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Bayesian Networks

Bayesian networks — DAG probabilistic models
৮ মিনিট পড়া উচ্চ · Advanced pgmpy

এই পাঠে যা শিখবেন

  • Probabilistic graphical model কী — Bayes-এর গভীর প্রয়োগ
  • DAG ও conditional probability table (CPD)
  • Chain rule decomposition — কেন compact
  • d-separation — independence reasoning
  • pgmpy দিয়ে Bangladesh medical diagnosis (TB diagnosis network)

১ · কেন Bayesian Network?

১০টি binary variable-এর joint distribution — $2^{10} = 1024$ probability। ৫০টি variable — $2^{50} \approx 10^{15}$। কোনোভাবেই storage-এ আঁটে না। কিন্তু — বাস্তবে variable-গুলো সব pairwise dependent না। "Smoking → cancer → cough" — এই structure থাকলে শুধু conditional probability $P(\text{cough} \mid \text{cancer})$ store করলেই হলো।

Judea Pearl (১৯৮৮) — Bayesian Network formalize। ২০১১-তে Turing award। Causal AI-র গোড়াপত্তন। Bayesian Network মূল idea — independence structure exploit করে exponential → polynomial।

Bayesian Network = DAG + CPDs

Node = variable। Edge $X \to Y$ মানে $X$ "causes" বা directly influences $Y$। Cycle না (acyclic)। প্রতিটি node-এ একটি conditional probability distribution $P(X_i \mid \text{Pa}(X_i))$ — parent given তার distribution।

২ · Joint distribution decomposition

Chain rule of probability:

$$P(X_1, \ldots, X_n) = \prod_{i=1}^n P(X_i \mid X_1, \ldots, X_{i-1})$$

Bayesian Network-এর মূল সরলীকরণ — $X_i$ শুধু তার parents-এর উপর conditional, অন্য ancestor-এর সাথে independent (given parents):

$$P(X_1, \ldots, X_n) = \prod_{i=1}^n P(X_i \mid \text{Pa}(X_i))$$

৫টি binary variable, যদি প্রতি node-এর সর্বোচ্চ ২ parent — store মাত্র $5 \times 2^3 = 40$ probability (full joint $2^5 = 32$, তুলনীয়)। ১০ node, max ৩ parent → $10 \times 2^4 = 160$ vs $2^{10} = 1024$ — ৬× সাশ্রয়। ৫০ node-এ পার্থক্য astronomic।

৩ · উদাহরণ — Sprinkler network

চার variable:

  • Cloudy (C)
  • Sprinkler (S) — depends on Cloudy (cloudy হলে sprinkler off)।
  • Rain (R) — depends on Cloudy।
  • WetGrass (W) — depends on Sprinkler ও Rain।

Joint:

$$P(C, S, R, W) = P(C) \cdot P(S \mid C) \cdot P(R \mid C) \cdot P(W \mid S, R)$$

Full joint হত $2^4 = 16$ entry। CPD-এ — $P(C)$ ১, $P(S|C)$ ২, $P(R|C)$ ২, $P(W|S,R)$ ৪ → মোট ৯ entry।

৪ · Conditional independence ও d-separation

Bayesian Network-এ structural information directly conditional independence indicate করে। দু'টি node X ও Y — given Z — independent কি? এই উত্তর "d-separation" rule দিয়ে।

তিন basic structure:

  • Chain $X \to Z \to Y$: Z observed হলে X ⊥ Y। (Smoking → cancer → cough; cancer জানলে smoking ও cough independent।)
  • Fork $X \leftarrow Z \to Y$: Z observed হলে X ⊥ Y। (Common cause — সূর্য ☀️ — উভয় ice cream sale ও sunburn।)
  • V-structure (collider) $X \to Z \leftarrow Y$: Z observed হলে X ও Y dependent! (Burglary, earthquake → alarm; alarm shunlei দু'টোই possible cause।)

এই counter-intuitive collider behavior — Pearl-এর breakthrough। "Explain away" reasoning — alarm বাজলে যদি earthquake confirm — burglary-র probability কমে।

৫ · Inference — query answer

"Wet grass হয়েছে — cloudy ছিল কি?" — এটা $P(C \mid W)$ compute। Methods:

  • Exact inference:
    • Variable elimination — sum out non-query variables।
    • Junction tree algorithm — DAG → tree → message passing।
    • Polynomial in tree-width, exponential worst-case।
  • Approximate inference:
    • Sampling — rejection, importance, Gibbs (L40)।
    • Variational methods — distribution approximation।
    • Loopy belief propagation।

