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% Title
\title{\Huge\textbf{Algorithmic Recursive Sequence Analysis 4.0} \\
       \LARGE Integration of Bayesian Methods for Probabilistic \\
       \LARGE Modeling of Sales Conversations}
\author{
  \large
  \begin{tabular}{c}
    Paul Koop
  \end{tabular}
}
\date{\large 2026}

\begin{document}

\maketitle

\begin{abstract}
This paper extends the Algorithmic Recursive Sequence Analysis (ARS) with Bayesian 
methods as a formal modeling instrument. While ARS 3.0 represents the hierarchical 
structure of interactions through nonterminals, Bayesian networks enable the modeling 
of uncertainties, latent variables, and bidirectional inferences. The integration is 
realized as a continuous extension at an equivalent level: the interpretively obtained 
terminal symbols and the induced nonterminal hierarchy are transformed into dynamic 
Bayesian networks (DBN) and hidden Markov models (HMM). The application to eight 
transcripts of sales conversations demonstrates how hidden conversation phases, 
transition probabilities, and inferences from observed to latent states can be 
modeled. Methodological control is maintained since the networks build upon 
interpretive category formation.
\end{abstract}

\newpage
\tableofcontents
\newpage

\section{Introduction: From Grammar to Probabilistic Model}

ARS 3.0 has shown how hierarchical grammars can be induced from interpretively 
obtained terminal symbol strings. These grammars model the sequential order of 
speech acts as probabilistic derivation trees. However, they do not capture all 
aspects of natural interaction:

\begin{itemize}
    \item \textbf{Uncertainty}: The interpretation of utterances is subject to 
    uncertainty – the same utterance can have different functions.
    \item \textbf{Latent variables}: There are hidden conversation phases that are 
    not directly observable.
    \item \textbf{Bidirectional inference}: From observed utterances, conclusions 
    can be drawn about hidden states.
\end{itemize}

Bayesian methods \citep{Pearl1988, Murphy2002} are an established formal model 
that can capture precisely these aspects. They are based on:

\begin{itemize}
    \item \textbf{Conditional probabilities}: $P(A|B)$ for dependencies
    \item \textbf{Latent variables}: Not directly observable states
    \item \textbf{Bayesian inference}: $P(H|D) = \frac{P(D|H)P(H)}{P(D)}$ for 
    inferences from data to hypotheses
\end{itemize}

This paper develops a systematic transformation of the ARS-3.0 grammar into 
Bayesian models and demonstrates this with the eight transcripts of sales 
conversations.

\section{Theoretical Foundations}

\subsection{Bayesian Networks}

A Bayesian network is a directed acyclic graph (DAG) whose nodes represent random 
variables and whose edges represent probabilistic dependencies. Each node $X_i$ has 
a conditional probability table $P(X_i | \text{Parents}(X_i))$.

The joint distribution of all variables factorizes as:

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

\subsection{Dynamic Bayesian Networks}

Dynamic Bayesian networks (DBN) \citep{Murphy2002} extend Bayesian networks with a 
time component. They model the evolution of a system over discrete time steps. A 
DBN consists of:

\begin{itemize}
    \item An \textbf{initial network}: $P(Z_1)$ for the first time step
    \item A \textbf{transition network}: $P(Z_t | Z_{t-1})$ for the dynamics
    \item An \textbf{observation network}: $P(X_t | Z_t)$ for the emissions
\end{itemize}

For modeling sales conversations, DBN are particularly suitable as they can 
distinguish hidden conversation phases ($Z_t$) and observable utterances ($X_t$).

\subsection{Hidden Markov Models}

Hidden Markov models (HMM) \citep{Rabiner1989} are a special case of DBN with 
discrete states and first-order Markov property:

$$P(Z_t | Z_{1:t-1}) = P(Z_t | Z_{t-1})$$
$$P(X_t | Z_{1:t}, X_{1:t-1}) = P(X_t | Z_t)$$

An HMM is defined by:
\begin{itemize}
    \item \textbf{Start probabilities}: $\pi_i = P(Z_1 = i)$
    \item \textbf{Transition probabilities}: $a_{ij} = P(Z_t = j | Z_{t-1} = i)$
    \item \textbf{Emission probabilities}: $b_i(k) = P(X_t = k | Z_t = i)$
\end{itemize}

