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HMM — Hidden Markov Models

Hidden Markov Models — sequential probabilistic models
৮ মিনিট পড়া উচ্চ · Advanced hmmlearn

এই পাঠে যা শিখবেন

  • HMM-এর তিন core component — initial, transition, emission
  • Markov assumption ও সীমাবদ্ধতা
  • HMM-এর তিন core problem — likelihood, decoding, learning
  • hmmlearn দিয়ে hands-on — Bangla weather/sentiment sequence
  • NLP-তে HMM ও তার আধুনিক বিকল্প

১ · Sequential data — কেন HMM?

বাস্তবে অনেক ডেটাই sequential — শব্দের sequence (sentence), সময়ের সাথে stock price, ECG signal, DNA sequence। প্রতিটি element-কে independently treat করলে — temporal pattern হারাই। HMM (Baum-Petrie, ১৯৬৬; Rabiner ১৯৮৯ tutorial) — sequence-এর underlying state Markov চেইন assume করে।

Hidden: state সরাসরি দেখা যায় না। যেমন — speech-এ phoneme observable নয়, audio signal observable। POS tagging-এ — শব্দ visible, tag (noun/verb) hidden।

Markov: next state শুধু current-এর উপর depends — past-এর full history-র উপর নয়।

HMM-এর পাঁচ component

১) States $S = \{s_1, \ldots, s_N\}$ — hidden states।
২) Observations $O = \{o_1, \ldots, o_M\}$ — possible visible outputs।
৩) Transition matrix $\mathbf{A}$ — $a_{ij} = P(s_j \mid s_i)$।
৪) Emission matrix $\mathbf{B}$ — $b_i(o) = P(o \mid s_i)$।
৫) Initial distribution $\boldsymbol{\pi}$ — $\pi_i = P(s_i \text{ at } t=1)$।

২ · Joint probability

State sequence $\mathbf{s} = (s_1, \ldots, s_T)$, observation sequence $\mathbf{o} = (o_1, \ldots, o_T)$। Joint:

$$P(\mathbf{s}, \mathbf{o}) = \pi_{s_1} \cdot b_{s_1}(o_1) \cdot \prod_{t=2}^T a_{s_{t-1} s_t} \cdot b_{s_t}(o_t)$$

প্রতিটি step — previous state থেকে transition + current state-এ emission। Bayesian Network notation-এ — state ও observation node-এর DAG (L36)।

৩ · তিন core problem

Rabiner-এর classic ১৯৮৯ paper — তিন প্রশ্নই HMM-এর সবকিছু:

  • Problem 1 — Likelihood (Evaluation):
    Model $\lambda = (\mathbf{A}, \mathbf{B}, \boldsymbol{\pi})$ ও observation $\mathbf{o}$ given — $P(\mathbf{o} \mid \lambda)$ কত?
    Solution: Forward algorithm (dynamic programming) — exponential থেকে polynomial।
  • Problem 2 — Decoding:
    $\mathbf{o}$ ও $\lambda$ given — সবচেয়ে likely state sequence $\mathbf{s}^*$?
    Solution: Viterbi algorithm (L38)। Most-likely path find।
  • Problem 3 — Learning:
    $\mathbf{o}$ given — best $\lambda$ কী?
    Solution: Baum-Welch algorithm (L38) — EM-এর বিশেষ রূপ।

৪ · Forward algorithm — likelihood

Naive: সব $N^T$ state sequence enumerate — exponential, infeasible।

Forward DP — $\alpha_t(i) = P(o_1, \ldots, o_t, s_t = i \mid \lambda)$:

$$\alpha_1(i) = \pi_i b_i(o_1)$$

$$\alpha_{t+1}(j) = \left[ \sum_i \alpha_t(i) \cdot a_{ij} \right] b_j(o_{t+1})$$

$$P(\mathbf{o} \mid \lambda) = \sum_i \alpha_T(i)$$

Complexity $O(N^2 T)$ — feasible।

৫ · উদাহরণ — Dhaka weather → wear

Hidden states: Weather $\in$ {Sunny, Rainy}।

Observations: যা মানুষ পরছে $\in$ {Umbrella, Sunglasses, Raincoat}।

Transition $\mathbf{A}$: Sunny→Sunny ০.৭, Sunny→Rainy ০.৩, Rainy→Sunny ০.৪, Rainy→Rainy ০.৬।

Emission $\mathbf{B}$: Sunny → Umbrella ০.১, Sunglasses ০.৭, Raincoat ০.২। Rainy → Umbrella ০.৬, Sunglasses ০.১, Raincoat ০.৩।

