.. |GVar| replace:: :class:`gvar.GVar`
.. |nonlinear_fit| replace:: :class:`lsqfit.nonlinear_fit`
.. |BufferDict| replace:: :class:`gvar.BufferDict`
.. |~| unicode:: U+00A0
   :trim:

.. highlight:: python

.. _non-gaussian-behavior:

Non-Gaussian Behavior; Testing Fits
=====================================

Introduction
--------------------
The various analyses in the Tutorial assume implicitly that every
probability distribution relevant to a fit is Gaussian. The input
data and priors are assumed Gaussian. The ``chi**2`` function is
assumed to be well approximated by a Gaussian in the vicinity of
its minimum, in order to estimate uncertainties for the best-fit
parameters. Functions of those parameters are assumed to yield
results that are described by Gaussian random variables. These assumptions
are usually pretty good for high-statistics data, when standard deviations
are small, but can lead to problems with low statistics.

Here we present three methods for testing these assumptions.
Some of these techniques, like the *statistical bootstrap* and
Bayesian integration, can also be used to analyze non-Gaussian
results.


Bootstrap Error Analysis; Non-Gaussian Output
-------------------------------------------------
The bootstrap provides an efficient way to check on a fit's
validity, and also a method for analyzing non-Gaussian outputs.
The strategy is to:

    1.  make a large number of "bootstrap copies" of the
        original input data and prior that differ from each other
        by random amounts characteristic of the underlying randomness
        in the original data and prior (see
        the documentation for 
        
            :meth:`lsqfit.nonlinear_fit.bootstrapped_fit_iter`

        for more information);

    2.  repeat the entire fit analysis for each bootstrap copy of
        the data and prior, extracting fit results from each;

    3.  use the variation of the fit results from bootstrap copy
        to bootstrap copy to determine an approximate probability
        distribution (possibly non-Gaussian) for the each result.

To illustrate, we return to our fit in the section
on :ref:`correlated-parameters`, where the uncertainties on the
final parameters were relatively large.
We will use a booststrap analysis to check the error
estimates coming out of that fit.
We do this by adding code right after the
fit, in the ``main()`` function::

    import numpy as np
    import gvar as gv
    import lsqfit

    def main():
        x, y = make_data()
        prior = make_prior()
        fit = lsqfit.nonlinear_fit(prior=prior, data=(x,y), fcn=fcn)
        print(fit)
        print('p1/p0 =', fit.p[1] / fit.p[0], 'p3/p2 =', fit.p[3] / fit.p[2])
        print('corr(p0,p1) =', gv.evalcorr(fit.p[:2])[1,0])

        # boostrap analysis: collect bootstrap data
        print('\nBootstrap Analysis:')
        Nbs = 40                # number of bootstrap copies
        output = {'p':[], 'p1/p0':[], 'p3/p2':[]}
        for bsfit in fit.bootstrapped_fit_iter(Nbs):
            p = bsfit.pmean
            output['p'].append(p)
            output['p1/p0'].append(p[1] / p[0])
            output['p3/p2'].append(p[3] / p[2])

        # average over bootstrap copies and tabulate results
        output = gv.dataset.avg_data(output, bstrap=True)
        print(gv.tabulate(output))
        print('corr(p0,p1) =', gv.evalcorr(output['p'][:2])[1,0])

    def make_data():
        x = np.array([
            4., 2., 1., 0.5, 0.25, 0.167, 0.125, 0.1, 0.0833, 0.0714, 0.0625
            ])
        y = gv.gvar([
            '0.198(14)', '0.216(15)', '0.184(23)', '0.156(44)', '0.099(49)',
            '0.142(40)', '0.108(32)', '0.065(26)', '0.044(22)', '0.041(19)',
            '0.044(16)'
            ])
        return x, y

    def make_prior():
        p = gv.gvar(['0(1)', '0(1)', '0(1)', '0(1)'])
        p[1] = 20 * p[0] + gv.gvar('0.0(1)')     # p[1] correlated with p[0]
        return p

    def fcn(x, p):
        return (p[0] * (x**2 + p[1] * x)) / (x**2 + x * p[2] + p[3])

    if __name__ == '__main__':
        main()

The ``bootstrapped_fit_iter`` produces fits ``bsfit`` for each of
``Nbs=40`` different bootstrap copies of the input data (``y`` and the prior).
We collect the mean values for the various parameters and functions of
parameters from each fit, ignoring the uncertainties, and
then calculate averages and covariance matrices from these results using
:func:`gvar.dataset.avg_data`.