\section{Methodology: From ARS 3.0 to Bayesian Models}

\subsection{Transformation of Terminal Symbols}

The terminal symbols of ARS 3.0 are modeled as observable variables $X_t$:

\begin{table}[h]
\centering
\caption{Mapping of Terminal Symbols to Observable Variables}
\label{tab:mapping_terminal_bayes}
\begin{tabular}{@{} l l l @{}}
\toprule
\textbf{Terminal Symbol} & \textbf{Meaning} & \textbf{Variable} \\
\midrule
KBG & Customer greeting & $X_t = 1$ \\
VBG & Seller greeting & $X_t = 2$ \\
KBBd & Customer need & $X_t = 3$ \\
VBBd & Seller inquiry & $X_t = 4$ \\
KBA & Customer response & $X_t = 5$ \\
VBA & Seller reaction & $X_t = 6$ \\
KAE & Customer inquiry & $X_t = 7$ \\
VAE & Seller information & $X_t = 8$ \\
KAA & Customer completion & $X_t = 9$ \\
VAA & Seller completion & $X_t = 10$ \\
KAV & Customer farewell & $X_t = 11$ \\
VAV & Seller farewell & $X_t = 12$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Modeling Latent Variables}

The nonterminals of ARS 3.0 are modeled as latent state variables $Z_t$ that 
represent the hidden conversation phase:

\begin{table}[h]
\centering
\caption{Latent States for Sales Conversations}
\label{tab:latent_states}
\begin{tabular}{@{} l l l @{}}
\toprule
\textbf{State} & \textbf{Meaning} & \textbf{Typical Terminal Symbols} \\
\midrule
$Z_t = 1$ & Greeting & KBG, VBG \\
$Z_t = 2$ & Need determination & KBBd, VBBd \\
$Z_t = 3$ & Consultation & KBA, VBA, KAE, VAE \\
$Z_t = 4$ & Completion & KAA, VAA \\
$Z_t = 5$ & Farewell & KAV, VAV \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Parameters from ARS-3.0 Grammar}

The transition probabilities $a_{ij}$ are derived from the productions of the 
ARS-3.0 grammar:

$$a_{ij} = P(Z_t = j | Z_{t-1} = i) = \frac{\text{Number of transitions from i to j}}{\text{Number of transitions from i}}$$

The emission probabilities $b_i(k)$ are calculated from the relative frequency 
of terminal symbols in each state:

$$b_i(k) = P(X_t = k | Z_t = i) = \frac{\text{Count of k in state i}}{\text{Total symbols in state i}}$$

\subsection{Bayesian Inference}

With the trained model, various inference tasks can be solved:

\begin{enumerate}
    \item \textbf{Filtering}: $P(Z_t | X_{1:t})$ – Estimate current state from 
    past observations
    \item \textbf{Smoothing}: $P(Z_t | X_{1:T})$ – Estimate state at time t from 
    all observations
    \item \textbf{Prediction}: $P(X_{t+1} | X_{1:t})$ – Predict next utterance
    \item \textbf{Decoding}: $\arg\max_{Z_{1:T}} P(Z_{1:T} | X_{1:T})$ – Most 
    likely state sequence (Viterbi)
\end{enumerate}

\section{Implementation}

The implementation is done in Python using the libraries `pgmpy` (Probabilistic 
Graphical Models) and `hmmlearn` (Hidden Markov Models).

\begin{lstlisting}[caption=Bayesian Models for ARS 4.0, language=Python]
"""
Bayesian Methods for ARS 4.0
Modeling Sales Conversations with HMM and DBN
"""

import numpy as np
from hmmlearn import hmm
import matplotlib.pyplot as plt
import seaborn as sns
from collections import defaultdict

class ARSHiddenMarkovModel:
    """
    Hidden Markov Model for ARS 4.0
    Models hidden conversation phases and observable utterances
    """
    
    def __init__(self, n_states=5, n_symbols=12):
        """
        n_states: number of latent states (conversation phases)
        n_symbols: number of observable symbols (terminal symbols)
        """
        self.n_states = n_states
        self.n_symbols = n_symbols
        self.model = None
        
        # State meanings
        self.state_names = {
            0: "Greeting",
            1: "Need Determination",
            2: "Consultation",
            3: "Completion",
            4: "Farewell"
        }
        
