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Bayesian networks: basic parameter learning Machine Intelligence Thomas D. Nielsen September 2008 Basic parameter learning September 2008 1 / 24 Estimation Example: Physical Measurements Mass of an atomic particle is measured in repeated

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Bayesian networks: basic parameter learning

Machine Intelligence Thomas D. Nielsen September 2008

Basic parameter learning September 2008 1 / 24

SLIDE 2

Estimation

Example: Physical Measurements Mass of an atomic particle is measured in repeated experiments. Measurement results = true mass + random error.

Basic parameter learning September 2008 2 / 24

SLIDE 3

Estimation

Example: Physical Measurements Mass of an atomic particle is measured in repeated experiments. Measurement results = true mass + random error.

true mass random error

Basic parameter learning September 2008 2 / 24

SLIDE 4

Estimation

Example: Physical Measurements Mass of an atomic particle is measured in repeated experiments. Measurement results = true mass + random error.

Basic parameter learning September 2008 2 / 24

SLIDE 5

Estimation

Example: Physical Measurements Mass of an atomic particle is measured in repeated experiments. Measurement results = true mass + random error. Estimate of true mass: mean value of normal distribution that best “fits” the data.

Basic parameter learning September 2008 2 / 24

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Estimation

Example: Coin Tossing Is the Euro fair?

Basic parameter learning September 2008 3 / 24

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Estimation

Example: Coin Tossing Is the Euro fair? Toss Euro 1000 times and count number of heads and tails: . . .

Basic parameter learning September 2008 3 / 24

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Estimation

Example: Coin Tossing Is the Euro fair? Result: heads: 521, tails: 479. Probability of Euro falling heads (estimate): value that best “fits” the data: 521/1000.

Basic parameter learning September 2008 3 / 24

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Estimation: Classical

Structure of Estimation Problem Given: Data produced by some random process that is characterized by one or several numerical parameters. Wanted: Infer value of (some) parameters. (Classical) Method: Obtain estimate for parameter by a function that maps possible data sets into the parameter space.

Basic parameter learning September 2008 4 / 24

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Estimation: Classical

Parametric Family Let W be a set, Θ ⊆ Rk for some k ≥ 1. For every θ ∈ Θ let Pθ be a probability distribution on W. Then {Pθ | θ ∈ Θ} is called a parametric family (of distributions). Example 1: W = {h, t}, Θ = [0, 1]. Pθ: distribution with P(h) = θ (and P(t) = 1 − θ).

Basic parameter learning September 2008 5 / 24

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Estimation: Classical

Parametric Family Let W be a set, Θ ⊆ Rk for some k ≥ 1. For every θ ∈ Θ let Pθ be a probability distribution on W. Then {Pθ | θ ∈ Θ} is called a parametric family (of distributions). Example 1: W = {h, t}, Θ = [0, 1]. Pθ: distribution with P(h) = θ (and P(t) = 1 − θ). Example 2: W = {w1, . . . , wk}, Θ = {θ = (p1, . . . , pk) ∈ [0, 1]k | X pi = 1}. Pθ: distribution with P(wi) = pi.

Basic parameter learning September 2008 5 / 24

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Estimation: Classical

Parametric Family Let W be a set, Θ ⊆ Rk for some k ≥ 1. For every θ ∈ Θ let Pθ be a probability distribution on W. Then {Pθ | θ ∈ Θ} is called a parametric family (of distributions). Example 1: W = {h, t}, Θ = [0, 1]. Pθ: distribution with P(h) = θ (and P(t) = 1 − θ). Example 2: W = {w1, . . . , wk}, Θ = {θ = (p1, . . . , pk) ∈ [0, 1]k | X pi = 1}. Pθ: distribution with P(wi) = pi. Example 3: W = R, Θ = R × R+. For θ = (µ, σ) ∈ Θ: Pθ normal distribution with mean µ and standard deviation σ.

Basic parameter learning September 2008 5 / 24

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Estimation: Classical

Sample A family X1, . . . , XN of random variables is called independent identically distributed (iid) if the family is independent, and P(Xi) = P(Xj) for all i, j.

Basic parameter learning September 2008 6 / 24

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Estimation: Classical

Sample A family X1, . . . , XN of random variables is called independent identically distributed (iid) if the family is independent, and P(Xi) = P(Xj) for all i, j. A sample s1, . . . , sN ∈ W of observations (or data items) is interpreted as the observed values of an iid family of random variables with distribution P(Xi) = Pθ.

Basic parameter learning September 2008 6 / 24

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Estimation: Classical

Sample A family X1, . . . , XN of random variables is called independent identically distributed (iid) if the family is independent, and P(Xi) = P(Xj) for all i, j. A sample s1, . . . , sN ∈ W of observations (or data items) is interpreted as the observed values of an iid family of random variables with distribution P(Xi) = Pθ. Likelihood Function Given a parametric family {Pθ | θ ∈ Θ} of distributions on W, and a sample s = (s1, . . . , sN) ∈ W N. The function θ → Pθ(s) :=

N

Y

i=1

Pθ(si),

resp. θ → log Pθ(s) =

N

X

i=1

log Pθ(si), is called the likelihood function (resp. log-likelihood function) for θ given s.

Basic parameter learning September 2008 6 / 24

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Estimation: Classical

Maximum Likelihood Estimator Given: parametric family and sample s. Every θ∗ ∈ Θ with θ∗ = arg max

θ∈Θ Pθ(s)

is called a maximum likelihood estimate for θ (given s).

