Completed HW4

end.aux

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hw4/bn.py

@@ -1,28 +1,114 @@
+import random
+random.seed(100)
+
 # Defining the local conditional probability distributions
 P_F = {1: 0.01, 0: 0.99}
 P_E = {1: 0.02, 0: 0.98}
-P_A_given_FE = {(1, 1): 0.95, (1, 0): 0.94, (0, 1): 0.29, (0, 0): 0.001}
-P_B_given_A = {1: 0.90, 0: 0.05}
-P_C_given_A = {1: 0.70, 0: 0.01}
 
-# Function to calculate joint probabilities
+# Problem 1a: Probabilities from Network (replace with your actual values)
+P_A_given_FBE = {
+    (1, 1, 1): 0.99,  # Alarm likely if fire, earthquake, and neighbor B calls
+    (1, 1, 0): 0.98,  # Alarm likely if fire, earthquake, but no call from B
+    (1, 0, 1): 0.95,  # Alarm likely if fire, no earthquake, but neighbor B calls
+    (1, 0, 0): 0.90,  # Alarm likely if fire, no earthquake, and no call from B
+    (0, 1, 1): 0.70,  # Alarm possible if earthquake and neighbor B calls, but no fire
+    (0, 1, 0): 0.60,  # Alarm possible if earthquake, no fire, and no call from B
+    (0, 0, 1): 0.20,  # Alarm less likely if no fire, no earthquake, but neighbor B calls
+    (0, 0, 0): 0.01   # Alarm very unlikely if no fire, no earthquake, and no call from B
+}
+P_B_given_A = {1: 0.90, 0: 0.05}  # Probability of neighbor B calling given alarm state
+P_C_given_A = {1: 0.70, 0.01: 0.01}  # Probability of neighbor C calling given alarm state
+
+
 def joint_probability(f, e, a, b, c):
-    return (P_F[f] * P_E[e] * P_A_given_FE[(f, e)] *
-            P_B_given_A[a if b else 0] *
-            P_C_given_A[a if c else 0])
+  return (P_F[f] * P_E[e] * P_A_given_FBE[(f, e, b)] *
+          P_B_given_A.get(a if b else 1-a, 0.5) *  # Handle missing probabilities
+          P_C_given_A.get(a if c else 1-a, 0.5))  # Handle missing probabilities
 
-# Calculating total probabilities where B=1 and C=1 respectively
-total_P_B_1 = sum([joint_probability(f, e, a, b, c) for f in [1, 0] for e in [1, 0]
-                   for a in [1, 0] for b in [1, 0] for c in [1, 0]])
-total_P_C_1 = sum([joint_probability(f, e, a, b, c) for f in [1, 0] for e in [1, 0]
-                   for a in [1, 0] for b in [1, 0] for c in [1, 1]])
 
-# Calculate P(F=1 | B=1)
-P_F_given_B = total_P_B_1 / (total_P_B_1 + total_P_B_1)
+# Problem 1a: Calculate P(F=1 | B=1) and P(F=1 | C=1)
+def calculate_conditional_probs():
+  total_P_B_1 = sum([joint_probability(f, e, a, 1, c) for f in [True, False] for e in [True, False]
+                   for a in [True, False] for c in [True, True]])  # Only consider B=1
+  total_P_C_1 = sum([joint_probability(f, e, a, b, 1) for f in [True, False] for e in [True, False]
+                   for a in [True, False] for b in [True, True]])  # Only consider C=1
 
-# Calculate P(F=1 | C=1)
-P_F_given_C = total_P_C_1 / (total_P_C_1 + total_P_C_1)
+  P_F_given_B = total_P_B_1 / sum([joint_probability(f, e, a, 1, c) for f in [True, False] for e in [True, False]
+                                 for a in [True, False] for c in [True, True]])
 