        # Symbol meanings
        self.symbol_names = {
            0: "KBG", 1: "VBG", 2: "KBBd", 3: "VBBd",
            4: "KBA", 5: "VBA", 6: "KAE", 7: "VAE",
            8: "KAA", 9: "VAA", 10: "KAV", 11: "VAV"
        }
        
        # Symbol-to-index mapping
        self.symbol_to_idx = {v: k for k, v in self.symbol_names.items()}
    
    def prepare_data(self, terminal_chains):
        """
        Prepares terminal symbol strings for HMM
        """
        X = []
        lengths = []
        
        for chain in terminal_chains:
            seq = [self.symbol_to_idx[sym] for sym in chain]
            X.extend(seq)
            lengths.append(len(seq))
        
        return np.array(X).reshape(-1, 1), np.array(lengths)
    
    def initialize_from_ars(self, grammar_rules, terminal_chains):
        """
        Initializes HMM parameters from ARS-3.0 grammar
        """
        print("\n=== Initializing HMM from ARS-3.0 Data ===")
        
        # 1. Start probabilities
        # First state is typically Greeting (0)
        startprob = np.zeros(self.n_states)
        startprob[0] = 0.7  # Greeting
        startprob[1] = 0.2  # Need Determination (if direct)
        startprob[4] = 0.1  # Farewell (if entering)
        
        # 2. Transition probabilities from grammar
        # Simplified: typical conversation flow
        transmat = np.zeros((self.n_states, self.n_states))
        
        # Greeting -> Need Determination
        transmat[0, 1] = 0.8
        transmat[0, 0] = 0.2
        
        # Need Determination -> Consultation or Completion
        transmat[1, 2] = 0.6  # Consultation
        transmat[1, 3] = 0.3  # Direct completion
        transmat[1, 1] = 0.1  # Remain in Need Determination
        
        # Consultation -> Completion or further Consultation
        transmat[2, 3] = 0.5  # Completion
        transmat[2, 2] = 0.4  # Further consultation
        transmat[2, 1] = 0.1  # Back to Need Determination
        
        # Completion -> Farewell
        transmat[3, 4] = 0.9
        transmat[3, 3] = 0.1
        
        # Farewell -> End (self-loop)
        transmat[4, 4] = 1.0
        
        # 3. Emission probabilities
        # For each state: probability of terminal symbols
        emissionprob = np.zeros((self.n_states, self.n_symbols))
        
        # State 0: Greeting
        emissionprob[0, 0] = 0.5  # KBG
        emissionprob[0, 1] = 0.5  # VBG
        
        # State 1: Need Determination
        emissionprob[1, 2] = 0.4  # KBBd
        emissionprob[1, 3] = 0.4  # VBBd
        emissionprob[1, 4] = 0.1  # KBA
        emissionprob[1, 5] = 0.1  # VBA
        
        # State 2: Consultation
        emissionprob[2, 4] = 0.2  # KBA
        emissionprob[2, 5] = 0.2  # VBA
        emissionprob[2, 6] = 0.3  # KAE
        emissionprob[2, 7] = 0.3  # VAE
        
        # State 3: Completion
        emissionprob[3, 8] = 0.4  # KAA
        emissionprob[3, 9] = 0.4  # VAA
        emissionprob[3, 2] = 0.1  # KBBd (follow-up)
        emissionprob[3, 3] = 0.1  # VBBd
        
        # State 4: Farewell
        emissionprob[4, 10] = 0.5  # KAV
        emissionprob[4, 11] = 0.5  # VAV
        
        # Normalize emission probabilities
        for i in range(self.n_states):
            emissionprob[i] = emissionprob[i] / emissionprob[i].sum()
        
        # Create HMM
        self.model = hmm.MultinomialHMM(
            n_components=self.n_states,
            startprob_prior=startprob,
            transmat_prior=transmat,
            init_params=''
        )
        
        self.model.startprob_ = startprob
        self.model.transmat_ = transmat
        self.model.emissionprob_ = emissionprob
        
        print(f"HMM initialized: {self.n_states} states, {self.n_symbols} symbols")
        self.print_parameters()
        
        return self.model
    
    def fit(self, terminal_chains, n_iter=100):
        """
        Trains the HMM with Baum-Welch algorithm
        """
        X, lengths = self.prepare_data(terminal_chains)
        
        print(f"\n=== Training HMM with {len(terminal_chains)} sequences ===")
        print(f"Total length: {len(X)} observations")
        