মানুষ পরপর Umbrella, Raincoat, Umbrella পরেছে — weather sequence কী ছিল? এটাই decoding problem।

HMM — hidden states + observations through time Markov chain on states, emission per step t = 1 t = 2 t = 3 t = 4 s₁ s₂ s₃ s₄ hidden a (transition) o₁ o₂ o₃ o₄ visible b (emission) 3 core problems: Likelihood (forward), Decoding (Viterbi), Learning (Baum-Welch)
HMM — hidden state Markov chain (উপরে), প্রতি state থেকে observation emit (নিচে)। আমরা শুধু observation দেখি; state inference দরকার।

৬ · hmmlearn — Gaussian HMM

Discrete observation নয়, continuous (e.g., audio MFCC, stock price)? — Gaussian emission।

Python · hmmlearn 
# pip install hmmlearn
import numpy as np
from hmmlearn import hmm

# 2 hidden regime: bull / bear stock market
np.random.seed(0)
n_samples = 500

# bull regime: positive return mean 0.5%
# bear regime: negative return mean -0.5%
# alternating 50-step blocks
returns = []
for i in range(10):
    if i % 2 == 0:
        returns.append(np.random.normal(0.005, 0.01, 50))
    else:
        returns.append(np.random.normal(-0.005, 0.02, 50))
X = np.concatenate(returns).reshape(-1, 1)

model = hmm.GaussianHMM(n_components=2, covariance_type="full",
                        n_iter=100, random_state=0)
model.fit(X)

states = model.predict(X)
print(f"Inferred regime sequence (first 60): {states[:60]}")
print(f"\nRegime means: {model.means_.ravel().round(4)}")
print(f"Transition matrix:\n{model.transmat_.round(3)}")
print(f"Score (log-likelihood): {model.score(X):.2f}")

    


    

      Model দু'টি regime detect — positive ও negative mean। Transition matrix-এ diagonal probability বেশি (regime persist)। Stock market regime detection — DSE (Dhaka Stock Exchange) analysis-এ classical।
    


    
৭ · NLP-তে HMM — POS tagging

    

      Bangla sentence — শব্দ visible, POS tag (বিশেষ্য, ক্রিয়া) hidden। HMM-এর textbook application।
    


    

      

        
        Python · hmmlearn (categorical)
      

      
from hmmlearn import hmm

# 3 POS tag, 5 word vocabulary (toy)
# states: 0=Noun, 1=Verb, 2=Adj
# words : 0=cat, 1=run, 2=red, 3=fast, 4=dog

# transition: noun→verb common, verb→noun common
A = np.array([[0.1, 0.6, 0.3],   # Noun → ?
              [0.7, 0.1, 0.2],   # Verb → ?
              [0.6, 0.2, 0.2]])  # Adj  → ?

# emission: noun emits cat/dog, verb emits run, adj emits red/fast
B = np.array([[0.4, 0.05, 0.05, 0.1, 0.4],   # Noun
              [0.1, 0.7,  0.1,  0.05, 0.05], # Verb
              [0.05, 0.05, 0.45, 0.4, 0.05]])# Adj
pi = np.array([0.6, 0.2, 0.2])

model = hmm.CategoricalHMM(n_components=3)
model.startprob_ = pi
model.transmat_ = A
model.emissionprob_ = B

# observe word sequence: cat run fast dog
obs = np.array([[0], [1], [3], [4]])
logprob, states = model.decode(obs, algorithm="viterbi")

tag_names = ["Noun", "Verb", "Adj"]
print("Sequence: cat run fast dog")
print("Tags:    ", " ".join(tag_names[s] for s in states))
print(f"Log-probability: {logprob:.3f}")

      

        
        
      

      

    


    

      Viterbi decode — most-likely tag sequence। "cat → Noun, run → Verb, fast → Adj, dog → Noun" — natural। Transition + emission combine, দু'টি signal balance।
    


    
৮ · HMM-এর সীমাবদ্ধতা

    
    
      
  • Markov: long-distance dependency miss। "He, who studied hard last year, now ___" — gap বড় হলে।
    • 
      
  • Independence: emission শুধু current state-এ depend — context-aware emission impossible।
    • 
      
  • Discrete state: continuous latent (mixed)।
    • 
      
  • Local optima: Baum-Welch local maximum।
    • 
      
  • Modern alternatives: RNN, LSTM, Transformer — context-aware, better।
    • 
    


    
৯ · কখন HMM আজও relevant

    
    