৬ · Learning — structure ও parameters

Parameter learning: structure given হলে — CPT-গুলো MLE বা Bayesian estimation থেকে।

Structure learning: সবচেয়ে কঠিন। DAG-এর exponential search space।

  • Score-based — BIC, BDe — search best DAG।
  • Constraint-based — conditional independence tests।
  • Hybrid — structure prior + data।

Practice-এ — domain expert structure দেন, parameter data থেকে learn।

Bayesian Network — Sprinkler example Joint = ∏ P(node | parents) Cloudy P(C) Sprinkler P(S|C) Rain P(R|C) WetGrass P(W|S,R) P(C) T 0.5 / F 0.5 P(S|C) C=T → S=T 0.1 C=F → S=T 0.5 P(R|C) C=T → R=T 0.8 C=F → R=T 0.2 P(W|S,R) — 4 entries
Sprinkler Bayesian Network — Cloudy → {Sprinkler, Rain} → WetGrass। Joint distribution ৪টি local CPD থেকে। Inference — query "WetGrass given, Cloudy probability?"।

৭ · pgmpy দিয়ে Bangladesh TB diagnosis

Smoking → TB → Cough; Pollution (Dhaka air) → TB-এর independent contributor। Goal — cough দেখে TB-এর probability।

Python · pgmpy 
# pip install pgmpy
from pgmpy.models import DiscreteBayesianNetwork
from pgmpy.factors.discrete import TabularCPD
from pgmpy.inference import VariableElimination

model = DiscreteBayesianNetwork([
    ("Smoking", "TB"),
    ("Pollution", "TB"),
    ("TB", "Cough"),
    ("TB", "XrayPositive"),
])

cpd_smoke = TabularCPD("Smoking", 2, [[0.7], [0.3]])  # 30% smoke
cpd_poll  = TabularCPD("Pollution", 2, [[0.4], [0.6]])  # 60% Dhaka heavy
cpd_tb    = TabularCPD("TB", 2,
    # P(TB | Smoking, Pollution) — order: (S=0,P=0), (S=0,P=1), (S=1,P=0), (S=1,P=1)
    values=[[0.99, 0.95, 0.90, 0.70],   # TB = no
            [0.01, 0.05, 0.10, 0.30]],  # TB = yes
    evidence=["Smoking", "Pollution"], evidence_card=[2, 2])
cpd_cough = TabularCPD("Cough", 2,
    values=[[0.80, 0.20],
            [0.20, 0.80]],
    evidence=["TB"], evidence_card=[2])
cpd_xray  = TabularCPD("XrayPositive", 2,
    values=[[0.95, 0.10],
            [0.05, 0.90]],
    evidence=["TB"], evidence_card=[2])

model.add_cpds(cpd_smoke, cpd_poll, cpd_tb, cpd_cough, cpd_xray)
print("Model valid:", model.check_model())

# Inference
infer = VariableElimination(model)

# Q1: P(TB | Cough = yes)
q1 = infer.query(["TB"], evidence={"Cough": 1})
print("\nP(TB | Cough=yes):", q1.values)

# Q2: P(TB | Cough = yes, XrayPositive = yes)
q2 = infer.query(["TB"], evidence={"Cough": 1, "XrayPositive": 1})
print("P(TB | Cough=yes, Xray+):", q2.values)

    


    

      Cough alone — TB-এর probability marginally বাড়ে (~১৫%)। কিন্তু X-ray positive যোগ করলে — probability dramatically rise (~৭৫%)। Bayesian network multi-evidence smoothly integrate — doctor-এর reasoning-এর mathematical model।
    


    
৮ · Bayesian Network বনাম Naive Bayes

    

      Naive Bayes (L18) একটি বিশেষ Bayesian Network — class node অন্য সব feature-এর parent, feature-গুলো একে অপরের সাথে independent (given class)।
    


    
    
      
  • Naive Bayes: simple, fast, high-bias। Class → all features (star structure)।
    • 
      
  • Tree-Augmented Naive Bayes (TAN): features-এর মধ্যে limited dependency।
    • 
      
  • General Bayesian Network: arbitrary DAG, any structure।
    • 
    


    
৯ · কখন Bayesian Network

    
    