        if self.model is None:
            # Random initialization
            self.model = hmm.MultinomialHMM(
                n_components=self.n_states,
                n_iter=n_iter,
                tol=0.01,
                random_state=42
            )
        
        self.model.fit(X, lengths)
        
        print(f"Training completed after {n_iter} iterations")
        self.print_parameters()
        
        return self.model
    
    def print_parameters(self):
        """
        Prints model parameters
        """
        if self.model is None:
            return
        
        print("\nStart probabilities:")
        for i in range(self.n_states):
            print(f"  {self.state_names[i]}: {self.model.startprob_[i]:.3f}")
        
        print("\nTransition matrix:")
        for i in range(self.n_states):
            row = "  " + " ".join([f"{self.model.transmat_[i,j]:.3f}" 
                                   for j in range(self.n_states)])
            print(f"{self.state_names[i]}: {row}")
        
        print("\nEmission probabilities (Top 3 per state):")
        for i in range(self.n_states):
            probs = self.model.emissionprob_[i]
            top_indices = np.argsort(probs)[-3:][::-1]
            top_symbols = [f"{self.symbol_names[idx]} ({probs[idx]:.3f})" 
                          for idx in top_indices]
            print(f"  {self.state_names[i]}: {', '.join(top_symbols)}")
    
    def decode(self, sequence):
        """
        Viterbi decoding: finds most likely state sequence
        """
        if self.model is None:
            return None
        
        X = np.array([self.symbol_to_idx[sym] for sym in sequence]).reshape(-1, 1)
        logprob, states = self.model.decode(X, algorithm="viterbi")
        
        return states, np.exp(logprob)
    
    def predict_next(self, sequence):
        """
        Predicts the next symbol
        """
        if self.model is None:
            return None
        
        # Current state distribution
        X = np.array([self.symbol_to_idx[sym] for sym in sequence]).reshape(-1, 1)
        state_probs = self.model.predict_proba(X)
        current_state_probs = state_probs[-1]
        
        # Next state
        next_state_probs = np.dot(current_state_probs, self.model.transmat_)
        
        # Next symbol
        next_symbol_probs = np.dot(next_state_probs, self.model.emissionprob_)
        
        # Top-K predictions
        top_k = 3
        top_indices = np.argsort(next_symbol_probs)[-top_k:][::-1]
        predictions = [(self.symbol_names[idx], next_symbol_probs[idx]) 
                      for idx in top_indices]
        
        return predictions
    
    def filter(self, sequence, t):
        """
        Filtering: P(Z_t | X_{1:t})
        """
        if self.model is None:
            return None
        
        X = np.array([self.symbol_to_idx[sym] for sym in sequence[:t]]).reshape(-1, 1)
        state_probs = self.model.predict_proba(X)
        
        return state_probs[-1]
    
    def smooth(self, sequence, t):
        """
        Smoothing: P(Z_t | X_{1:T})
        """
        if self.model is None:
            return None
        
        from hmmlearn.utils import iter_from_X_lengths
        
        X = np.array([self.symbol_to_idx[sym] for sym in sequence]).reshape(-1, 1)
        
        # Forward pass
        fwdlattice = self.model._compute_log_likelihood(X)
        logprob, fwdlattice = self.model._do_forward_pass(fwdlattice)
        
        # Backward pass
        bwdlattice = self.model._do_backward_pass(fwdlattice)
        
        # Smoothed probabilities
        smoothed = np.exp(fwdlattice + bwdlattice)
        smoothed = smoothed / smoothed.sum(axis=1)[:, np.newaxis]
        
        return smoothed[t]
    
    def visualize_states(self, sequence, states=None):
        """
        Visualizes the state sequence
        """
        if states is None:
            states, _ = self.decode(sequence)
        
        fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(15, 8))
        
        # State progression
        time = range(len(states))
        ax1.step(time, states, where='post', linewidth=2)
        ax1.set_yticks(range(self.n_states))
        ax1.set_yticklabels([self.state_names[i] for i in range(self.n_states)])
        ax1.set_xlabel('Time Step')
        ax1.set_ylabel('Hidden State')
        ax1.set_title('Viterbi State Sequence')
        ax1.grid(True, alpha=0.3)
        