Basic parameter learning September 2008 7 / 24

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Estimation: Classical

Maximum Likelihood Estimator Given: parametric family and sample s. Every θ∗ ∈ Θ with θ∗ = arg max

θ∈Θ Pθ(s)

is called a maximum likelihood estimate for θ (given s). Since the logarithm is a strictly monotone function, maximum likelihood estimatates are also

btained by maximizing the log-likelihood:

θ∗ = arg max

θ∈Θ log Pθ(s)

Basic parameter learning September 2008 7 / 24

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Estimation: classical

Thumbtack example We have tossed a thumb tack 100 times. It has landed pin up 80 times, and we now look for the model that best fits the observations/data: T T T 0.1 0.2 0.3 M0.1 M0.2 M0.3 Model Structure Probability, P(pinup) =

Basic parameter learning September 2008 8 / 24

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Estimation: classical

Thumbtack example We have tossed a thumb tack 100 times. It has landed pin up 80 times, and we now look for the model that best fits the observations/data: T T T 0.1 0.2 0.3 M0.1 M0.2 M0.3 Model Structure Probability, P(pinup) = We can measure how well a model fits the data using: P(D|Mθ) = P(pinup, pinup, pindown, . . . , pinup|Mθ) = P(pinup|Mθ)P(pinup|Mθ)P(pindown|Mθ) · . . . · P(pinup|Mθ) This is also called the likelihood of Mθ given D.

Basic parameter learning September 2008 8 / 24

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Estimation: classical

Thumbtack example We have tossed a thumb tack 100 times. It has landed pin up 80 times, and we now look for the model that best fits the observations/data: T T T 0.1 0.2 0.3 M0.1 M0.2 M0.3 Model Structure Probability, P(pinup) = We select the parameter ˆ θ that maximizes: ˆ θ = arg max

θ

P(D|Mθ) = arg max

θ 100

Y

i=1

P(di|Mθ) = arg max

θ

µ · θ80(1 − θ)20.

Basic parameter learning September 2008 8 / 24

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Estimation: classical

Thumbtack example We have tossed a thumb tack 100 times. It has landed pin up 80 times, and we now look for the model that best fits the observations/data: T T T 0.1 0.2 0.3 M0.1 M0.2 M0.3 Model Structure Probability, P(pinup) = By setting: d dθ µ · θ80(1 − θ)20 = 0 we get the maximum likelihood estimate: ˆ θ = 0.8.

Basic parameter learning September 2008 8 / 24

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Estimation: Classical

Maximum Likelihood Estimates for Multinomial Distribution Consider the family of multinomial distribution defined as W = {w1, . . . , wk}, Θ = {θ = (p1, . . . , pk) ∈ [0, 1]k | X pi = 1}. Pθ: distribution with P(wi) = pi. For {Pθ | θ ∈ Θ} and s ∈ W N: there exists exactly one maximum likelihood estimate θ∗ = (p∗

1 , . . . , p∗ k ) given by

p∗

i = 1

N |{j ∈ {1, . . . , N} | sj = wi}| [i.e. θ∗ is just the empirical distribution defined by the data on W]

Basic parameter learning September 2008 9 / 24

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Estimation: Classical

Proof: (for W = {w1, w2}): p∗

1

= 1/N|{j ∈ {1, . . . , N} | sj = w1}| p∗

2

= 1/N|{j ∈ {1, . . . , N} | sj = w2}| (= 1 − p∗

1 )

log Pθ(s) =

N

X

j=1

log Pθ(sj) = N · (p∗

1 log(p1) + p∗ 2 log(p2))

= N · (p∗

1 log(p1) + (1 − p∗ 1 ) log(1 − p1))

Differentiated w.r.t. p1: N · (p∗

1 /p1 − (1 − p∗ 1)/(1 − p1))

Only root: p1 = p∗

1 .

Basic parameter learning September 2008 10 / 24

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Estimation: Classical

Consistency Let W = {w1, . . . , wk}, and the data s1, s2, . . . , sN be generated by the distribution Pθ with parameters θ = (p1, . . . , pk). Then for all ǫ > 0 and i = 1, . . . , k: lim

N→∞ Pθ(|p∗ i − pi| ≥ ǫ) = 0

Note: p∗ is a function of s. The probability Pθ(|p∗

i − pi| ≥ ǫ) is the probability that by sampling

from Pθ a sample s will be obtained, so that for the p∗ computed from s the inequality |p∗

i − pi| ≥ ǫ holds.

Similar consistency properties hold for many other types of maximum likelihood estimates.

Basic parameter learning September 2008 11 / 24

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Estimation: Classical

Chebyshev’s Inequality A quantitative bound: Pθ(|p∗

i − pi| ≥ ǫ) ≤

1 ǫ2N pi(1 − pi)

Basic parameter learning September 2008 12 / 24

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Estimation: Bayesian

Prior Beliefs Classical approach: toss Euro 3 times and always obtain heads infer P(heads) = 1. Inference solely based on observed data – no use of prior knowledge, beliefs, etc. Bayesian methods: start with an encoding of prior beliefs, and show how to modify them on the basis of observed data.

Basic parameter learning September 2008 13 / 24

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Estimation: Bayesian

Approach Model a-priori assumptions on parameter by probability distribution on Θ. Based on observed sample s update to a-posteriori distribution. Example 1: Apparently symmetric coin. W = {heads, tails}. θ = P(heads). Observation: sample with 3 heads and 7 tails 1 1 prior posterior

Basic parameter learning September 2008 14 / 24

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Estimation: Bayesian

Example 2: Result of first clinical tests of new type of drug. W = {pos, neg}. θ = P(pos). Observation: sample with 3 pos and 7 neg 1 prior posterior 1 0.3

Basic parameter learning September 2008 15 / 24

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