／var／logmarcus chiu

Noisy-OR Model as CPD

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Let:

  • π‘Œ have parentsΒ {𝑋1, …, 𝑋𝑛}
  • {π‘Œ, 𝑋1, …, 𝑋𝑛}Β be a binary-valued random variables

The CPD 𝐏(π‘Œ|𝑋1, …, 𝑋𝑛)Β is aΒ noisy-orΒ if there are 𝑛+1Β noise parametersΒ {πœ†0, πœ†1, …, πœ†π‘›}Β such that:

  • 𝐏(π‘Œ=0|𝑋1, …, 𝑋𝑛) = (1 - πœ†0) Β· 𝛱𝑖:𝑋𝑖=1(1 - πœ†π‘–)
  • 𝐏(π‘Œ=1|𝑋1, …, 𝑋𝑛) = 1 - [(1 -Β πœ†0) Β· 𝛱𝑖:𝑋𝑖=1(1 -Β πœ†π‘–)]

Thus, in aΒ Noisy-OR ModelΒ each 𝑋𝑖 influencesΒ π‘ŒΒ independently

noisy-or-model.drawio

Example CPT (πœ†0= 0)

𝑋1

𝑋2

𝑋3

𝐏(π‘Œ=0|𝑋1, 𝑋2, 𝑋3)

𝐏(π‘Œ=1|𝑋1, 𝑋2, 𝑋3)

0

0

0

1

0

0

0

1

1 - πœ†3

πœ†3

0

1

0

1 - πœ†2

πœ†2

1

0

0

1 - πœ†1

πœ†1

0

1

1

(1 - πœ†2)(1 - πœ†3)

1 - [(1 - πœ†2)(1 - πœ†3)]

1

0

1

(1 - πœ†1)(1 - πœ†3)

1 - [(1 - πœ†1)(1 - πœ†3)]

1

1

0

(1 - πœ†1)(1 - πœ†2)

1 - [(1 - πœ†1)(1 - πœ†2)]

1

1

1

(1 - πœ†1)(1 - πœ†2)(1 - πœ†3)

1 - [(1 - πœ†1)(1 - πœ†2)(1 - πœ†3)]