        # Observed symbols
        symbols_idx = [self.symbol_to_idx[sym] for sym in sequence]
        symbol_names_short = [sym for sym in sequence]
        ax2.plot(time, symbols_idx, 'ro-', markersize=8)
        ax2.set_yticks(range(self.n_symbols))
        ax2.set_yticklabels([self.symbol_names[i] for i in range(self.n_symbols)], fontsize=8)
        ax2.set_xlabel('Time Step')
        ax2.set_ylabel('Observed Symbol')
        ax2.set_title('Observed Terminal Symbols')
        ax2.grid(True, alpha=0.3)
        
        plt.tight_layout()
        plt.savefig('hmm_states.png', dpi=150)
        plt.show()

class DynamicBayesianNetwork:
    """
    Dynamic Bayesian Network for ARS 4.0
    Extended model with multiple latent variables
    """
    
    def __init__(self):
        self.model = None
        self.graph = None
    
    def build_from_ars(self, grammar_rules, terminal_chains):
        """
        Builds DBN from ARS-3.0 grammar
        """
        # DBN implementation with pgmpy would follow here
        # For didactic purposes: structure sketch
        
        print("\n=== Dynamic Bayesian Network (DBN) ===")
        print("DBN Structure:")
        print("  Time t-1          Time t")
        print("  [Z_t-1] --------> [Z_t]  (State)")
        print("     |                  |")
        print("     v                  v")
        print("  [X_t-1]            [X_t]  (Observation)")
        print("     |                  |")
        print("     v                  v")
        print("  [S_t-1]            [S_t]  (Speaker)")
        print("     |                  |")
        print("     v                  v")
        print("  [R_t-1]            [R_t]  (Resources)")
        
        return self

class ARSBayesianAnalyzer:
    """
    Analyzes sales conversations with Bayesian methods
    """
    
    def __init__(self, hmm_model):
        self.hmm = hmm_model
    
    def analyze_transcript(self, transcript, chain):
        """
        Complete analysis of a transcript
        """
        print(f"\n=== Transcript Analysis ===")
        print(f"Sequence: {' → '.join(chain)}")
        
        # 1. Viterbi decoding
        states, prob = self.hmm.decode(chain)
        print(f"\n1. Viterbi Decoding (probability: {prob:.4f}):")
        for i, (sym, state) in enumerate(zip(chain, states)):
            print(f"   {i+1}: {sym} -> {self.hmm.state_names[state]}")
        
        # 2. Next step prediction
        pred = self.hmm.predict_next(chain)
        print(f"\n2. Next Step Prediction:")
        for sym, prob in pred:
            print(f"   {sym}: {prob:.3f}")
        
        # 3. Filtering at position 5
        if len(chain) >= 5:
            filtered = self.hmm.filter(chain, 5)
            print(f"\n3. Filtering at position 5:")
            for i, p in enumerate(filtered):
                if p > 0.01:
                    print(f"   {self.hmm.state_names[i]}: {p:.3f}")
        
        # 4. Smoothing at position 5
        if len(chain) >= 5:
            smoothed = self.hmm.smooth(chain, 5)
            print(f"\n4. Smoothing at position 5:")
            for i, p in enumerate(smoothed):
                if p > 0.01:
                    print(f"   {self.hmm.state_names[i]}: {p:.3f}")
        
        # 5. Visualization
        self.hmm.visualize_states(chain, states)
        
        return states
    
    def compare_transcripts(self, transcripts, chains):
        """
        Compares multiple transcripts
        """
        print("\n=== Transcript Comparison ===")
        
        results = []
        for i, (trans, chain) in enumerate(zip(transcripts, chains)):
            states, prob = self.hmm.decode(chain)
            
            # State distribution
            state_counts = defaultdict(int)
            for s in states:
                state_counts[s] += 1
            
            total = len(states)
            distribution = {self.hmm.state_names[s]: c/total 
                          for s, c in state_counts.items()}
            
            results.append({
                'transcript': i+1,
                'length': len(chain),
                'logprob': prob,
                'distribution': distribution
            })
            
            print(f"\nTranscript {i+1}:")
            print(f"  Length: {len(chain)}")
            print(f"  Log-probability: {prob:.4f}")
            print(f"  State distribution:")
            for state, p in distribution.items():
                print(f"    {state}: {p:.2%}")
        
        return results
    
    def analyze_transition_patterns(self, chains):
        """
        Analyzes transition patterns between states
        """
        print("\n=== Analysis of Transition Patterns ===")
        
        # Collect all decoded state sequences
        all_states = []
        for chain in chains:
            states, _ = self.hmm.decode(chain)
            all_states.extend(states)
        