Most of the bootstrap results agree with the results coming directly from
the fit:

.. literalinclude:: eg3c.out

In particular, the bootstrap analysis confirms the previous error estimates
(to within 10-30%, since ``Nbs=40``) except for ``p3/p2``, where the error
is substantially larger in the bootstrap analysis.

If ``p3/p2`` is important, one might want to look
more closely at its distribution.
We use the bootstrap to create histograms of the probability distributions
of ``p3/p2`` and ``p1/p01``
by adding the following code to the end of the ``main()``
function::

        print('Histogram Analysis:')
        count = {'p1/p0':[], 'p3/p2':[]}
        hist = {
            'p1/p0':gv.PDFHistogram(fit.p[1] / fit.p[0]),
            'p3/p2':gv.PDFHistogram(fit.p[3] / fit.p[2]),
            }

        # collect bootstrap data
        for bsfit in fit.bootstrapped_fit_iter(n=1000):
            p = bsfit.pmean
            count['p1/p0'].append(hist['p1/p0'].count(p[1] / p[0]))
            count['p3/p2'].append(hist['p3/p2'].count(p[3] / p[2]))

        # calculate averages and covariances
        count = gv.dataset.avg_data(count)

        # print histogram statistics and show plots
        import matplotlib.pyplot as plt
        pltnum = 1
        for k in count:
            print(k + ':')
            print(hist[k].analyze(count[k]).stats)
            plt.subplot(1, 2, pltnum)
            plt.xlabel(k)
            hist[k].make_plot(count[k], plot=plt)
            if pltnum == 2:
                plt.ylabel('')
            pltnum += 1
        plt.show()

Here we do 1000 bootstrap copies (rather than 40)
to improve the accuracy of the bootstrap
results. The output from this code shows statistical analyses of the
histogram data for ``p1/p0`` and ``p3/p2``:

.. literalinclude:: eg3d.out

The code also displays histograms of the probability distributions, where the
dashed lines show the results expected directly from the fit (that is,
in the Gaussian approximation):

    .. image:: eg3d.png
        :width: 100%

While the distribution for ``p1/p0`` is consistent with the fit
results (dashed line) and Gaussian,
the distribution for ``p3/p2`` is significantly skewed, with
a much longer tail to the right. The final result for ``p3/p2`` might
more accurately be summarized as 0.48 with errors of +0.31 and |~| -0.15,
although the Gaussian estimate of 0.48±0.22 would suffice for many
applications.
The skewed distribution for ``p3/p2`` is not particularly
surprising given the ±50% uncertainty in the
denominator |~| ``p2``.


Bayesian Integrals
-------------------
Bayesian expectation values provide an alternative to least-squares fits.
These expectation values are integrals over the fit parameters that are
weighted by the probability density function (PDF for the parameters)
proportional
to ``exp(-chi**2/2)``, where ``chi**2`` includes contributions from both
the data and the priors. They can be used to
calculate mean values of the parameters, their covariances, and the means
and covariances of any function of the parameters.
These will agree with the
best-fit results of our least-squares fits provided ``chi**2`` is well
approximated by its quadratic expansion in the parameters --- that is,
insofar as
``exp(-chi**2/2)`` is well approximated
by the Gaussian distribution in the parameters
specified by their best-fit means and covariance matrix (from ``fit.p``).

