From 8fd41af9cd1c6f5862c34090252622dad11f32ca Mon Sep 17 00:00:00 2001 From: Austin Schaffer Date: Sun, 18 Feb 2024 12:33:35 -0500 Subject: [PATCH] Module 6 notes in progress --- .../AI/Module_06/Module 6 - Bayes Nets.md | 258 ++++++++++++++++++ .../images/Pasted image 20240217144529.png | Bin 0 -> 33238 bytes .../images/Pasted image 20240217144613.png | Bin 0 -> 194127 bytes .../images/Pasted image 20240217150019.png | Bin 0 -> 33158 bytes .../images/Pasted image 20240217153405.png | Bin 0 -> 98054 bytes .../images/Pasted image 20240217160022.png | Bin 0 -> 383086 bytes 6 files changed, 258 insertions(+) create mode 100644 OMSCS/Courses/AI/Module_06/Module 6 - Bayes Nets.md create mode 100644 OMSCS/Courses/AI/Module_06/images/Pasted image 20240217144529.png create mode 100644 OMSCS/Courses/AI/Module_06/images/Pasted image 20240217144613.png create mode 100644 OMSCS/Courses/AI/Module_06/images/Pasted image 20240217150019.png create mode 100644 OMSCS/Courses/AI/Module_06/images/Pasted image 20240217153405.png create mode 100644 OMSCS/Courses/AI/Module_06/images/Pasted image 20240217160022.png diff --git a/OMSCS/Courses/AI/Module_06/Module 6 - Bayes Nets.md b/OMSCS/Courses/AI/Module_06/Module 6 - Bayes Nets.md new file mode 100644 index 0000000..7b7422e --- /dev/null +++ b/OMSCS/Courses/AI/Module_06/Module 6 - Bayes Nets.md @@ -0,0 +1,258 @@ +--- +tags: + - OMSCS + - AI + - Probability + - Bayes +--- +# Module 6 - Bayes Nets + +## Bayes Rule Graphically +![[Pasted image 20240217144613.png]] + +Diagnostic Reasoning +- $P(A | B)$ +- $P(A|\neg B)$ + +You need 3 parameters to specify a bayes network +- $P(A)$ +- $P(B|A)$ +- $P(B|\neg A)$ + +## Bayes Rule Refresher +$$ +P(A|B)=\frac{P(B|A)P(A)}{P(B)} +$$ + +$$ +P(\neg A | B) = \frac{P(B|\neg A)P(\neg A)}{P(B)} +$$ + +$$ +P(A|B)+P(\neg A|B)=1 +$$ + +- $P'(A|B)=P(B|A)P(A)$ +- $P'(\neg A|B)=P(B|\neg A)P(\neg A)$ +- For some normalizer value $\eta$ + - $P(A|B)=\eta P'(A|B)$ + - $P(\neg A|B)=\eta P'(\neg A|B)$ + - $\eta=(P'(A|B)+P'(\neg A|B))^{-1}$ +- The advantage of using these pseudo probabilities is that you don't have to calculate $P(B)$ directly, which can be pretty complicated in some problems. + +### 2-Test Cancer Example +![[Pasted image 20240217150019.png]] +- $P(C)=0.01$ +- $P(\neg C)=0.99$ +- $P(T_x=+|C)=0.9$ +- $P(T_x=-|C)=0.1$ +- $P(T_x=-|\neg C)=0.8$ +- $P(T_x=+|\neg C)=0.2$ +- $P(C|T_1=+, T_2=+)=?$ + +Working out +- Calculating $P(++|C)$ + - $P(T_1=+,T_2=+|C)=P(++|C)$ + - $P(++|C)=P(T_1=+|C)P(T_2=+|C)=0.9^2$ + - $P(++|C)=0.81$ +- Calculating $P(++|\neg C)$ + - $P(T_1=+,T_2=+|\neg C)=P(++|\neg C)$ + - $P(++|\neg C)=P(T_1=+|\neg C)P(T_2=+|\neg C)=0.2^2$ + - $P(++|\neg C)=0.04$ +- $P(C)=0.01$ +- $P(++)$ $\leftarrow$ we can avoid calculating this + - Calculating $P'(C|++)$ + - $P'(C|++)=P(++|C)P(C)$ + - $P'(C|++)=(0.81)(0.01)$ + - $P'(C|++)=0.0081$ + - Calculating $P'(\neg C|++)$ + - $P'(\neg C|++)=P(++|\neg C)P(\neg C)$ + - $P'(\neg C|++)=(0.04)(0.99)$ + - $P'(\neg C|++)=0.0396$ + - $\eta = (0.0081+0.0396)^{-1} = 20.9643...