      
  • Domain knowledge structured: medical, fault diagnosis।
    • 
      
  • Causal reasoning দরকার: "what if" interventions।
    • 
      
  • Missing data robust: evidence partial — inference কাজ করে।
    • 
      
  • Interpretable: graph + CPD — human readable।
    • 
      
  • Avoid: high-dim feature-এ overkill, deep learning ভাল।
    • 
    


    

      Bayesian Network "correlation ≠ causation" সমস্যা সমাধান করে না — শুধু conditional independence encode। প্রকৃত causal inference-এ Pearl-এর do-calculus, structural causal models লাগে। কিন্তু Bayesian Network — সেই পথের প্রথম স্তর।
    


  


  

    
    
  


  
  

    
 ভাবনার প্রশ্ন

    
প্রতিটি প্রশ্ন নিজে কিছুক্ষণ ভাবুন — তারপর "→ উত্তর" চাপুন।


    

      
        প্র ০১
        "Explain away" — V-structure-এর counter-intuitive behavior কী? Real-world implication?
      
      

        

          চমৎকার Bayesian Network-এর সবচেয়ে gripping concept।
        

        

          Setup — Burglary alarm:
        

        
    
          
  • Burglary (B) — independent।
    • 
          
  • Earthquake (E) — independent।
    • 
          
  • Alarm (A) — depends on B এবং E।
    • 
          
  • B ও E marginally independent (no edge)।
    • 
        

        

          Marginal:
        

        
    
          
  • P(B) = 0.001।
    • 
          
  • P(E) = 0.002।
    • 
          
  • P(B, E) = P(B) × P(E) = 0.000002 (independent)।
    • 
        

        

          Alarm given:
        

        
    
          
  • Alarm বাজছে — B অথবা E (অথবা দু'টি)।
    • 
          
  • P(B | A) — B-এর probability বাড়ে (এটা causal contributor)।
    • 
          
  • P(B | A, E) — E দেখানে যাচ্ছে cause। B-এর probability কমে — "explained away"।
    • 
          
  • Counter-intuitive: B ও E marginally independent ছিল, কিন্তু A দেখে — দু'টি negatively correlated।
    • 
        

        

          Mathematical:
        

        
    
          
  • P(B | A, E) < P(B | A) — explaining away।
    • 
          
  • একটি cause confirm — অন্যটির need কমে।
    • 
        

        

          Real-world implications:
        

        

          (১) Medical diagnosis:
        

        
    
          
  • Patient cough — TB or pneumonia?
    • 
          
  • Pneumonia confirmed → TB-এর posterior কমে।
    • 
          
  • একটি সঠিক diagnosis অন্যকে rule out।
    • 
        

        

          (২) Fault detection:
        

        
    
          
  • Server down — disk failure or network issue?
    • 
          
  • Network confirmed bad — disk-এর probability decline।
    • 
        

        

          (৩) Causal misinterpretation:
        

        
    
          
  • "Smart students attend top universities" — talent ও hard work both contribute।
    • 
          
  • Top university seen — talent confirmed → hard work-এর likelihood কমে?
    • 
          
  • দু'টি skill-ই দেখা যায় না collider-এর through।
    • 
        

        

          Berkson's paradox:
        

        
    
          
  • Hospital data — sick + injured patients overrepresented।
    • 
          
  • Hospitalization status (collider) selecting on।
    • 
          
  • Sickness ও injury seem negatively correlated in hospital, but not in population।
    • 
        

        

          Causal AI critical:
        

        
    
          
  • Conditioning on collider creates spurious correlation।
    • 
          
  • "Controlling for everything" can introduce bias, not remove।
    • 
          
  • Pearl's "Book of Why" — central theme।
    • 
        

        

          Practical:
        

        
    
          
  • Statistical model — be careful selecting variables।
    • 
          
  • Causal DAG draw — collider identify।
    • 
          
  • Don't condition on collider.
    • 
        

        

          Bangladesh case:
        

        
    
          
  • Loan approval data — only approved loans visible।
    • 
          
  • Approval (collider) → income ও credit score correlation distorted।
    • 
          
  • Counter-intuitive — high income borrowers default more in dataset (selection bias)।
    • 
        

        

          মূল উপলব্ধি: Marginal independence ≠ conditional independence। Bayesian Network — এই subtle relationship visualize ও exploit। Real causal reasoning-এর foundation।
        

      

    


    

      
        প্র ০২
        Bayesian Network বনাম neural network — কোথায় কোনটা? Hybrid possible?
      
