def gfit(X, sigma, p=5, nbin=200, unif_fraction=0.1):
    """
    Fit empirical Bayes prior in the hierarchical model [Efron2014]_.

    .. math::

        mu ~ G, X ~ N(mu, sigma^2)

    Parameters
    ----------
    X: ndarray
        A 1D array of observations.
    sigma: float
        Noise estimate on X.
    p: int
        Number of parameters used to fit G. Default: 5
    nbin: int
        Number of bins used for discrete approximation.
        Default: 200
    unif_fraction: float
        Fraction of G modeled as "slab". Default: 0.1

    Returns
    -------
    An array of the posterior density estimate g.
    """
    min_x = min(min(X) - 2 * np.std(X, ddof=1), 0)
    max_x = max(max(X) + 2 * np.std(X, ddof=1),
                np.std(X, ddof=1))
    xvals = np.linspace(min_x, max_x, nbin)
    binw = (max_x - min_x) / (nbin - 1)

    zero_idx = max(np.where(xvals <= 0)[0])
    noise_kernel = norm().pdf(xvals / sigma) * binw / sigma

    if zero_idx > 0:
        noise_rotate = noise_kernel[list(np.arange(zero_idx, len(xvals))) +
                                    list(np.arange(0, zero_idx))]
    else:
        noise_rotate = noise_kernel

    XX = np.zeros((p, len(xvals)), dtype=np.float)
    for ind, exp in enumerate(range(1, p+1)):
        mask = np.ones_like(xvals)
        mask[np.where(xvals <= 0)[0]] = 0
        XX[ind, :] = pow(xvals, exp) * mask
    XX = XX.T

    def neg_loglik(eta):
        mask = np.ones_like(xvals)
        mask[np.where(xvals <= 0)[0]] = 0
        g_eta_raw = np.exp(np.dot(XX, eta)) * mask
        if ((np.sum(g_eta_raw) == np.inf) |
            (np.sum(g_eta_raw) <=
                100 * np.finfo(np.double).tiny)):
                return (1000 * (len(X) + sum(eta ** 2)))

        g_eta_main = g_eta_raw / sum(g_eta_raw)
        g_eta = ((1 - unif_fraction) * g_eta_main +
                 unif_fraction * mask / sum(mask))
        f_eta = fftconvolve(g_eta, noise_rotate, mode='same')
        return np.sum(np.interp(X, xvals,
                      -np.log(np.maximum(f_eta, 0.0000001))))

    eta_hat = minimize(neg_loglik,
                       list(itertools.repeat(-1, p))).x
    g_eta_raw = np.exp(np.dot(XX, eta_hat)) * mask
    g_eta_main = g_eta_raw / sum(g_eta_raw)
    g_eta = ((1 - unif_fraction) * g_eta_main +
             unif_fraction * mask) / sum(mask)

    return xvals, g_eta
def gfit(X, sigma, p=5, nbin=200, unif_fraction=0.1):
    """
    Fit empirical Bayes prior in the hierarchical model [Efron2014]_.

    .. math::

        mu ~ G, X ~ N(mu, sigma^2)

    Parameters
    ----------
    X: ndarray
        A 1D array of observations.
    sigma: float
        Noise estimate on X.
    p: int
        Number of parameters used to fit G. Default: 5
    nbin: int
        Number of bins used for discrete approximation.
        Default: 200
    unif_fraction: float
        Fraction of G modeled as "slab". Default: 0.1

    Returns
    -------
    An array of the posterior density estimate g.
    """
    min_x = min(min(X) - 2 * np.std(X, ddof=1), 0)
    max_x = max(max(X) + 2 * np.std(X, ddof=1),
                np.std(X, ddof=1))
    xvals = np.linspace(min_x, max_x, nbin)
    binw = (max_x - min_x) / (nbin - 1)

    zero_idx = max(np.where(xvals <= 0)[0])
    noise_kernel = norm().pdf(xvals / sigma) * binw / sigma

    if zero_idx > 0:
        noise_rotate = noise_kernel[list(np.arange(zero_idx, len(xvals))) +
                                    list(np.arange(0, zero_idx))]
    else:
        noise_rotate = noise_kernel