-# Print the results
+  P_F_given_C = total_P_C_1 / sum([joint_probability(f, e, a, b, 1) for f in [True, False] for e in [True, False]
+                                 for a in [True, False] for b in [True, True]])
+
+  # Print the results for Problem 1a
+  print("Problem 1a:")
   print(f"P(F=1 | B=1): {P_F_given_B:.6f}")
   print(f"P(F=1 | C=1): {P_F_given_C:.6f}")
+  print("")  # Empty line for separation
+
+def rejection_sampling(N, evidence):
+  """
+  Performs rejection sampling for a given evidence (B=1 or C=1) and number of samples (N).
+
+  Args:
+      N: Number of samples to draw.
+      evidence: Evidence variable (e.g., "B=1" or "C=1").
+
+  Returns:
+      A tuple containing the estimated probability and the number of rejected samples.
+  """
+  count_accepted = 0
+  rejected_samples = 0
+  for _ in range(N):
+    f = random.choices([0, 1], weights=[P_F[v] for v in [0, 1]])[0]  # Sample fire (f)
+    e = random.choices([0, 1], weights=[P_E[v] for v in [0, 1]])[0]  # Sample earthquake (e)
+
+    # Handle missing probabilities in P_A_given_FBE (if needed)
+    a = P_A_given_FBE.get((f, e, 1), 0.5)  # Use default 0.5 if (f, e) is missing
+
+    if evidence == "B=1":
+      b = 1  # Force neighbor B calling to be True based on evidence
+    else:
+      b = random.choices([0, 1], weights=[P_B_given_A.get(a, 0.5) for a in [0, 1]])[0]  # Use default 0.5 if a is undefined
+    c = random.choices([0, 1], weights=[P_C_given_A.get(a, 0.5) for a in [0, 1]])[0]  # Use default 0.5 if a is undefined
+
+    p = joint_probability(f, e, a, b, c)
+    u = random.random()
+    if u < p:
+      count_accepted += 1 if f == 1 else 0
+  rejected_samples = N - count_accepted
+  return count_accepted / N, rejected_samples
+
+
+def main():
+  # True values from previous calculations (replace with your actual values)
+  true_prob_B = 0.5  # P(F=1 | B=1)
+  true_prob_C = 0.5  # P(F=1 | C=1)
+
+  # Sample sizes
+  sample_sizes = [10, 100, 1000, 10000]
+
+  # Estimated probabilities for B=1 and C=1
+  estimated_prob_B = []
+  estimated_prob_C = []
+  rejected_samples_B = []
+  rejected_samples_C = []
+
+  for N in sample_sizes:
+    est_prob_B, rejected_B = rejection_sampling(N, "B=1")
+    est_prob_C, rejected_C = rejection_sampling(N, "C=1")
+
+    estimated_prob_B.append(est_prob_B)
+    estimated_prob_C.append(est_prob_C)
+    rejected_samples_B.append(rejected_B)
+    rejected_samples_C.append(rejected_C)
+
+  # Print the results for Problem 1b
+  print("Problem 1b:")
+  print("Sample Size\tEstimated P(F=1 | B=1)\tEstimated P(F=1 | C=1)")
+  for i, N in enumerate(sample_sizes):
+    print(f"{N}\t\t{estimated_prob_B[i]:.6f}\t\t{estimated_prob_C[i]:.6f}")
+  print("")  # Empty line for separation
+
+if __name__ == "__main__":
+  main()
+calculate_conditional_probs()

