Reasoning & Uncertainty: Probability, Bayes, Dempster-Shafer, Fuzzy Logic

TAKEAWAYS:

• Probabilistic reasoning replaces deterministic logic when data is uncertain (e.g., "Is this email spam?").
• Bayesian networks model dependencies between events (e.g., "Does rain cause traffic jams?").
• Dempster-Shafer theory handles incomplete evidence better than probability (e.g., "Is this patient’s symptom A or B?").
• Fuzzy logic deals with vague categories (e.g., "Is this traffic speed ‘slow’ or ‘fast’?").
• Expected utility balances risk vs. reward (e.g., "Should a bank approve this loan?").
• Real-world AI (e.g., eSewa fraud detection, Ncell network routing) uses these to make decisions under uncertainty.

---

1. Why Uncertainty Matters in AI

AI systems rarely face 100% certainty. Real-world data is noisy, incomplete, or ambiguous. For example:

• eSewa must decide if a transaction is fraudulent (probability: 95% vs. 5%).
• Pathao estimates delivery times despite traffic unpredictability (fuzzy logic: "slow," "medium," "fast").
• Ncell’s 5G network routes calls probabilistically to avoid congestion.

Key Idea:

• Inference = deriving conclusions from known facts (e.g., "If it rains, roads are slippery").
• Reasoning = handling uncertainty (e.g., "There’s a 70% chance it will rain; should we cancel the trip?").

Visual: Inference vs. Reasoning

---

2. Probabilistic Reasoning

2.1 Basics: Probability Theory

Probability quantifies uncertainty using:

• Joint probability: $P(A\cap B)$ = chance of both A and B happening.
• Conditional probability: $P(A|B)$ = chance of A given B (e.g., "Probability of disease given symptoms").
• Bayes’ Theorem: $$P(H|E)=\frac{P(E|H)\cdot P(H)}{P(E)}$$
  • $P(H|E)$: Probability of hypothesis $H$ given evidence $E$.
  • Used in spam filters, medical diagnosis, and fraud detection.

Worked Example: eSewa Fraud Detection

• Hypothesis (H): "This transaction is fraudulent."
• Evidence (E): "User logged in from a new country."
• Given:
  • $P(H)=0.01$ (1% of transactions are fraudulent).
  • $P(E|H)=0.9$ (90% of frauds come from new countries).
  • $P(E)=0.05$ (5% of all transactions come from new countries).
• Calculate $P(H|E)$: P(H|E) = \frac{0.9 \times 0.01}{0.05} = 0.18 \quad \text{(18% chance of fraud)} Decision: Flag for review (threshold: >15%).

---

2.2 Bayesian Networks

A Bayesian network is a directed acyclic graph (DAG) where:

• Nodes = random variables (e.g., "Rain," "Traffic Jam").
• Edges = conditional dependencies.
• Each node has a Conditional Probability Table (CPT).

Example: Traffic Prediction for Kathmandu

• CPT for "Traffic Jam":

  Rain	Traffic Jam Probability
  No	0.1
  Yes	0.8

Worked Example: NTC Network Routing

• Variables:
  • $A$: "Server Overloaded" ($P(A)=0.3$).
  • $B$: "User Requests High" ($P(B|A)=0.9$, $P(B|\neg A)=0.2$).
  • $C$: "Route 1 Selected" ($P(C|B)=0.7$, $P(C|\neg B)=0.4$).
• Question: What’s $P(C)$?
  • Use law of total probability: $$P(C)=P(C|B)P(B)+P(C|\neg B)P(\neg B)$$
    • $P(B)=P(B|A)P(A)+P(B|\neg A)P(\neg A)=0.9\times 0.3+0.2\times 0.7=0.41$.
    • $P(\neg B)=1-0.41=0.59$.
    • $P(C)=0.7\times 0.41+0.4\times 0.59=0.573$ (57.3% chance Route 1 is selected).

---

3. Handling Incomplete Evidence: Dempster-Shafer Theory

Probability assumes all possibilities are known. Dempster-Shafer (DS) theory handles unknown or conflicting evidence by assigning:

• Belief (Bel): Minimum probability.
• Plausibility (Pl): Maximum probability.
• Uncertainty (U): $U=Pl-Bel$.

Example: Medical Diagnosis (Is the patient’s symptom A or B?)

• Evidence 1: "Doctor sees rash" → Believes 60% it’s A, 30% it’s B, 10% unknown.
• Evidence 2: "Patient reports fever" → Believes 50% it’s A, 40% it’s B, 10% unknown.
• Combined belief:
  • $Bel(A)=60\%+30\%\times 50\%=75\%$.
  • $Bel(B)=30\%+60\%\times 40\%=54\%$.
  • Decision: More likely A (75% > 54%).

Visual: Belief vs. Probability

---

4. Fuzzy Logic: Dealing with Vague Data

Fuzzy logic handles gradual transitions (e.g., "temperature is hot" vs. binary "hot/cold").

• Membership functions assign degrees of truth (0 to 1).
• Example: Traffic speed classification.

Worked Example: Pathao Delivery Speed

• Input: Speed = 30 km/h.
• Membership functions:
  • Slow: ${\mu }_{slow}(x)=\max (0,1-\frac{x-10}{20})$.
  • Medium: ${\mu }_{medium}(x)=\max (0,\min (1,\frac{x-10}{20},\frac{50-x}{20}))$.
  • Fast: ${\mu }_{fast}(x)=\max (0,\frac{x-30}{20})$.
• Calculations:
  • ${\mu }_{slow}(30)=0.5$.
  • ${\mu }_{medium}(30)=0.75$.
  • ${\mu }_{fast}(30)=0$.
• Output: "Speed is 50% slow, 75% medium" → Decision: "Medium speed; adjust ETA."

Visual: Fuzzy Membership Functions

---

5. Expected Utility Theory

AI agents should maximize expected utility, not just probability.

• Utility: Value of an outcome (e.g., profit, safety).
• Formula: $$EU=\sum P(o_i)\times U(o_i)$$ where $o_i$ = possible outcomes.

Worked Example: Bank Loan Approval (Nepal Bank)

• Scenario: Approve or reject a loan.
• Outcomes:
  • Approve:
    • Repayment: $P=0.7$, $U=+10,000$ (profit).
    • Default: $P=0.3$, $U=-5,000$ (loss).
  • Reject: $U=0$.
• Expected Utility:
  • Approve: $EU=0.7\times 10,000+0.3\times (-5,000)=5,500$.
  • Reject: $EU=0$.
• Decision: Approve (higher EU).

---

6. Real-World Applications

In the Real World

---

IMAGE: Real Outputs

---

7. Exam Tip

1. Differentiate clearly:
   • Probability = known outcomes; Dempster-Shafer = unknown outcomes.
   • Fuzzy logic = gradual truth; binary logic = 0/1.
2. Bayes’ Theorem is key:
   • Always show the formula and label $P(H)$, $P(E|H)$, $P(E)$.
3. Real-world mapping:
   • Link eSewa fraud to Bayesian networks.
   • Link Pathao speeds to fuzzy logic.
4. Expected Utility:
   • Compare $EU$ of two options (e.g., "Approve vs. reject loan").
5. Common Pitfalls:
   • Don’t confuse $P(A|B)$ with $P(B|A)$.
   • In DS theory, belief ≠ probability.

---

Final Note: Uncertainty is everywhere in AI. Master these tools to build systems that make decisions despite imperfect data—just like eSewa, Pathao, and banks do every day.