$ +- Calculating $P(C|++)$ + - $P(C|++)=\eta P'(C|++)$ + - $P(C|++)=(20.9643...)(0.0081)$ + - $P(C|++)=0.1698$ + +I initially got the wrong answer for this due to using the wrong value of $P(T_1=+|\neg C)$ in the "Calculating $P(++|\neg C)$" step. I used 0.1 instead of 0.2, which multiplied the errors downstream. + +| | prior | + | + | P' | $P(C\|++)$ | +| ---- | ---- | ---- | ---- | ---- | ---- | +| $C$ | 0.01 | 0.9 | 0.9 | 0.0081 | 0.1698 | +| $\neg C$ | 0.99 | 0.2 | 0.2 | 0.0396 | 0.8301 | +### 2-Test Cancer Example 2 + +- $P(C)=0.01$ +- $P(\neg C)=0.99$ +- $P(T_x=+|C)=0.9$ +- $P(T_x=-|C)=0.1$ +- $P(T_x=-|\neg C)=0.8$ +- $P(T_x=+|\neg C)=0.2$ +- $P(C|T_1=+, T_2=-)=?$ + +Working out +- Calculating $P(+-|C)$ + - $P(T_1=+,T_2=-|C)=P(+-|C)$ + - $P(+-|C)=P(T_1=+|C)P(T_2=-|C)=(0.9)(0.1)$ + - $P(+-|C)=0.09$ +- Calculating $P(+-|\neg C)$ + - $P(T_1=+,T_2=-|\neg C)=P(+-|\neg C)$ + - $P(+-|\neg C)=P(T_1=+|\neg C)P(T_2=-|\neg C)=(0.8)(0.2)$ + - $P(+-|\neg C)=0.16$ +- $P(C)=0.01$ +- $P(+-)$ $\leftarrow$ we can avoid calculating this + - Calculating $P'(C|+-)$ + - $P'(C|+-)=P(+-|C)P(C)$ + - $P'(C|+-)=(0.09)(0.01)$ + - $P'(C|++)=0.0009$ + - Calculating $P'(\neg C|+-)$ + - $P'(\neg C|+-)=P(+-|\neg C)P(\neg C)$ + - $P'(\neg C|+-)=(0.16)(0.99)$ + - $P'(\neg C|+-)=0.1584$ + - $\eta = (0.1584+0.0009)^{-1} = 6.2774...$ +- Calculating $P(C|+-)$ + - $P(C|+-)=\eta P'(C|+-)$ + - $P(C|+-)=(6.2774...)(0.0009)$ + - $P(C|++)=0.005649...$ + +| | prior | + | - | P' | $P(C\|++)$ | +| ---- | ---- | ---- | ---- | ---- | ---- | +| $C$ | 0.01 | 0.9 | 0.1 | 0.0009 | 0.0056 | +| $\neg C$ | 0.99 | 0.2 | 0.8 | 0.1584 | 0.9943 | + +## Conditional Independence +Useful assumptions in the examples above. +- Both $T_x$ variables are conditionally independent +- $P(T_2|C,T_1)=P(T_2|C)$ is one way of specifying conditional independence. + +![[Pasted image 20240217153405.png]] + +$B \perp C \space | \space A \ne B \perp C$ + +> Getting a positive test result about cancer (B) increases the probability of having cancer (A) above the prior probability for A, which changes the probability that another test will result in a positive (C). + +### Cancer Example 3 + +- $P(C)=0.01$ +- $P(\neg C)=0.99$ +- $P(T_x=+|C)=0.9$ +- $P(T_x=-|C)=0.1$ +- $P(T_x=-|\neg C)=0.8$ +- $P(T_x=+|\neg C)=0.2$ +- $P(T_2=+ \space | \space T_1=+)=?