def simulate(initial_sigma, motion_sigma, gps_sigma, n_steps):
    """Simulate a sequence of locations and noisy GPS measurements

    Args:
        initial_sigma, motion_sigma, gps_sigma: parameters of the simulation
        n_steps: number of timesteps

    Returns:
        a DataFrame with columns 'x' and 'gps' giving the true location and
        gps readouts.
    """

    # Sample an initial location from the distribution ovetr the initial loc
    x = 0 # sstats.norm.rvs(scale=np.sqrt(initial_sigma))
    loc_hist = []
    for s in range(n_steps):
        # TODO: sample a new x and gps readout
        x = sstats.norm.rvs(loc=x, scale=np.sqrt(motion_sigma))
        gps_readout = sstats.norm.rvs(loc=x, scale=np.sqrt(gps_sigma))
        loc_hist.append((x, gps_readout))
    loc_df = pd.DataFrame(loc_hist, columns=['x', 'gps'])
    return loc_df


def kalman_predict(loc_df, initial_sigma, motion_sigma, gps_sigma):
    # Set our initial belief about our location
    prior_mu = 0
    prior_sigma = initial_sigma
    predictions = []
    for gps_readout in loc_df.gps:
        # expand the prior by the movement
        prior_sigma = prior_sigma + motion_sigma
        # now do the bayes update
        k = prior_sigma / (prior_sigma + gps_sigma)
        posterior_mu = prior_mu + k * (gps_readout - prior_mu)
        posterior_sigma = prior_sigma - k * prior_sigma
        prior_mu = posterior_mu
        prior_sigma = posterior_sigma
        predictions.append((posterior_mu, posterior_sigma))

    predictions_df = pd.DataFrame(predictions, columns=['mu', 'sigma'])
    return predictions_df