inference - Advanced Quantitative Research Methodology...

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Advanced Quantitative Research Methodology, Lecture Notes: Theories of Inference 1 Gary King http://GKing.Harvard.Edu January 31, 2010 1 c Copyright 2010 Gary King, All Rights Reserved. Gary King () Advanced Quantitative Research Methodology, Lecture Notes: January 31, 2010 1 / 1

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The Problem of Inference Gary King () Inference 2 / 1
The Problem of Inference 1. Probability: P( y | M ) = P( known | unknown) Gary King () Inference 2 / 1

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The Problem of Inference 1. Probability: P( y | M ) = P( known | unknown) 2. The goal of inverse probability: P( M | y ) = P(unknown | known ). Gary King () Inference 2 / 1
The Problem of Inference 1. Probability: P( y | M ) = P( known | unknown) 2. The goal of inverse probability: P( M | y ) = P(unknown | known ). 3. Define: M = { M * , θ } , where M * is assumed and θ is to be estimated Gary King () Inference 2 / 1

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The Problem of Inference 1. Probability: P( y | M ) = P( known | unknown) 2. The goal of inverse probability: P( M | y ) = P(unknown | known ). 3. Define: M = { M * , θ } , where M * is assumed and θ is to be estimated 4. More limited goal: P( θ | y , M * ) P( θ | y ). Gary King () Inference 2 / 1
The Problem of Inference 5. Bayes Theorem (no additional assumptions): Gary King () Inference 3 / 1

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The Problem of Inference 5. Bayes Theorem (no additional assumptions): P( θ | y ) = P( θ, y ) P( y ) [Defn. of conditional probability] Gary King () Inference 3 / 1
The Problem of Inference 5. Bayes Theorem (no additional assumptions): P( θ | y ) = P( θ, y ) P( y ) [Defn. of conditional probability] = P( θ )P( y | θ ) P( y ) [P( AB ) = P( B )P( A | B )] Gary King () Inference 3 / 1

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The Problem of Inference 5. Bayes Theorem (no additional assumptions): P( θ | y ) = P( θ, y ) P( y ) [Defn. of conditional probability] = P( θ )P( y | θ ) P( y ) [P( AB ) = P( B )P( A | B )] = P( θ )P( y | θ ) P( θ )P( y | θ ) d θ [P( A ) = P( AB ) dB ] Gary King () Inference 3 / 1
The Problem of Inference 5. Bayes Theorem (no additional assumptions): P( θ | y ) = P( θ, y ) P( y ) [Defn. of conditional probability] = P( θ )P( y | θ ) P( y ) [P( AB ) = P( B )P( A | B )] = P( θ )P( y | θ ) P( θ )P( y | θ ) d θ [P( A ) = P( AB ) dB ] 6. If we knew the right side, we could compute the inverse probability. Gary King () Inference 3 / 1

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The Problem of Inference 5. Bayes Theorem (no additional assumptions): P( θ | y ) = P( θ, y ) P( y ) [Defn. of conditional probability] = P( θ )P( y | θ ) P( y ) [P( AB ) = P( B )P( A | B )] = P( θ )P( y | θ ) P( θ )P( y | θ ) d θ [P( A ) = P( AB ) dB ] 6. If we knew the right side, we could compute the inverse probability. 7. 2 theories of inference arose to interpret this result: likelihood and Bayesian Gary King () Inference 3 / 1
The Problem of Inference 5. Bayes Theorem (no additional assumptions): P( θ | y ) = P( θ, y ) P( y ) [Defn. of conditional probability] = P( θ )P( y | θ ) P( y ) [P( AB ) = P( B )P( A | B )] = P( θ )P( y | θ ) P( θ )P( y | θ ) d θ [P( A ) = P( AB ) dB ] 6. If we knew the right side, we could compute the inverse probability. 7. 2 theories of inference arose to interpret this result: likelihood and Bayesian 8. In both, P( y | θ ) is a traditional probability density ; what’s the rest? Gary King () Inference 3 / 1

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The Problem of Inference 5. Bayes Theorem (no additional assumptions): P( θ | y ) = P( θ, y ) P( y ) [Defn. of conditional probability] = P( θ )P( y | θ ) P( y ) [P( AB ) = P( B )P( A | B )] = P( θ )P( y | θ ) P( θ )P( y | θ ) d θ [P( A ) = P( AB ) dB ] 6. If we knew the right side, we could compute the inverse probability.
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