docs: def mfpt(T, target, origin=None, tau=1, mu=None):
      r"""Mean first passage times (from a set of starting states - optional)
      to a set of target states.
  
      Parameters
      ----------
      T : ndarray or scipy.sparse matrix, shape=(n,n)
          Transition matrix.
      target : int or list of int
          Target states for mfpt calculation.
      origin : int or list of int (optional)
          Set of starting states.
      tau : int (optional)
          The time-lag (in elementary time steps of the microstate
          trajectory) at which the given transition matrix was
          constructed.
      mu : (n,) ndarray (optional)
          The stationary distribution of the transition matrix T.
  
      Returns
      -------
      m_t : ndarray, shape=(n,) or shape(1,)
          Mean first passage time or vector of mean first passage times.
  
      Notes
      -----
      The mean first passage time :math:`\mathbf{E}_x[T_Y]` is the expected
      hitting time of one state :math:`y` in :math:`Y` when starting in state :math:`x`.
  
      For a fixed target state :math:`y` it is given by
  
      .. math :: \mathbb{E}_x[T_y] = \left \{  \begin{array}{cc}
                                               0 & x=y \\
                                               1+\sum_{z} T_{x,z} \mathbb{E}_z[T_y] & x \neq y
                                               \end{array}  \right.
  
      For a set of target states :math:`Y` it is given by
  
      .. math :: \mathbb{E}_x[T_Y] = \left \{  \begin{array}{cc}
                                               0 & x \in Y \\
                                               1+\sum_{z} T_{x,z} \mathbb{E}_z[T_Y] & x \notin Y
                                               \end{array}  \right.
  
      The mean first passage time between sets, :math:`\mathbf{E}_X[T_Y]`, is given by
  
      .. math :: \mathbb{E}_X[T_Y] = \sum_{x \in X}
                  \frac{\mu_x \mathbb{E}_x[T_Y]}{\sum_{z \in X} \mu_z}
  
      References
      ----------
      .. [1] Hoel, P G and S C Port and C J Stone. 1972. Introduction to
          Stochastic Processes.
  
      Examples
      --------
  
      >>> import numpy as np
      >>> from msmtools.analysis import mfpt
  
      >>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
      >>> m_t = mfpt(T, 0)
      >>> m_t
      array([  0.,  12.,  22.])
  
      """
      # check inputs
      T = _types.ensure_ndarray_or_sparse(T, ndim=2, uniform=True, kind='numeric')
      target = _types.ensure_int_vector(target)
      origin = _types.ensure_int_vector_or_None(origin)
      # go
      if _issparse(T):
          if origin is None:
              t_tau = sparse.mean_first_passage_time.mfpt(T, target)
          else:
              t_tau = sparse.mean_first_passage_time.mfpt_between_sets(T, target, origin, mu=mu)
      else:
          if origin is None:
              t_tau = dense.mean_first_passage_time.mfpt(T, target)
          else:
              t_tau = dense.mean_first_passage_time.mfpt_between_sets(T, target, origin, mu=mu)
  
      # scale answer by lag time used.
      return tau * t_tau