Here we use
:class:`lsqfit.BayesIntegrator` to evaluate Bayesian expectation values.
:class:`lsqfit.BayesIntegrator` uses the :mod:`vegas` module for adaptive
multi-dimensional integration to evaluate expectation values. It integrates
arbitrary functions of the parameters, multiplied by the probability
density function, over the entire parameter space. (Module :mod:`vegas`
must be installed for :class:`lsqfit.BayesIntegrator`.)

To illustrate :class:`lsqfit.BayesIntegrator`,
we again revisit the analysis in the section
on :ref:`correlated-parameters`. We modify the end of the ``main()`` function
of our original code
to evaluate the means and covariances of the parameters, and also
their probability histograms, using a Bayesian integral::

    import matplotlib.pyplot as plt
    import numpy as np
    import gvar as gv
    import lsqfit

    def main():
        x, y = make_data()
        prior = make_prior()
        fit = lsqfit.nonlinear_fit(prior=prior, data=(x,y), fcn=fcn)
        print(fit)

        # Bayesian integrator
        expval = lsqfit.BayesIntegrator(fit)

        # adapt integrator expval to PDF from fit
        neval = 1000
        nitn = 10
        expval(neval=neval, nitn=nitn)

        # <g(p)> gives mean and covariance matrix, and counts for histograms
        hist = [
            gv.PDFHistogram(fit.p[0]), gv.PDFHistogram(fit.p[1]),
            gv.PDFHistogram(fit.p[2]), gv.PDFHistogram(fit.p[3]),
            ]
        def g(p):
            return dict(
                mean=p,
                outer=np.outer(p, p),
                count=[
                    hist[0].count(p[0]), hist[1].count(p[1]),
                    hist[2].count(p[2]), hist[3].count(p[3]),
                    ],
                )

        # evaluate expectation value of g(p)
        results = expval(g, neval=neval, nitn=nitn, adapt=False)

        # analyze results
        print('\nIterations:')
        print(results.summary())
        print('Integration Results:')
        pmean = results['mean']
        pcov =  results['outer'] - np.outer(pmean, pmean)
        print('    mean(p) =', pmean)
        print('    cov(p) =\n', pcov)

        # create GVars from results
        p = gv.gvar(gv.mean(pmean), gv.mean(pcov))
        print('\nBayesian Parameters:')
        print(gv.tabulate(p))

        # show histograms
        print('\nHistogram Statistics:')
        count = results['count']
        for i in range(4):
            # print histogram statistics
            print('p[{}]:'.format(i))
            print(hist[i].analyze(count[i]).stats)
            # make histogram plots
            plt.subplot(2, 2, i + 1)
            plt.xlabel('p[{}]'.format(i))
            hist[i].make_plot(count[i], plot=plt)
            if i % 2 != 0:
                plt.ylabel('')
        plt.show()

    def make_data():
        x = np.array([
            4.    ,  2.    ,  1.    ,  0.5   ,  0.25  ,  0.167 ,  0.125 ,
            0.1   ,  0.0833,  0.0714,  0.0625
            ])
        y = gv.gvar([
            '0.198(14)', '0.216(15)', '0.184(23)', '0.156(44)', '0.099(49)',
            '0.142(40)', '0.108(32)', '0.065(26)', '0.044(22)', '0.041(19)',
            '0.044(16)'
            ])
        return x, y

    def make_prior():
        p = gv.gvar(['0(1)', '0(1)', '0(1)', '0(1)'])
        p[1] = 20 * p[0] + gv.gvar('0.0(1)')     # p[1] correlated with p[0]
        return p

    def fcn(x, p):
        return (p[0] * (x**2 + p[1] * x)) / (x**2 + x * p[2] + p[3])

    if __name__ == '__main__':
        main()


Here ``expval`` is an integrator that is used to evaluate expectation
values of arbitrary functions of the fit parameters.
:class:`BayesIntegrator` uses output (``fit``) from a least-squares fit to
design a :mod:`vegas` integrator
optimized for calculating expectation values.
The integrator uses an
iterative Monte Carlo algorithm that adapts to the
probability density function after each iteration.
See the :mod:`vegas` documentation for much more information.