      
  • Small data: deep learning data-hungry — HMM ১০০ sample-এও কাজ।
    • 
      
  • Interpretable: state, transition explicit।
    • 
      
  • Online/streaming: real-time decode।
    • 
      
  • Domain knowledge: known structure encode।
    • 
      
  • Bioinformatics: CpG island, gene structure — HMM gold standard।
    • 
      
  • Speech: classical recognizer; modern hybrid HMM-DNN।
    • 
    


    

      Bangla NLP-তে HMM POS tagger এখনও baseline হিসেবে ব্যবহার হয় — কিন্তু production-এ BERT-based model superior। HMM শিখুন intuition-এর জন্য, deploy production-এ বিরল।
    


  


  

    
    
  


  
  

    
 ভাবনার প্রশ্ন

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


    

      
        প্র ০১
        Markov assumption কেন এত strong? Higher-order Markov কাজ করে কি?
      
      

        

          Markov assumption — HMM-এর হৃদয়। কিন্তু এটা সরলীকরণ — বাস্তবে প্রায়ই violated।
        

        

          Markov assumption:
        

        
    
          
  • $P(s_t | s_{t-1}, s_{t-2}, \ldots) = P(s_t | s_{t-1})$।
    • 
          
  • Future শুধু present-এ depend।
    • 
          
  • "Memoryless"।
    • 
        

        

          কেন strong:
        

        
    
          
  • Real sequence-এ long-range dependency: "If yesterday আমি sick, then today still recovering"।
    • 
          
  • Language: "The cats that I saw — verb plural"।
    • 
          
  • Music: motif return after many bars।
    • 
          
  • Stock: trend persistence beyond single-day।
    • 
        

        

          Higher-order Markov:
        

        
    
          
  • $P(s_t | s_{t-1}, s_{t-2})$ — 2nd order।
    • 
          
  • $P(s_t | s_{t-1}, \ldots, s_{t-k})$ — $k$-th order।
    • 
          
  • State space exponentially blows up — $N^k$।
    • 
          
  • Practically rare beyond 2nd-order।
    • 
        

        

          Trick — augmented state:
        

        
    
          
  • $k$-th order HMM equivalent to 1st order with state = $(s_{t-k+1}, \ldots, s_t)$।
    • 
          
  • Theory same; computation cost-এ shift।
    • 
        

        

          Beyond Markov:
        

        
    
          
  • RNN: hidden state encode arbitrary past — theoretically infinite-order।
    • 
          
  • LSTM: selective memory — long-range dependency।
    • 
          
  • Transformer: attention-এ যেকোনো past position-এ direct access।
    • 
          
  • State Space Model (Mamba, ২০২৩): efficient long-context।
    • 
        

        

          Practical considerations:
        

        
    
          
  • Markov enough for short sequences।
    • 
          
  • 2nd-order helps marginally।
    • 
          
  • Long sequence + complex dependency → neural model।
    • 
        

        

          Hidden semi-Markov:
        

        
    
          
  • State duration explicit modeling।
    • 
          
  • "Speech phoneme average ৫০ms" — classical HMM exponential, semi-Markov gamma।
    • 
        

        

          Auto-regressive HMM:
        

        
    
          
  • Emission depends on previous emission too।
    • 
          
  • $P(o_t | s_t, o_{t-1})$।
    • 
          
  • Time series modeling enhanced।
    • 
        

        

          Bangladesh examples:
        

        
    
          
  • Bangla sentence — long-range agreement (subject-verb)। HMM struggle। BERT excels।
    • 
          
  • Climate modeling — monthly weather Markov OK; weekly fine-grain insufficient।
    • 
          
  • Healthcare — symptom progression weeks-long, HMM 1st-order limited।
    • 
        

        

          মূল উপলব্ধি: Markov simplifying — practical, computational। Long-range dependency need → modern neural। HMM understand করেই পরবর্তী step appreciate।
        

      

    


    

      
        প্র ০২
        HMM বনাম RNN/Transformer — performance gap কত? HMM-এর জায়গা আছে কি আজকের দিনে?
      