$ + +Ok so the theory here is that by having a positive result for $T_1$, that raises the probability of $C$ above it's prior value, which affects the calculations for $T_2$. + +> For this problem, we want to apply the principle of total probability. + +- $P(T_2=+ \space | \space T_1=+) = P(+_2 | +_1)$ +- $P(+_2 | +_1)=P(+_2|+_1, C)P(C|+_1)+P(+_2|+_1, \neg C)P(\neg C | +_1)$ +- $P(C|+_1)=0.043$ +- $P(\neg C|+_1)=1-0.043=0.957$ +- Thanks to conditional independence + - $P(+_2|+_1, C)=P(+_2|C)=0.9$ + - $P(+_2|+_1, \neg C)=P(+_2|\neg C)=0.2$ +- $P(+_2 | +_1)=(0.9)(0.043)+(0.2)(0.957)$ +- $P(+_2 | +_1)=0.2301$ +- Baseline: $P(+_x)=0.207$ + +## Absolute and Conditional Independence +- $A \perp B$ - means that A and B have "absolute independence" +- $A \perp B \space | \space C$ - means that A and B are "conditionally independent" on C +- Absolute independence does not imply conditional independence. +- Conditional independence does not imply absolute independence. + +![[Pasted image 20240217160022.png]] + +- $P(R | S) = P(R)$ + - We don't know anything about $H$, so $S$ has no effect on $R$. + +### Quiz 1 + +- $P(R|H,S)=?$ + +> Using Bayes Rule we can transform this question. + +$$ +P(R|H,S)=\frac{P(H|R,S)P(R|S)}{P(H|S)} +$$ + +Where the hell did $P(H|S)$ come from? + +> thanks to the rules of conditional independence: $P(R|S)=P(R)$ +> +> thanks to the rules of total probability: +> $P(H|S)=P(H|R,S)P(R)+P(H|\neg R,S)P(\neg R)$ +> +> Substituting these values in to the equation above. +$$ +\frac{P(H|R,S)P(R|S)}{P(H|S)}=\frac{P(H|R,S)P(R)}{P(H|R,S)P(R)+P(H|\neg R,S)P(\neg R)} +$$ + +### Quiz 2 +- $P(R|H)=?$ + +Using bayes theorem + +$$ +P(R|H)=\frac{P(H|R)P(R)}{P(H)} +$$ + +- $P(H|R)$ + - $= P(H|\neg S,R)P(\neg S) + P(H|S,R)P(S)$ + - $=(0.9)(0.3)+(1)(0.7)$ + - $=0.97$ +- $P(R) = 0.01$ +- $P(H)$ + - $= P(H|S,R)P(S, R)$ + - $=P(H|S,R)P(S)P(R)$ + - $=(1)(0.7)(0.01)$ + - $=0.007$ + - $+ \space P(H|\neg S, R)P(\neg S, R)$ + - $=P(H|\neg S, R)P(\neg S)P(R)$ + - $= (0.9)(0.3)(0.01)$ + - $=0.0027$ + - $+ \space P(H|S,\neg R)P(S, \neg R)$ + - $=P(H|S,\neg R)P(S)P(\neg R)$ + - $=(0.7)(0.7)(0.99)$ + - $=0.4851$ + - $+ \space P(H|\neg S, \neg R)P(\neg S, \neg R)$ + - $=P(H|\neg S, \neg R)P(\neg S)P(\neg R)$ + - $=(0.1)(0.3)(0.99)$ + - $=0.0297$ + - $=0.007+0.0027+0.4851+0.0297$ + - $=0.5245$ +- $\frac{P(H|R)P(R)}{P(H)}$ + - $=(0.97)(0.01)(0.5245)^{-1}$ + - $=(0.97)(0.01)(1.9065...)$ + - $=0.01849...$ + +Using the "pseudo-probabilities" method + +$$ +P(R|H)=\frac{P(H|R)P(R)}{P(H|R)P(R)+P(H|\neg R)P(\neg R)} +$$ + +- $P(H|R)$ + - $= P(H|\neg S, R)P(\neg S) + P(H|S,R)P(S)$ + - $=(0.9)(0.3)+(1)(0.7)$ + - $=0.97$ +- $P(H|\neg R)$ + - $= P(H|\neg S, \neg R)P(\neg S) + P(H|S,\neg R)P(S)$ + - $= (0.1)(0.3) + (0.7)(0.7)$ + - $= 0.03+0.49$ + - $=0.52$ +- $P'(R|H)=P(H|R)P(R)$ + - $=(0.97)(0.01)$ + - $=0.0097$ +- $P'(\neg R|H)=P(H|\neg R)P(\neg R)$ + - $=(0.52)(0.99)$ + - $=0.5148$ +- 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