      

        

          ভিন্ন paradigm — শক্তি ও দুর্বলতা complementary।
        

        

          Bayesian Network strength:
        

        
    
          
  • Interpretable — graph human-readable।
    • 
          
  • Causal reasoning — interventions।
    • 
          
  • Missing data graceful — partial evidence inference।
    • 
          
  • Uncertainty native — probability output।
    • 
          
  • Domain knowledge integrate — expert prior।
    • 
          
  • Small data effective।
    • 
        

        

          Bayesian Network weakness:
        

        
    
          
  • Discrete/few-state continuous — high-D continuous struggle।
    • 
          
  • Structure learning intractable।
    • 
          
  • Inference exponential worst-case।
    • 
          
  • Scaling — million-row hard।
    • 
        

        

          Neural network strength:
        

        
    
          
  • Big data shine।
    • 
          
  • High-D, continuous — image, text।
    • 
          
  • Feature learning automatic।
    • 
          
  • End-to-end optimization।
    • 
          
  • State-of-art performance।
    • 
        

        

          Neural network weakness:
        

        
    
          
  • Black box — interpretability difficult।
    • 
          
  • Causal — correlation only।
    • 
          
  • Missing data — imputation needed।
    • 
          
  • Uncertainty — calibration extra effort।
    • 
          
  • Domain knowledge integrate hard।
    • 
          
  • Small data — overfit risk।
    • 
        

        

          কখন Bayesian Network:
        

        
    
          
  • Medical diagnosis — interpretability critical।
    • 
          
  • Risk assessment — uncertainty quantified।
    • 
          
  • Fault diagnosis — causal chain।
    • 
          
  • Decision support — domain expert involve।
    • 
          
  • Small structured data।
    • 
        

        

          কখন neural network:
        

        
    
          
  • Image classification — pixels থেকে category।
    • 
          
  • Speech recognition।
    • 
          
  • NLP — translation, summarization।
    • 
          
  • Recommendation system — million users।
    • 
          
  • End-to-end learning।
    • 
        

        

          Hybrid approaches:
        

        

          (১) Bayesian deep learning:
        

        
    
          
  • Neural network weights — distribution, not point।
    • 
          
  • Variational inference, MCMC।
    • 
          
  • Uncertainty in deep models।
    • 
          
  • BNN, Bayes by Backprop।
    • 
        

        

          (২) Probabilistic programming:
        

        
    
          
  • PyMC, Pyro, Stan।
    • 
          
  • Bayesian model — neural network components allowed।
    • 
          
  • Best of both।
    • 
        

        

          (৩) Deep generative model:
        

        
    
          
  • VAE — neural encoder + probabilistic latent।
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  • Normalizing flows।
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  • Diffusion models।
    • 
        

        

          (৪) Causal neural network:
        

        
    
          
  • Causal discovery via NN (NOTEARS algorithm)।
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  • Counterfactual reasoning ML।
    • 
          
  • Pearl's framework + deep learning।
    • 
        

        

          (৫) Neuro-symbolic AI:
        

        
    
          
  • Symbolic structure (Bayesian Network) + neural feature।
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  • DeepProbLog।
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  • Promising frontier।
    • 
        

        

          Bangladesh applications:
        

        
    
          
  • Loan default — Bayesian Network for feature relationship + neural for transaction sequence।
    • 
          
  • Disease diagnosis — knowledge graph + image CNN।
    • 
          
  • Crop yield — weather BN + drone image NN।
    • 
        

        

          মূল উপলব্ধি: Bayesian Network knowledge engineering। Neural network end-to-end learning। Real-world AI — both, not either-or।
        

      

    


    

      
        প্র ০৩
        Structure learning কেন NP-hard? Practical heuristics কী?
      
      

        

          DAG-এর search space combinatorial — তাই NP-hardness।
        

        

          Search space size:
        

        
    
          
  • $n$ node-এর জন্য DAG count super-exponential।
    • 
          
  • $n=5$ → 29281, $n=10$ → 4.18 × 10¹⁸, $n=20$ → 2.34 × 10⁷²।
    • 
          
  • Robinson formula।
    • 
        

        

          NP-hardness proof:
        

        
    
          
  • Chickering (১৯৯৬) — minimum description length DAG NP-hard।
    • 
          
  • Bayesian score optimal DAG NP-complete।
    • 
        

        

          Approach categories:
        

        

          (১) Score-based:
        

        
    
