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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def mbar( ttrajs, dtrajs, bias, maxiter=100000, maxerr=1.0E-15, save_convergence_info=0, dt_traj='1 step', direct_space=False):
 r""" Multi-state Bennet acceptance ratio Parameters ttrajs : numpy.ndarray(T) of int, or list of numpy.ndarray(T_i) of int A single discrete trajectory or a list of discrete trajectories. The integers are in at any time. dtrajs : numpy.ndarray(T) of int, or list of numpy.ndarray(T_i) of int A single discrete trajectory or a list of discrete trajectories. The integers are indexes trajectory is in at any time. bias : numpy.ndarray(T, num_therm_states), or list of numpy.ndarray(T_i, num_therm_states) A single reduced bias energy trajectory or a list of reduced bias energy trajectories. For every simulation frame seen in trajectory i and time step t, btrajs[i][t, k] is the reduced bias energy of that frame evaluated in the k'th thermodynamic state (i.e. at the k'th umbrella/Hamiltonian/temperature) maxiter : int, optional, default=10000 The maximum number of dTRAM iterations before the estimator exits unsuccessfully. maxerr : float, optional, default=1e-15 Convergence criterion based on the maximal free energy change in a self-consistent iteration step. save_convergence_info : int, optional, default=0 Every save_convergence_info iteration steps, store the actual increment and the actual loglikelihood; 0 means no storage. dt_traj : str, optional, default='1 step' Description of the physical time corresponding to the lag. May be used by analysis algorithms such as plotting tools to pretty-print the axes. By default '1 step', i.e. there is no physical time unit. Specify by a number, whitespace and unit. Permitted units are (* is an arbitrary string):
 | 'fs', 'femtosecond*' | 'ps', 'picosecond*' | 'ns', 'nanosecond*' | 'us', 'microsecond*' | 'ms', 'millisecond*' | 's', 'second*' direct_space : bool, optional, default=False Whether to perform the self-consitent iteration with Boltzmann factors (direct space) or free energies (log-space). When analyzing data from multi-temperature simulations, direct-space is not recommended. Returns ------- A :class:`MultiThermModel <pyemma.thermo.models.multi_therm.MultiThermModel>` object A stationary model which consists of thermodynamic quantities at all temperatures/thermodynamic states. Example ------- **Umbrella sampling**: Suppose we simulate in K umbrellas, centered at .. math:: b_k(x) = \frac{c_k}{2 \textrm{kT}} \cdot (x - y_k)^2 Suppose we have one simulation of length T in each umbrella, and they are ordered from 0 to K-1. We have discretized the x-coordinate into 100 bins. Then dtrajs and ttrajs should each be a list of :math:`K` arrays. dtrajs would look for example like this:: where each array has length T, and is the sequence of bins (in the range 0 to 99) visited along the trajectory. ttrajs would look like this:: Because trajectory 1 stays in umbrella 1 (index 0), trajectory 2 stays in umbrella 2 (index 1), and so forth. The bias would be a list of :math:`T \times K` arrays which specify each frame's bias energy in all thermodynamic states: Let us try the above example: See :class:`MultiThermModel <pyemma.thermo.models.multi_therm.MultiThermModel>` for a full documentation. .. autoclass:: pyemma.thermo.models.multi_therm.MultiThermModel :members: :undoc-members: .. rubric:: Methods .. autoautosummary:: pyemma.thermo.models.multi_therm.MultiThermModel :methods: .. rubric:: Attributes .. autoautosummary:: pyemma.thermo.models.multi_therm.MultiThermModel :attributes: References .. [1] Shirts, M.R. and Chodera, J.D. 2008 Statistically optimal analysis of samples from multiple equilibrium states J. Chem. Phys. 129, 124105 """

    # check trajectories
    ttrajs = _types.ensure_dtraj_list(ttrajs)
    dtrajs = _types.ensure_dtraj_list(dtrajs)
    if len(ttrajs) != len(dtrajs):
        raise ValueError("Unmatching number of dtraj/ttraj elements: %d!=%d" % (
            len(dtrajs), len(ttrajs)))
    if len(ttrajs) != len(bias):
        raise ValueError("Unmatching number of ttraj/bias elements: %d!=%d" % (
            len(ttrajs), len(bias)))
    for ttraj, dtraj, btraj in zip(ttrajs, dtrajs, bias):
        if len(ttraj) != len(dtraj):
            raise ValueError("Unmatching number of data points in ttraj/dtraj: %d!=%d" % (
                len(ttraj), len(dtraj)))
        if len(ttraj) != btraj.shape[0]:
            raise ValueError("Unmatching number of data points in ttraj/bias trajectory: %d!=%d" % (
                len(ttraj), len(btraj)))
    # build MBAR
    from pyemma.thermo import MBAR
    mbar_estimator = MBAR(
        maxiter=maxiter, maxerr=maxerr, save_convergence_info=save_convergence_info,
        dt_traj=dt_traj, direct_space=direct_space)
    # run estimation
    return mbar_estimator.estimate((ttrajs, dtrajs, bias))