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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 _backward_step(self, deltaY, deltaY_sigma2inv_rho_weightT, sigma2_e, weight, mu, mu_Gamma_inv, Gamma_inv, Lambda_0, Lambda_1, H):
 """ backward step for HMM, assuming both the hidden state and noise have 1-step dependence on the previous value. """

        n_T = len(Gamma_inv)
        # All the terms with hat before are parameters of posterior
        # distributions of X conditioned on data from all time points,
        # whereas the ones without hat calculated by _forward_step
        # are mean and covariance of posterior of X conditioned on
        # data up to the time point.
        Gamma_inv_hat = [None] * n_T
        mu_Gamma_inv_hat = [None] * n_T
        mu_hat = [None] * n_T

        mu_hat[-1] = mu[-1].copy()
        mu_Gamma_inv_hat[-1] = mu_Gamma_inv[-1].copy()
        Gamma_inv_hat[-1] = Gamma_inv[-1].copy()

        for t in np.arange(n_T - 2, -1, -1):
            tmp = np.linalg.solve(Gamma_inv_hat[t + 1] - Gamma_inv[t + 1]
                                  + Lambda_1, H)
            Gamma_inv_hat[t] = Gamma_inv[t] + Lambda_0 - np.dot(H.T, tmp)
            mu_Gamma_inv_hat[t] = mu_Gamma_inv[t] \
                - deltaY_sigma2inv_rho_weightT[t, :] + np.dot(
                    mu_Gamma_inv_hat[t + 1] - mu_Gamma_inv[t + 1]
                    + np.dot(deltaY[t, :] / sigma2_e, weight.T), tmp)
            mu_hat[t] = np.linalg.solve(Gamma_inv_hat[t],
                                        mu_Gamma_inv_hat[t])
        return mu_hat, mu_Gamma_inv_hat, Gamma_inv_hat