          
  • BIC, AIC, BDe — DAG-এর "fit" score।
    • 
          
  • Greedy hill climbing — random DAG → local edge addition/removal।
    • 
          
  • Tabu search, simulated annealing।
    • 
          
  • Local optima trap risk।
    • 
        

        

          (২) Constraint-based:
        

        
    
          
  • PC algorithm (Spirtes-Glymour-Scheines)।
    • 
          
  • Conditional independence tests (chi-square, mutual info)।
    • 
          
  • Independence pattern → DAG skeleton।
    • 
          
  • V-structure orientation rules।
    • 
        

        

          (৩) Hybrid:
        

        
    
          
  • MMHC — Max-Min Hill Climbing।
    • 
          
  • Constraint-based skeleton + score-based orientation।
    • 
        

        

          (৪) Continuous optimization:
        

        
    
          
  • NOTEARS (২০১৮) — DAG constraint as smooth penalty।
    • 
          
  • Gradient-based optimization।
    • 
          
  • Modern, scalable।
    • 
        

        

          (৫) Bayesian:
        

        
    
          
  • Posterior over DAGs।
    • 
          
  • MCMC sampling structures।
    • 
          
  • Order MCMC — efficient।
    • 
        

        

          Practical tricks:
        

        
    
          
  • Domain knowledge: structural prior — known edges।
    • 
          
  • Variable ordering: if known, search reduces।
    • 
          
  • Sparsity prior: few-edge DAG preferred।
    • 
          
  • Bootstrap: resampling stable edges।
    • 
        

        

          Software:
        

        
    
          
  • bnlearn (R) — comprehensive।
    • 
          
  • pgmpy — Python।
    • 
          
  • causal-learn — modern।
    • 
          
  • NOTEARS — neural causal discovery।
    • 
        

        

          Validation:
        

        
    
          
  • Held-out likelihood।
    • 
          
  • Domain expert review।
    • 
          
  • Causal interpretation sanity check।
    • 
        

        

          Open problems:
        

        
    
          
  • Latent confounder — fundamental ambiguity।
    • 
          
  • Causal direction from observational data — sometimes impossible।
    • 
          
  • Equivalence class — multiple DAGs same independencies।
    • 
        

        

          মূল উপলব্ধি: Structure learning hard, but practical heuristics work decently। Domain expert সবচেয়ে valuable input। Pure data-driven discovery এখনও open research।
        

      

    


    

      
        প্র ০৪
        Bangladesh-এর rural health centers-এ Bayesian Network deploy — TB, dengue, typhoid diagnosis support। Pipeline ও challenges?
      
      

        

          Real Bangladesh public health impact — clinical decision support system।
        

        

          Use case:
        

        
    
          
  • Rural community health worker — patient evaluate।
    • 
          
  • Symptom + basic test → likely diagnosis suggestion।
    • 
          
  • Doctor unavailable area — first-line guidance।
    • 
        

        

          (১) Diseases ও symptoms:
        

        
    
          
  • TB: cough, fever, weight loss, night sweat, X-ray।
    • 
          
  • Dengue: high fever, headache, rash, platelet drop।
    • 
          
  • Typhoid: prolonged fever, abdominal pain, Widal test।
    • 
          
  • Common cold, malaria — confounders।
    • 
        

        

          (২) Bayesian Network structure:
        

        
    
          
  • Risk factors (smoking, mosquito exposure, season) → diseases।
    • 
          
  • Disease → symptoms।
    • 
          
  • Disease → test results।
    • 
          
  • Tree-like structure mostly।
    • 
        

        

          (৩) CPT acquisition:
        

        
    
          
  • Medical literature — base rates।
    • 
          
  • Hospital data — Bangladesh prevalence।
    • 
          
  • Expert elicitation — doctor input।
    • 
          
  • Local prior — district-specific (dengue urban, malaria hill area)।
    • 
        

        

          (৪) Inference:
        

        
    
          
  • Variable elimination — exact, suitable for ৫-১০ disease network।
    • 
          
  • Real-time on tablet feasible।
    • 
        

        

          (৫) UI/UX:
        

        
    
          
  • Bangla-only interface।
    • 
          
  • Symptom checkbox।
    • 
          
  • Test result entry।
    • 
          
  • Top-3 diagnosis with probability।
    • 
          
  • Explanation — "fever + low platelet → dengue likely"।
    • 
        

        

          (৬) Validation:
        

        
    
          
  • Retrospective hospital records — accuracy benchmark।
    • 
          
  • Doctor agreement study।
    • 
          
  • Pilot at health center — feedback।
    • 
        

        

