code: def sample_path(alpha, A, pobs, T=None, dtype=np.float32):
      
      N = pobs.shape[1]
      # set T
      if T is None:
          T = pobs.shape[0]  # if not set, use the length of pobs as trajectory length
      elif T > pobs.shape[0] or T > alpha.shape[0]:
          raise ValueError('T must be at most the length of pobs and alpha.')
  
      # initialize path
      S = np.zeros(T, dtype=int)
  
      # Sample final state.
      psel = alpha[T-1, :]
      psel /= psel.sum()  # make sure it's normalized
      # Draw from this distribution.
      S[T-1] = np.random.choice(range(N), size=1, p=psel)
  
      # Work backwards from T-2 to 0.
      for t in range(T-2, -1, -1):
          # Compute P(s_t = i | s_{t+1}..s_T).
          psel = alpha[t, :] * A[:, S[t+1]]
          psel /= psel.sum()  # make sure it's normalized
          # Draw from this distribution.
          S[t] = np.random.choice(range(N), size=1, p=psel)
  
      return S
docs: alpha : ndarray((T,N), dtype = float), optional, default = None
          alpha[t,i] is the ith forward coefficient of time t.
      beta : ndarray((T,N), dtype = float), optional, default = None
          beta[t,i] is the ith forward coefficient of time t.
      A : ndarray((N,N), dtype = float)
          transition matrix of the hidden states
      pobs : ndarray((T,N), dtype = float)
          pobs[t,i] is the observation probability for observation at time t given hidden state i