We first call the integrator without a function. This allows it
to adapt to the probability density function from the fit without the extra
overhead of evaluating a function of the parameters. The integrator
uses ``nitn=10`` iterations of the :mod:`vegas` algorithm, with at most
``neval=1000`` evaluations of the probability density function for each
iteration.

We then use the optimized integrator to evaluate the expectation value
of function ``g(p)`` (turning adaptation off with ``adapt=False``).
The expectation value of ``g(p)`` is returned
in dictionary ``results``.

The results from this script are:

.. literalinclude:: eg3e.out

The iterations table shows results from each of the ``nitn=10``
vegas iterations used to
evaluate the expectation values. Estimates for the integral of the probability
density function are listed for each iteration. (Results from the integrator
are approximate, with error estimates.) These are consistent with each other
and with the (more accurate) overall average.

The integration results show that the Bayesian estimates for the means
of the parameters are accurate to roughly 1% or better, which is
sufficiently accurate here given the size of the standard deviations.
Estimates for
the covariance matrix elements are less accurate, which is typical.
This information is converted into |GVar|\s for the parameters and
tabulated under "Bayesian Parameters," for comparison with the
original fit results --- the agreement is pretty good.
This is further confirmed by the
(posterior) probability distributions
for each parameter:

.. image:: eg3e.png
    :width: 85%

The means are shifted slightly from the fit results and there
is modest skewing, but the differences are not great.

As a second example of Bayesian integration, we return briefly to
the problem described in :ref:`positive-parameters`: we want the
average ``a`` of noisy data subject the constraint that the average
must be positive. The constraint is likely to
introduce strong distortions in the probability density function (PDF)
given that the fit analysis suggests a value of 0.011±0.013.
We plot the actual PDF using the following code, beginning with
a fit that uses a flat prior (between 0 and |~| 0.04)::

    import gvar as gv
    import lsqfit

    # data, prior, and fit function
    y = gv.gvar([
       '-0.17(20)', '-0.03(20)', '-0.39(20)', '0.10(20)', '-0.03(20)',
       '0.06(20)', '-0.23(20)', '-0.23(20)', '-0.15(20)', '-0.01(20)',
       '-0.12(20)', '0.05(20)', '-0.09(20)', '-0.36(20)', '0.09(20)',
       '-0.07(20)', '-0.31(20)', '0.12(20)', '0.11(20)', '0.13(20)'
       ])

    prior = {}
    prior['erfinv(50*a-1)'] = gv.gvar('0(1)') / gv.sqrt(2)

    def fcn(p, N=len(y)):
       a = (1 + p['50*a-1']) / 50.
       return N * [a]

    # least-squares fit
    fit = lsqfit.nonlinear_fit(prior=prior, data=y, fcn=fcn)
    print(fit)
    a = (1 + fit.p['50*a-1']) / 50
    print('a =', a)

    # Bayesian analysis: histogram for a
    hist = gv.PDFHistogram(a, nbin=16, binwidth=0.5)

    def g(p):
        a = (1 + p['50*a-1']) / 50
        return hist.count(a)

    expval = lsqfit.BayesIntegrator(fit)
    expval(neval=1009, nitn=10)
    count = expval(g, neval=1000, nitn=10, adapt=False)

    # print out results and show plot
    print('\nHistogram Analysis:')
    print (hist.analyze(count).stats)

    hist.make_plot(count, show=True)

The output from this script is

.. literalinclude:: eg6-hist.out

and the probability distribution for ``a`` looks like

.. image:: eg6.png
    :width: 70%

This distribution is distorted between ``a=0`` and the mean
value, but otherwise is fairly similar to the Gaussian result
0.11±0.13 (dashed line). A more accurate summary of the result
for ``a`` would be 0.12 with an error of +0.14 and -0.09, though
again the Gaussian result is not terribly misleading even in this
case.