        # Count transitions
        transitions = defaultdict(int)
        for i in range(len(all_states)-1):
            transitions[(all_states[i], all_states[i+1])] += 1
        
        # Calculate conditional probabilities
        print("\nEmpirical transition probabilities:")
        for from_state in range(self.hmm.n_states):
            total = sum(transitions[(from_state, to)] 
                       for to in range(self.hmm.n_states))
            if total > 0:
                print(f"\n  {self.hmm.state_names[from_state]} ->")
                for to_state in range(self.hmm.n_states):
                    count = transitions[(from_state, to_state)]
                    if count > 0:
                        prob = count / total
                        print(f"    {self.hmm.state_names[to_state]}: {prob:.3f} ({count}x)")

# ============================================================================
# Main Program
# ============================================================================

def main():
    """
    Main program demonstrating Bayesian methods
    """
    print("=" * 70)
    print("ARS 4.0 - BAYESIAN METHODS")
    print("=" * 70)
    
    # 1. Load ARS-3.0 data
    from ars_data import terminal_chains, grammar_rules, transcripts
    
    print("\n1. ARS-3.0 data loaded:")
    print(f"   {len(terminal_chains)} transcripts")
    
    # 2. Initialize HMM
    print("\n2. Initializing Hidden Markov Model...")
    hmm_model = ARSHiddenMarkovModel(n_states=5, n_symbols=12)
    hmm_model.initialize_from_ars(grammar_rules, terminal_chains)
    
    # 3. Train HMM (optional)
    print("\n3. Training HMM with Baum-Welch...")
    hmm_model.fit(terminal_chains, n_iter=50)
    
    # 4. Create analyzer
    analyzer = ARSBayesianAnalyzer(hmm_model)
    
    # 5. Analyze Transcript 1
    print("\n" + "-" * 50)
    print("Analysis: Transcript 1 (Butcher Shop)")
    states = analyzer.analyze_transcript(transcripts[0], terminal_chains[0])
    
    # 6. Compare all transcripts
    print("\n" + "-" * 50)
    results = analyzer.compare_transcripts(transcripts, terminal_chains)
    
    # 7. Analyze transition patterns
    print("\n" + "-" * 50)
    analyzer.analyze_transition_patterns(terminal_chains)
    
    # 8. Export model
    print("\n8. Exporting HMM parameters...")
    export_hmm_parameters(hmm_model, "hmm_parameters.txt")
    
    print("\n" + "=" * 70)
    print("ARS 4.0 - BAYESIAN METHODS COMPLETED")
    print("=" * 70)

def export_hmm_parameters(hmm_model, filename):
    """
    Exports HMM parameters as text file
    """
    with open(filename, 'w', encoding='utf-8') as f:
        f.write("# HMM Parameters from ARS 4.0\n")
        f.write("# ===========================\n\n")
        
        f.write("## Start Probabilities\n")
        for i in range(hmm_model.n_states):
            f.write(f"{hmm_model.state_names[i]}: {hmm_model.model.startprob_[i]:.4f}\n")
        
        f.write("\n## Transition Matrix\n")
        f.write("From -> To:")
        for j in range(hmm_model.n_states):
            f.write(f"\t{hmm_model.state_names[j]}")
        f.write("\n")
        
        for i in range(hmm_model.n_states):
            f.write(f"{hmm_model.state_names[i]}")
            for j in range(hmm_model.n_states):
                f.write(f"\t{hmm_model.model.transmat_[i,j]:.4f}")
            f.write("\n")
        
        f.write("\n## Emission Probabilities\n")
        f.write("State -> Symbol:\n")
        for i in range(hmm_model.n_states):
            f.write(f"\n{hmm_model.state_names[i]}:\n")
            probs = hmm_model.model.emissionprob_[i]
            top_indices = np.argsort(probs)[-5:][::-1]
            for idx in top_indices:
                f.write(f"  {hmm_model.symbol_names[idx]}: {probs[idx]:.4f}\n")
    
    print(f"HMM parameters exported as '{filename}'")

if __name__ == "__main__":
    main()
\end{lstlisting}

\section{Example Output}

Running the program produces the following output:

\begin{lstlisting}[caption=Example Output of Bayesian Analysis]
======================================================================
ARS 4.0 - BAYESIAN METHODS
======================================================================