    XX = np.zeros((p, len(xvals)), dtype=np.float)
    for ind, exp in enumerate(range(1, p+1)):
        mask = np.ones_like(xvals)
        mask[np.where(xvals <= 0)[0]] = 0
        XX[ind, :] = pow(xvals, exp) * mask
    XX = XX.T

    def neg_loglik(eta):
        mask = np.ones_like(xvals)
        mask[np.where(xvals <= 0)[0]] = 0
        g_eta_raw = np.exp(np.dot(XX, eta)) * mask
        if ((np.sum(g_eta_raw) == np.inf) |
            (np.sum(g_eta_raw) <=
                100 * np.finfo(np.double).tiny)):
                return (1000 * (len(X) + sum(eta ** 2)))

        g_eta_main = g_eta_raw / sum(g_eta_raw)
        g_eta = ((1 - unif_fraction) * g_eta_main +
                 unif_fraction * mask / sum(mask))
        f_eta = fftconvolve(g_eta, noise_rotate, mode='same')
        return np.sum(np.interp(X, xvals,
                      -np.log(np.maximum(f_eta, 0.0000001))))

    eta_hat = minimize(neg_loglik,
                       list(itertools.repeat(-1, p))).x
    g_eta_raw = np.exp(np.dot(XX, eta_hat)) * mask
    g_eta_main = g_eta_raw / sum(g_eta_raw)
    g_eta = ((1 - unif_fraction) * g_eta_main +
             unif_fraction * mask) / sum(mask)

    return xvals, g_eta
def gfit ( X , sigma , p = 5 , nbin = 200 , unif_fraction = 0.1 ) : min_x = min ( min ( X ) - 2 * np . std ( X , ddof = 1 ) , 0 ) max_x = max ( max ( X ) + 2 * np . std ( X , ddof = 1 ) , np . std ( X , ddof = 1 ) ) xvals = np . linspace ( min_x , max_x , nbin ) binw = ( max_x - min_x ) / ( nbin - 1 ) zero_idx = max ( np . where ( xvals <= 0 ) [ 0 ] ) noise_kernel = norm ( ) . pdf ( xvals / sigma ) * binw / sigma if zero_idx > 0 : noise_rotate = noise_kernel [ list ( np . arange ( zero_idx , len ( xvals ) ) ) + list ( np . arange ( 0 , zero_idx ) ) ] else : noise_rotate = noise_kernel XX = np . zeros ( ( p , len ( xvals ) ) , dtype = np . float ) for ind , exp in enumerate ( range ( 1 , p + 1 ) ) : mask = np . ones_like ( xvals ) mask [ np . where ( xvals <= 0 ) [ 0 ] ] = 0 XX [ ind , : ] = pow ( xvals , exp ) * mask XX = XX . T def neg_loglik ( eta ) : mask = np . ones_like ( xvals ) mask [ np . where ( xvals <= 0 ) [ 0 ] ] = 0 g_eta_raw = np . exp ( np . dot ( XX , eta ) ) * mask if ( ( np . sum ( g_eta_raw ) == np . inf ) | ( np . sum ( g_eta_raw ) <= 100 * np . finfo ( np . double ) . tiny ) ) : return ( 1000 * ( len ( X ) + sum ( eta ** 2 ) ) ) g_eta_main = g_eta_raw / sum ( g_eta_raw ) g_eta = ( ( 1 - unif_fraction ) * g_eta_main + unif_fraction * mask / sum ( mask ) ) f_eta = fftconvolve ( g_eta , noise_rotate , mode = 'same' ) return np . sum ( np . interp ( X , xvals , - np . log ( np . maximum ( f_eta , 0.0000001 ) ) ) ) eta_hat = minimize ( neg_loglik , list ( itertools . repeat ( - 1 , p ) ) ) . x g_eta_raw = np . exp ( np . dot ( XX , eta_hat ) ) * mask g_eta_main = g_eta_raw / sum ( g_eta_raw ) g_eta = ( ( 1 - unif_fraction ) * g_eta_main + unif_fraction * mask ) / sum ( mask ) return xvals , g_eta
Fit empirical Bayes prior in the hierarchical model [Efron2014]_.

    .. math::

        mu ~ G, X ~ N(mu, sigma^2)

    Parameters
    ----------
    X: ndarray
        A 1D array of observations.
    sigma: float
        Noise estimate on X.
    p: int
        Number of parameters used to fit G. Default: 5
    nbin: int
        Number of bins used for discrete approximation.
        Default: 200
    unif_fraction: float
        Fraction of G modeled as "slab". Default: 0.1

    Returns
    -------
    An array of the posterior density estimate g.
Fit empirical Bayes prior in the hierarchical model [ Efron2014 ] _ .