The Bayesian integrals are relatively simple in these example. More
complicated problems can require much more computer time to evaluate
the integrals, with hundreds of thousands or millions of integrand
evaluations per iteration (``neval``). This is particularly true
as the number of parameters increases. :class:`BayesIntegrator` uses
information from the least-squares fit to simplify the integration
for :mod:`vegas` by optimizing the integration variables used, but
integrals over tens of variables are intrinsically challenging.
:class:`BayesIntegrator` can be used with MPI to run such integrals
on multiple processors, for a considerable speed-up.

We used Bayesian integrals here to deal with non-Gaussian behavior in
fit outputs. The case study :ref:`outliers` shows how to use them
when the input data is not quite Gaussian.

.. _testing-fits:

Testing Fits with Simulated Data
--------------------------------
Ideally we would test a fitting protocol by doing fits of data similar to
our actual fit but where we know the correct values for the fit parameters
ahead of the fit. Method

    :meth:`lsqfit.nonlinear_fit.simulated_fit_iter` 

returns an iterator that
creates any number of such simulations of the original fit.

A key assumption underlying least-squares fitting is that the fit
data ``y[i]`` are random samples from a distribution whose mean
is the fit function ``fcn(x, fitp)`` evaluated with the best-fit
values ``fitp`` for the parameters. ``simulated_fit_iter`` iterators
generate simulated data by drawing other random samples from the
same distribution, assigning them the same covariance matrix as the
original data. The simulated data are fit
using the same priors and fitter settings
as in the original fit, and the results (an :class:`lsqfit.nonlinear_fit`
object) are returned by the iterator. The fits with simulated data should
have good ``chi**2`` values, and the results from these fits
should agree, within errors, with the original fit results since the
simulated data are from the same distribution as the original data. There
is a problem with the fitting protocol if this is not the case most of the
time.

To illustrate we again examine the fits
in the section on :ref:`correlated-parameters`:
we add three fit simulations at the end of the ``main()`` function::

    import numpy as np
    import gvar as gv
    import lsqfit

    def main():
        x, y = make_data()
        prior = make_prior()
        fit = lsqfit.nonlinear_fit(prior=prior, data=(x,y), fcn=fcn)
        print(40 * '*' + ' real fit')
        print(fit.format(True))

        # 3 simulated fits
        for sfit in fit.simulated_fit_iter(n=3):
            # print simulated fit details
            print(40 * '=' + ' simulation')
            print(sfit.format(True))

            # compare simulated fit results with exact values (pexact=fit.pmean)
            diff = sfit.p - sfit.pexact
            print('\nsfit.p - pexact =', diff)
            print(gv.fmt_chi2(gv.chi2(diff)))
            print

    def make_data():
        x = np.array([
            4.    ,  2.    ,  1.    ,  0.5   ,  0.25  ,  0.167 ,  0.125 ,
            0.1   ,  0.0833,  0.0714,  0.0625
            ])
        y = gv.gvar([
            '0.198(14)', '0.216(15)', '0.184(23)', '0.156(44)', '0.099(49)',
            '0.142(40)', '0.108(32)', '0.065(26)', '0.044(22)', '0.041(19)',
            '0.044(16)'
            ])
        return x, y

    def make_prior():
        p = gv.gvar(['0(1)', '0(1)', '0(1)', '0(1)'])
        p[1] = 20 * p[0] + gv.gvar('0.0(1)')        # p[1] correlated with p[0]
        return p

    def fcn(x, p):
        return (p[0] * (x**2 + p[1] * x)) / (x**2 + x * p[2] + p[3])

    if __name__ == '__main__':
        main()

This code produces the following output, showing how the input data
fluctuate from simulation to simulation:

.. literalinclude:: eg3f.out

The parameters ``sfit.p`` produced by the simulated fits agree well
with the original fit parameters ``pexact=fit.pmean``, with good
fits in each case. We calculate the ``chi**2`` for the difference
``sfit.p - pexact`` in
each case; good ``chi**2`` values validate the parameter values, standard
deviations, and correlations.