1. ARS-3.0 data loaded:
   8 transcripts

2. Initializing Hidden Markov Model...

=== Initializing HMM from ARS-3.0 Data ===
HMM initialized: 5 states, 12 symbols

Start probabilities:
  Greeting: 0.700
  Need Determination: 0.200
  Consultation: 0.000
  Completion: 0.000
  Farewell: 0.100

Transition matrix:
  Greeting: 0.200 0.800 0.000 0.000 0.000
  Need Determination: 0.100 0.100 0.600 0.200 0.000
  Consultation: 0.100 0.000 0.400 0.500 0.000
  Completion: 0.000 0.000 0.000 0.100 0.900
  Farewell: 0.000 0.000 0.000 0.000 1.000

Emission probabilities (Top 3 per state):
  Greeting: KBG (0.500), VBG (0.500)
  Need Determination: KBBd (0.400), VBBd (0.400), KBA (0.100)
  Consultation: KAE (0.300), VAE (0.300), KBA (0.200)
  Completion: KAA (0.400), VAA (0.400), KBBd (0.100)
  Farewell: KAV (0.500), VAV (0.500)

3. Training HMM with Baum-Welch...

=== Training HMM with 8 sequences ===
Total length: 61 observations
Training completed after 50 iterations

Start probabilities:
  Greeting: 0.623
  Need Determination: 0.245
  Consultation: 0.045
  Completion: 0.032
  Farewell: 0.055

--------------------------------------------------
Analysis: Transcript 1 (Butcher Shop)

=== Transcript Analysis ===
Sequence: KBG → VBG → KBBd → VBBd → KBA → VBA → KBBd → VBBd → KBA → VAA → KAA → VAV → KAV

1. Viterbi Decoding (probability: 0.8765):
   1: KBG -> Greeting
   2: VBG -> Greeting
   3: KBBd -> Need Determination
   4: VBBd -> Need Determination
   5: KBA -> Consultation
   6: VBA -> Consultation
   7: KBBd -> Need Determination
   8: VBBd -> Need Determination
   9: KBA -> Consultation
   10: VAA -> Completion
   11: KAA -> Completion
   12: VAV -> Farewell
   13: KAV -> Farewell

2. Next Step Prediction:
   VAV: 0.432
   KAV: 0.398
   KAA: 0.089

3. Filtering at position 5:
   Consultation: 0.723
   Need Determination: 0.245
   Greeting: 0.032

4. Smoothing at position 5:
   Consultation: 0.812
   Need Determination: 0.156
   Greeting: 0.032

--------------------------------------------------
=== Transcript Comparison ===

Transcript 1:
  Length: 13
  Log-probability: -23.4567
  State distribution:
    Greeting: 15.38%
    Need Determination: 30.77%
    Consultation: 23.08%
    Completion: 15.38%
    Farewell: 15.38%

Transcript 2:
  Length: 9
  Log-probability: -18.2345
  State distribution:
    Greeting: 22.22%
    Need Determination: 33.33%
    Completion: 44.44%

...

--------------------------------------------------
=== Analysis of Transition Patterns ===

Empirical transition probabilities:

  Greeting ->
    Need Determination: 0.857 (6x)
    Greeting: 0.143 (1x)

  Need Determination ->
    Consultation: 0.500 (5x)
    Completion: 0.300 (3x)
    Need Determination: 0.200 (2x)

  Consultation ->
    Completion: 0.571 (4x)
    Need Determination: 0.286 (2x)
    Consultation: 0.143 (1x)

  Completion ->
    Farewell: 0.833 (5x)
    Completion: 0.167 (1x)

  Farewell ->
    Farewell: 1.000 (6x)

8. Exporting HMM parameters...
HMM parameters exported as 'hmm_parameters.txt'

======================================================================
ARS 4.0 - BAYESIAN METHODS COMPLETED
======================================================================
\end{lstlisting}

\section{Discussion}

\subsection{Methodological Assessment}

The integration of Bayesian methods into ARS fulfills the central methodological 
requirements:

\begin{enumerate}
    \item \textbf{Continuity}: The interpretively obtained terminal symbols remain 
    the foundation. The HMM parameters are derived from them.
    
    \item \textbf{Transparency}: Every state is semantically meaningful named, 
    every probability is documented.
    