      

        

          চমৎকার ranking question। এক যুগে state-of-art ছিল HMM, এখন neural network।
        

        

          HMM advantages:
        

        
    
          
  • Parameter efficient — small।
    • 
          
  • Fast training — Baum-Welch in seconds for 1000-sample sequence।
    • 
          
  • Inference dynamic programming — exact, polynomial।
    • 
          
  • Probability output principled।
    • 
          
  • State interpretable।
    • 
          
  • Domain knowledge integrate (transition prior)।
    • 
          
  • No GPU needed।
    • 
        

        

          RNN/LSTM advantages:
        

        
    
          
  • Long-range dependency।
    • 
          
  • Continuous high-D embedding।
    • 
          
  • Better with big data।
    • 
          
  • Feature learn automatic।
    • 
          
  • End-to-end task-specific।
    • 
        

        

          Transformer advantages:
        

        
    
          
  • Parallel training (vs RNN sequential)।
    • 
          
  • Attention captures any-distance dependency।
    • 
          
  • Pretrained model transfer learning।
    • 
          
  • State-of-art everywhere।
    • 
        

        

          Performance benchmark:
        

        

          POS tagging (English):
        

        
    
          
  • HMM — ~95% accuracy।
    • 
          
  • BiLSTM-CRF — ~97%।
    • 
          
  • BERT — ~98%।
    • 
          
  • Gap small but consistent।
    • 
        

        

          Speech recognition:
        

        
    
          
  • HMM-GMM — ~80% WER (১৯৯০s)।
    • 
          
  • HMM-DNN hybrid — ~70% (২০১২)।
    • 
          
  • End-to-end neural (Transformer/Conformer) — ~5% (Whisper, ২০২২)।
    • 
          
  • Massive jump।
    • 
        

        

          Bioinformatics gene finding:
        

        
    
          
  • HMM — still dominant (HMMER, Glimmer)।
    • 
          
  • Neural approach challenging — interpretability + small data।
    • 
          
  • HMM আজও gold standard।
    • 
        

        

          Time series regime:
        

        
    
          
  • HMM — financial classical।
    • 
          
  • Neural — better predictive power।
    • 
          
  • HMM interpret-able state, neural correlation।
    • 
        

        

          HMM আজও জায়গা:
        

        
    
          
  • Embedded systems: low compute, real-time।
    • 
          
  • Bioinformatics: domain-specific tools।
    • 
          
  • Education: sequence model intuition।
    • 
          
  • Speech (legacy): Kaldi, telephony।
    • 
          
  • Anomaly detection: regime detection — interpretable state।
    • 
          
  • Small/specialized data: fine-tuned-model বানাতে আগে।
    • 
        

        

          Hybrid HMM:
        

        
    
          
  • HMM-DNN — emission via neural, transition Markov।
    • 
          
  • Best of both for some tasks।
    • 
          
  • Speech recognition transition era।
    • 
        

        

          Bangladesh-specific:
        

        
    
          
  • Bangla speech — limited data, HMM-DNN useful।
    • 
          
  • Bangla POS — HMM baseline; BERT production।
    • 
          
  • Stock market regime — HMM interpretable।
    • 
        

        

          মূল উপলব্ধি: HMM "deprecated" সম্পূর্ণ ভুল — niche-এ champion। Big data + complex dependency = neural। Small/structured/interpretability = HMM। Tool selection — task-specific, not trend।
        

      

    


    

      
        প্র ০৩
        Number of states কীভাবে চয়ন? Underfitting vs overfitting trade-off।
      
      

        

          HMM-এর K-Means-similar challenge — $N$ (state count) আগে দিতে হয়।
        

        

          State count effect:
        

        

          (১) Too few states:
        

        
    
          
  • Underfitting — distinct regime merge।
    • 
          
  • "All weather is sunny or rainy" — অথচ overcast separate।
    • 
          
  • Likelihood low।
    • 
          
  • Decoding poor।
    • 
        

        

          (২) Too many states:
        

        
    
          
  • Overfitting — random fluctuation state হিসেবে।
    • 
          
  • Parameter count blows — $N^2 + NM$।
    • 
          
  • Likelihood high but generalization poor।
    • 
          
  • Local optima — Baum-Welch confuse।
    • 
          
  • State interpretation muddled।
    • 
        

        

          Selection methods:
        

        

          (১) Cross-validation:
        

        
    
          
  • Held-out log-likelihood।
    • 
          
  • Best $N$ on validation set।
    • 
          
  • Standard but slow।
    • 
        

        

          (২) Information criteria:
        

        
    
          
  • BIC (Bayesian Information Criterion):
    • 
          
  • $\text{BIC} = -2 \log L + p \log n$
    • 
          
  • Penalize parameters — automatically select।
    • 
          
  • AIC — less penalty।
    • 
        

        

          (৩) Domain knowledge:
        