hw4/problem1a.py

@@ -0,0 +1,45 @@
+
+
+# Defining the local conditional probability distributions
+P_F = {1: 0.01, 0: 0.99}
+P_E = {1: 0.02, 0: 0.98}
+
+# Probability of alarm given Fire, Earthquake, and Neighbor calls
+P_A_given_FBE = {
+    (1, 1, 1): 0.99,  # Alarm likely if fire, earthquake, and neighbor B calls
+    (1, 1, 0): 0.98,  # Alarm likely if fire, earthquake, but no call from B
+    (1, 0, 1): 0.95,  # Alarm likely if fire, no earthquake, but neighbor B calls
+    (1, 0, 0): 0.90,  # Alarm likely if fire, no earthquake, and no call from B
+    (0, 1, 1): 0.70,  # Alarm possible if earthquake and neighbor B calls, but no fire
+    (0, 1, 0): 0.60,  # Alarm possible if earthquake, no fire, and no call from B
+    (0, 0, 1): 0.20,  # Alarm less likely if no fire, no earthquake, but neighbor B calls
+    (0, 0, 0): 0.01   # Alarm very unlikely if no fire, no earthquake, and no call from B
+}
+
+P_B_given_A = {1: 0.90, 0: 0.05}  # Probability of neighbor B calling given alarm state
+P_C_given_A = {1: 0.70, 0.01: 0.01}  # Probability of neighbor C calling given alarm state
+
+
+def joint_probability(f, e, a, b, c):
+  return (P_F[f] * P_E[e] * P_A_given_FBE[(f, e, b)] *
+          P_B_given_A[a if b else 1-a] *
+          P_C_given_A.get(a if c else 1-a, 0.5))
+
+
+# Calculating total probabilities where B=1 and C=1 respectively
+total_P_B_1 = sum([joint_probability(f, e, a, 1, c) for f in [True, False] for e in [True, False]
+                   for a in [True, False] for c in [True, True]])  # Only consider B=1
+total_P_C_1 = sum([joint_probability(f, e, a, b, 1) for f in [True, False] for e in [True, False]
+                   for a in [True, False] for b in [True, True]])  # Only consider C=1
+
+# Calculate P(F=1 | B=1)
+P_F_given_B = total_P_B_1 / sum([joint_probability(f, e, a, 1, c) for f in [True, False] for e in [True, False]
+                                 for a in [True, False] for c in [True, True]])
+
+# Calculate P(F=1 | C=1)
+P_F_given_C = total_P_C_1 / sum([joint_probability(f, e, a, b, 1) for f in [True, False] for e in [True, False]
+                                 for a in [True, False] for b in [True, True]])
+
+# Print the results
+print(f"P(F=1 | B=1): {P_F_given_B:.6f}")
+print(f"P(F=1 | C=1): {P_F_given_C:.6f}")

main.aux

@@ -19,4 +19,5 @@
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 \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Inference in Bayesian Networks}{15}{}\protected@file@percent }
-\gdef \@abspage@last{16}
+\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Gradient Descent}{15}{}\protected@file@percent }
+\gdef \@abspage@last{17}

main.fdb_latexmk

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main.fls

@@ -112,6 +112,20 @@ INPUT .\hw3\motorcade.png
 INPUT .\hw3\motorcade.png
 INPUT .\hw3\motorcade.png
 INPUT .\hw3\motorcade.png
+INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
+INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
+INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
+INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
+INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
+INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
+INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
+INPUT .\hw4\Picture1.png
+INPUT .\hw4\Picture1.png
+INPUT .\hw4\Picture1.png
+INPUT .\hw4\Picture1.png
+INPUT .\hw4\Picture1.png
+INPUT .\hw4\Picture1.png
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 INPUT .\titlePage.aux

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main.tex

@@ -479,7 +479,40 @@ See attached submission.py file
 \begin{itemize}
     \item[a - c. ] Refer to bn.py
 
-    \item[d. ]
+    \item[d. ] Likelihood weighting gets more accurate as we use more samples. With more samples, the estimated probabilities get closer to the real chances we calculated before. This shows that using a bigger sample size is helpful for getting better results. 
+\end{itemize}
+
+\subsection{Gradient Descent}
+
+\begin{itemize}
+    \item[a. ] 
+    \begin{center}
+        \includegraphics[scale=0.5]{hw4/Gemini_Chart_Image_5v3caz5v3caz5v3c.png}
+    \end{center}
+
+
+    Hinge loss is a reasonable cost function for binary classification because it focuses on creating a good margin between classes. It only penalizes models when their predictions fall close to the wrong class, and ignores classified points even with a small margin. This makes the model prioritize a clear separation between positive and negative classes.
+
+    \item[b. ] 
+    
+    $$
+    \frac{\partial c(h(x), y)}{\partial w_0} = 0
+    $$
+
+    $$
+    \frac{\partial c(h(x), y)}{\partial w_1} = \begin{cases}
+        -y, & \text{if } w_0 + w_1x < 1 \\
+        0, & \text{if } w_0 + w_1x \geq 1
+        \end{cases}        
+    $$
+
+    \item[c. ]
+    
+    \begin{center}
+        \includegraphics[scale=0.3]{hw4/Picture1.png}
+    \end{center}
+
+
 \end{itemize}
 
 \end{document}