaining away!

• Excitatory links Rain Wet, Sprinkler
  Wet

75
Contrast w/ spreading activation
Rain
Sprinkler
Grass Wet

• Excitatory links Rain Wet, Sprinkler
  Wet
• Inhibitory link Rain Sprinkler

• Observing grass wet, Rain and Sprinkler become
  more active.
• Observing grass wet and sprinkler, Rain becomes
  less active explaining away.

76
Contrast w/ spreading activation
Rain
Burst pipe
Sprinkler
Grass Wet

• Each new variable requires more inhibitory
  connections.
• Interactions between variables are not causal.
• Not modular.
• Whether a connection exists depends on what other
  connections exist, in non-transparent ways.
• Big holism problem.
• Combinatorial explosion.

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Causality and the Markov property

• Markov property Any variable is conditionally
  independent of its non-descendants, given its
  parents.
• Example

78
Causality and the Markov property

• Markov property Any variable is conditionally
  independent of its non-descendants, given its
  parents.
• Example

79
Causality and the Markov property

• Markov property Any variable is conditionally
  independent of its non-descendants, given its
  parents.
• Example

80
Causality and the Markov property

• Markov property Any variable is conditionally
  independent of its non-descendants, given its
  parents.
• Example

81
Causality and the Markov property

• Markov property Any variable is conditionally
  independent of its non-descendants, given its
  parents.
• Example

82
Causality and the Markov property

• Markov property Any variable is conditionally
  independent of its non-descendants, given its
  parents.
• Example

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Causality and the Markov property

• Markov property Any variable is conditionally
  independent of its non-descendants, given its
  parents.
• Suppose we get the direction of causality wrong,
  thinking that symptoms causes diseases
• Does not capture the correlation between
  symptoms falsely believe P(Ache, Catch)
  P(Ache) P(Catch).

Ache
Catch
Cavity
84
Causality and the Markov property

• Markov property Any variable is conditionally
  independent of its non-descendants, given its
  parents.
• Suppose we get the direction of causality wrong,
  thinking that symptoms causes diseases
• Inserting a new arrow allows us to capture this
  correlation.
• This model is too complex do not believe that

Ache
Catch
Cavity
85
Causality and the Markov property

• Markov property Any variable is conditionally
  independent of its non-descendants, given its
  parents.
• Suppose we get the direction of causality wrong,
  thinking that symptoms causes diseases
• New symptoms require a combinatorial
  proliferation of new arrows. Too general, not
  modular, holism, yuck . . . .

Ache
X-ray
Catch
Cavity
86
Still to come

• Applications to models of categorization
• More on the relation between causality and
  probability
• Learning causal graph structures.
• Learning causal abstractions (diseases cause
  symptoms)
• Whats missing

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The end
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Mathcamp data raw
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Mathcamp data collapsed over parity
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Zenith radio data collapsed over parity
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