          (৭) Challenges:
        

        
    
          
  • Data quality: rural health record incomplete।
    • 
          
  • Local prevalence: haor area different from char area।
    • 
          
  • Atypical presentation: subset symptom only।
    • 
          
  • Co-infection: TB + diabetes common।
    • 
          
  • Cultural: patient symptom report bias।
    • 
          
  • Connectivity: offline-first requirement।
    • 
          
  • Trust: health worker AI skepticism।
    • 
          
  • Liability: wrong suggestion legal/medical।
    • 
        

        

          (৮) Ethical:
        

        
    
          
  • Decision support, not replacement।
    • 
          
  • Probabilistic uncertainty visible।
    • 
          
  • "When in doubt, refer to doctor" prominent।
    • 
          
  • Health worker training mandatory।
    • 
        

        

          (৯) Continuous improvement:
        

        
    
          
  • Regular CPT update — new prevalence data।
    • 
          
  • Feedback loop — confirmed diagnosis।
    • 
          
  • Drift detection — symptom pattern shift।
    • 
        

        

          (১০) Beyond Bayesian Network:
        

        
    
          
  • X-ray AI (CNN) — TB detection।
    • 
          
  • Smartphone fever camera — temperature।
    • 
          
  • Hybrid — Bayesian aggregation, neural feature।
    • 
        

        

          Stakeholders:
        

        
    
          
  • icddr,b — medical research।
    • 
          
  • BRAC — health worker network।
    • 
          
  • DGHS — government endorsement।
    • 
          
  • WHO Bangladesh — international support।
    • 
        

        

          মূল উপলব্ধি: Bayesian Network — Bangladesh rural health-এ tangible AI। Interpretability + uncertainty + small data — neural network-এর তুলনায় ideal। Implementation engineering + clinical partnership — সমান গুরুত্বপূর্ণ।
        

      

    

  


  

    
 অনুশীলন

    
    
      
  1. 
        হিসাব করুন: $A \to B \to C$ chain। P(A=T)=0.5, P(B=T|A=T)=0.8, P(B=T|A=F)=0.2, P(C=T|B=T)=0.9, P(C=T|B=F)=0.1। P(C=T) কত?
        
    
          
          
    
            
      
              
    • P(B=T) = P(B=T|A=T)P(A=T) + P(B=T|A=F)P(A=F) = 0.8×0.5 + 0.2×0.5 = 0.5।
      • 
              
    • P(C=T) = P(C=T|B=T)P(B=T) + P(C=T|B=F)P(B=F) = 0.9×0.5 + 0.1×0.5 = 0.5।
      • 
            
    
          
    
        
    
      
    2. 

      
  2. 
        pgmpy-এ চেষ্টা: উপরের chain network কোড করুন।
        
    
          
          
    
            
    from pgmpy.models import DiscreteBayesianNetwork
from pgmpy.factors.discrete import TabularCPD
from pgmpy.inference import VariableElimination

m = DiscreteBayesianNetwork([("A", "B"), ("B", "C")])
m.add_cpds(
    TabularCPD("A", 2, [[0.5], [0.5]]),
    TabularCPD("B", 2, [[0.8, 0.2], [0.2, 0.8]],
               evidence=["A"], evidence_card=[2]),
    TabularCPD("C", 2, [[0.1, 0.9], [0.9, 0.1]],
               evidence=["B"], evidence_card=[2]),
)
infer = VariableElimination(m)
print(infer.query(["C"]).values)
    
          
    
        
    
      
    3. 

      
  3. 
        ভাবুন: Bangladesh নৌকা-দুর্ঘটনা risk model — weather, overcrowding, pilot experience, equipment quality। Bayesian Network DAG আঁকুন।
        
    
          
          
    
            
      
              
    • Weather (storm) → Wave height।
      • 
              
    • Pilot experience → Pilot decisions।
      • 
              
    • Equipment quality → Mechanical failure risk।
      • 
              
    • Overcrowding → Stability।
      • 
              
    • {Wave height, Pilot decisions, Mechanical failure, Stability} → Accident।
      • 
              
    • Risk factors independent (mostly), accident V-structure collider।
      • 
              
    • Use: probability of accident inference, intervention prioritize।
      • 
              
    • Action: equipment improvement vs pilot training cost-benefit।
      • 
            
    
          
    
        
    
      
    4. 
    

  


  
  

    
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