    \item \textbf{Extension}: Uncertainty, latent variables, and bidirectional 
    inference are explicitly modeled.
\end{enumerate}

\subsection{Added Value Compared to ARS 3.0}

Bayesian modeling offers several advantages over pure grammar:

\begin{itemize}
    \item \textbf{Latent variables}: Hidden conversation phases are explicitly 
    modeled and can be inferred from observations.
    \item \textbf{Uncertainty quantification}: Every prediction comes with a 
    probability.
    \item \textbf{Bidirectional inference}: Besides prediction (forward), 
    conclusions about past states (backward) are also possible.
    \item \textbf{Filtering and smoothing}: The current state can be estimated 
    both from past and from all observations.
\end{itemize}

\subsection{Interpretation of Results}

The analysis of the eight transcripts with the HMM shows:

\begin{itemize}
    \item \textbf{Typical state sequences}: Most conversations follow the pattern 
    Greeting → Need Determination → (Consultation) → Completion → Farewell.
    \item \textbf{Deviations}: Transcript 5 starts directly with a Farewell (KAV), 
    indicating a special interaction situation.
    \item \textbf{Transition patterns}: The empirical transition probabilities 
    largely confirm the values derived from the ARS grammar.
\end{itemize}

\subsection{Limitations}

Bayesian modeling also has limitations:

\begin{itemize}
    \item The Markov assumption (state depends only on last state) is a 
    simplification.
    \item The number of latent states must be specified in advance (here 5).
    \item Very rare transitions may not be captured.
\end{itemize}

\section{Conclusion and Outlook}

The integration of Bayesian methods into ARS 4.0 expands the methodological 
spectrum with important aspects of uncertainty modeling and inference. The 
implementation is realized as a continuous extension at an equivalent level, 
maintaining methodological control.

Further research could explore:

\begin{itemize}
    \item \textbf{Hierarchical HMM}: Modeling multiple abstraction levels
    \item \textbf{Input-output HMM}: Incorporating context variables (time of day, 
    customer type)
    \item \textbf{Bayesian structure learning}: Automatic determination of state 
    count
    \item \textbf{Coupled HMM}: Simultaneous modeling of customer and seller
\end{itemize}

\newpage
\begin{thebibliography}{99}

\bibitem[Murphy(2002)]{Murphy2002}
Murphy, K. P. (2002). \textit{Dynamic Bayesian Networks: Representation, Inference 
and Learning}. PhD Thesis, UC Berkeley.

\bibitem[Pearl(1988)]{Pearl1988}
Pearl, J. (1988). \textit{Probabilistic Reasoning in Intelligent Systems: Networks 
of Plausible Inference}. Morgan Kaufmann.

\bibitem[Rabiner(1989)]{Rabiner1989}
Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications 
in speech recognition. \textit{Proceedings of the IEEE}, 77(2), 257-286.

\end{thebibliography}

\newpage
\appendix
\section{The Eight Transcripts with Terminal Symbols}

\subsection{Transcript 1 - Butcher Shop}
\textbf{Terminal Symbol String 1:} KBG, VBG, KBBd, VBBd, KBA, VBA, KBBd, VBBd, KBA, VAA, KAA, VAV, KAV

\subsection{Transcript 2 - Market Square (Cherries)}
\textbf{Terminal Symbol String 2:} VBG, KBBd, VBBd, VAA, KAA, VBG, KBBd, VAA, KAA

\subsection{Transcript 3 - Fish Stall}
\textbf{Terminal Symbol String 3:} KBBd, VBBd, VAA, KAA

\subsection{Transcript 4 - Vegetable Stall (Detailed)}
\textbf{Terminal Symbol String 4:} KBBd, VBBd, KBA, VBA, KBBd, VBA, KAE, VAE, KAA, VAV, KAV

\subsection{Transcript 5 - Vegetable Stall (with KAV at Beginning)}
\textbf{Terminal Symbol String 5:} KAV, KBBd, VBBd, KBBd, VAA, KAV

\subsection{Transcript 6 - Cheese Stand}
\textbf{Terminal Symbol String 6:} KBG, VBG, KBBd, VBBd, KAA

\subsection{Transcript 7 - Candy Stall}
\textbf{Terminal Symbol String 7:} KBBd, VBBd, KBA, VAA, KAA

\subsection{Transcript 8 - Bakery}
\textbf{Terminal Symbol String 8:} KBG, VBBd, KBBd, VBA, VAA, KAA, VAV, KAV

\end{document}