        
    
          
  • Speech: phoneme count fixed (~৪০ for English, ~৫০ for Bangla)।
    • 
          
  • Finance: bull/bear/sideways = 3 regime।
    • 
          
  • POS: tag set predefined।
    • 
        

        

          (৪) Hierarchical:
        

        
    
          
  • HDP-HMM (Hierarchical Dirichlet Process) — non-parametric।
    • 
          
  • State count auto-determined।
    • 
          
  • Bayesian — prior penalize complexity।
    • 
        

        

          (৫) Sticky HMM:
        

        
    
          
  • State persistence prior।
    • 
          
  • Avoid spurious quick switches।
    • 
        

        

          (৬) Initialization sensitivity:
        

        
    
          
  • Multiple random init।
    • 
          
  • K-Means init — emission similar to centroid।
    • 
          
  • Best likelihood selection।
    • 
        

        

          Practical workflow:
        

        
    
          
  1. Domain prior — start range।
    2. 
          
  2. BIC across $N \in [\text{min}, \text{max}]$।
    3. 
          
  3. Top-3 candidate validate manually।
    4. 
          
  4. Final selection — interpretability priority।
    5. 
        

        

          Caveats:
        

        
    
          
  • BIC strict — may underfit।
    • 
          
  • Domain expert often disagree with BIC।
    • 
          
  • State identifiability — different state-ordering same likelihood।
    • 
        

        

          Bangladesh case:
        

        
    
          
  • DSE regime: bull, bear, sideways = 3। BIC may suggest 2; domain says 3।
    • 
          
  • Bangla speech phoneme: ~৫০। Linguistic-driven।
    • 
          
  • Health symptom progression: domain expert input critical।
    • 
        

        

          মূল উপলব্ধি: $N$ selection mix of statistical criterion + domain knowledge + interpretability। No single right answer। Robustness check via multiple $N$ + qualitative review।
        

      

    


    

      
        প্র ০৪
        DSE (Dhaka Stock Exchange)-এ HMM দিয়ে regime detection — production system কেমন হবে?
      
      

        

          Real fintech application — Bangladesh equity research, asset management।
        

        

          (১) Use cases:
        

        
    
          
  • Bull/bear regime — portfolio adjustment।
    • 
          
  • Volatility regime — risk model।
    • 
          
  • Liquidity regime — execution strategy।
    • 
          
  • Market sentiment — research insight।
    • 
        

        

          (২) Data:
        

        
    
          
  • DSE Broad Index (DSEX) daily return।
    • 
          
  • Sector indices।
    • 
          
  • Volume, volatility।
    • 
          
  • Macro indicators (BB rate, inflation)।
    • 
          
  • Time horizon: ১০ year+ history।
    • 
        

        

          (৩) Model design:
        

        
    
          
  • Hidden states: 2-3 regime (bull/bear, optional sideways)।
    • 
          
  • Emission: Gaussian return + volatility।
    • 
          
  • Multivariate — multiple sector simultaneously।
    • 
        

        

          (৪) Training:
        

        
    
          
  • Baum-Welch — ১০০ iteration।
    • 
          
  • Multiple initialization — best likelihood।
    • 
          
  • Train on ৭০% history, test on ৩০%।
    • 
        

        

          (৫) Real-time inference:
        

        
    
          
  • Daily update — new return arrive।
    • 
          
  • Online forward algorithm।
    • 
          
  • Posterior probability each regime।
    • 
          
  • Threshold-based regime label।
    • 
        

        

          (৬) Output dashboard:
        

        
    
          
  • Current regime probability।
    • 
          
  • Historical regime timeline।
    • 
          
  • Transition probability (next regime)।
    • 
          
  • Sector-wise regime ভিন্নতা।
    • 
        

        

          (৭) Trading strategy integration:
        

        
    
          
  • Bull → equity overweight।
    • 
          
  • Bear → cash/bond overweight।
    • 
          
  • Sideways → mean-reversion।
    • 
          
  • Backtesting — risk-adjusted return।
    • 
        

        

          (৮) Validation:
        

        
    
          
  • Out-of-sample regime accuracy।
    • 
          
  • Sharpe ratio — regime-aware vs naive।
    • 
          
  • Drawdown reduction।
    • 
          
  • Comparison: HMM vs threshold-based, vs ML classifier।
    • 
        

        

          (৯) Challenges:
        

        
    
          
  • Bangladesh-specific: low liquidity, limited data depth।
    • 
          
  • Regime persistence: Bangladesh political event disrupt।
    • 
          
  • Black swan: COVID, war — out-of-distribution।
    • 
          
  • Data quality: missing day, holiday। DSE half-day session।
    • 
          
  • Survivorship bias: delisted stock dropped।
    • 
        

        

          (১০) Risk management:
        

        
    
          
  • HMM probability — uncertainty quantify।
    • 
          
  • Confidence interval position size।
    • 
          
  • Regime change probability — early warning।
    • 
          
  • VaR adjusted for regime।
    • 
        

        

          (১১) Ensemble:
        

        
    
          
  • HMM + threshold + ML classifier।
    • 
          
  • Voting/averaging regime probability।
    • 
          
  • Robust to single-model failure।
    • 
        

        

          (১২) Compliance:
        

        
    
          
  • BSEC regulation।
    • 
          
  • Backtest documented।
    • 
          
  • Model risk management framework।
    • 
          
  • Audit trail।
    • 
        

        

          (১৩) Continuous monitoring:
        

        
    
          
  • Performance drift detection।
    • 
          
  • Annual model retrain।
    • 
          
  • State count revisit (regime structure change?)।
    • 
          
  • Market microstructure shift adaptation।
    • 
        

        

          (১৪) Beyond HMM:
        

        
    
          
  • Transformer time-series forecasting (Informer, PatchTST)।
    • 
          
  • Mixture of HMM-NN।
    • 
          
  • Graph neural network for sector dependencies।
    • 
          
  • HMM-as-baseline, neural-as-champion।
    • 
        

        

          মূল উপলব্ধি: HMM-based regime detection Bangladesh asset management-এ practical, interpretable। Risk-adjusted return improvement via regime-aware allocation — proven globally, Bangladesh-এ underexplored opportunity। Engineering 80%, model 20%।
        

      

    

  


  

    
 অনুশীলন

    
    
      
  1. 
        হিসাব করুন: 2-state HMM, π=(0.6, 0.4), A=[[0.7, 0.3], [0.4, 0.6]], emission B[s][o]=[[0.5, 0.5], [0.1, 0.9]] (state 0: balance, state 1: o=1 likely)। Observe o=(0, 1)। P(o|λ)?
        
    
          
          
    
            
      
              
    • α₁(0) = 0.6 × 0.5 = 0.30।
      • 
              
    • α₁(1) = 0.4 × 0.1 = 0.04।
      • 
              
    • α₂(0) = (0.30×0.7 + 0.04×0.4) × 0.5 = 0.226 × 0.5 = 0.113।
      • 
              
    • α₂(1) = (0.30×0.3 + 0.04×0.6) × 0.9 = 0.114 × 0.9 = 0.1026।
      • 
              
    • P(o|λ) = 0.113 + 0.1026 = 0.2156।
      • 
            
    
          
    
        
    
      
    2. 

      
  2. 
        NumPy-তে চেষ্টা: Forward algorithm scratch।
        
    
          
          
    
            
    import numpy as np

def forward(obs, A, B, pi):
    T, N = len(obs), len(pi)
    alpha = np.zeros((T, N))
    alpha[0] = pi * B[:, obs[0]]
    for t in range(1, T):
        alpha[t] = (alpha[t-1] @ A) * B[:, obs[t]]
    return alpha, alpha[-1].sum()

A  = np.array([[0.7, 0.3], [0.4, 0.6]])
B  = np.array([[0.5, 0.5], [0.1, 0.9]])
pi = np.array([0.6, 0.4])
obs = [0, 1]

alpha, p = forward(obs, A, B, pi)
print("alpha:\n", alpha.round(4))
print("P(o|λ):", round(p, 4))
    
          
    
        
    
      
    3. 

      
  3. 
        ভাবুন: Bangla typing autocorrect-এ HMM কীভাবে কাজে দিতে পারে? Limitations?
        
    
          
          
    
            
      
              
    • Hidden state: intended word।
      • 
              
    • Observation: typed word (with errors)।
      • 
              
    • Transition: language model — common bigram।
      • 
              
    • Emission: typo probability — keyboard adjacency।
      • 
              
    • Decoding: Viterbi most-likely intended sequence।
      • 
              
    • Limit: word sequence Markov sufficient nয় — semantic context miss।
      • 
              
    • Modern alt: Transformer (BERT-based corrector) — context-aware।
      • 
              
    • Hybrid: HMM fast, neural fallback for complex case।
      • 
            
    
          
    
        
    
      
    4. 
    

  


  
  

    
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