.. _goodness-of-fit:

Goodness of Fit
------------------
The quality of a fit is often judged by the value of
``chi**2/N``, where ``N`` is the
number of degrees of freedom.
Conventionally we expect ``chi**2/N`` to be of order ``1 ± sqrt(2/N)``
since fluctuations in the mean values of the data are of order the
uncertainties in the data. More precisely the means are
assumed to be random samples drawn from a Gaussian distribution whose
means are given by the best-fit function and whose covariance matrix
comes from
the data. There are two situations where this measure of goodness-of-fit
becomes unreliable.

The first situation is when there is a large SVD cut on the data.
As discussed in :ref:`svd-cuts-statistics`, an SVD cut increases
the uncertainties
in the data without increasing the random fluctuations in the
data means (unless ``add_svdnoise=True``). As a result contributions
from the parts of the ``chi**2`` function affected by the SVD cut
tend to be much smaller than naively expected, artificially
pulling ``chi**2/N`` down.

The second situation that compromises ``chi**2`` is when some or all of
the priors used in a fit are broad --- that is, when a fit result for
a paramter has a much
smaller uncertainty than the corresponding prior, but a mean that
is artificially
close close to the prior's mean. This often arises when the means
used in the priors are not random samples (unlike the fit data), which is
frequently the case. Again contributions to ``chi**2``
associated with such priors tend to be much smaller than naively expected,
pulling ``chi**2`` down.

These complications can conspire to make ``chi**2/N``
signficantly less than |~| 1 when the fit is good. Of greater concern,
they can mask evidence of a bad fit:  ``chi**2/N ≈ 1`` is *not*
necessarily evidence of a good fit in such situations.

A simple way to address these situations is to redo the fit with
keyword parameters ``add_svdnoise=True`` and
``add_priornoise=True``. These cause :class:`lsqfit.nonlinear_fit` to
add extra fluctuations to the means in the prior and the data
that are characteristic of the probability distributions associated
with the priors and the SVD cut, respectively::

    prior =>  prior + (gv.sample(prior) - gv.mean(prior))
        y =>  y + gv.sample(y.svdcorrection)

These fluctuations
should leave fit results unchanged (within errors) but increase
``chi**2/N`` so it is of order one.

To add this test to the fit from :ref:`svd-cuts-statistics`, we modify the
code to include a second fit at the end::

    import numpy as np
    import gvar as gv
    import lsqfit

    def main():
        ysamples = [
            [0.0092441016, 0.0068974057, 0.0051480509, 0.0038431422, 0.0028690492], 
            [0.0092477405, 0.0069030565, 0.0051531383, 0.0038455855, 0.0028700587], 
            [0.0092558569, 0.0069102437, 0.0051596569, 0.0038514537, 0.0028749153], 
            [0.0092294581, 0.0068865156, 0.0051395262, 0.003835656, 0.0028630454], 
            [0.009240534, 0.0068961523, 0.0051480046, 0.0038424661, 0.0028675632],
            ]
        y = gv.dataset.avg_data(ysamples)
        x = np.array([15., 16., 17., 18., 19.])
        def fcn(p):
            return p['a'] * gv.exp(- p['b'] * x)
        prior = gv.gvar(dict(a='0.75(5)', b='0.30(3)'))
        fit = lsqfit.nonlinear_fit(data=y, prior=prior, fcn=fcn, svdcut=0.0028)
        print(fit.format(True))

        print('\n================ Add noise to prior, SVD')
        noisyfit = lsqfit.nonlinear_fit(
            data=y, prior=prior, fcn=fcn, svdcut=0.0028,
            add_svdnoise=True, add_priornoise=True,
            )
        print(noisyfit.format(True))

    if __name__ == '__main__':
        main()

Running this code gives the following output:

.. literalinclude:: eg10e.out

The fit with extra noise has a larger ``chi**2``, as expected,
but is still a good fit. It also
gives fit parameters that agree within errors
with those from the
original fit. In general, there is probably something wrong with
the original fit (e.g., ``svdcut``
too small, or priors inconsistent with the fit data)
if adding noise makes ``chi**2/N`` signficantly larger than one,
or changes the best-fit values of the parameters significantly.

.. _fit-residuals:

Fit Residuals and Q-Q Plots
---------------------------
It can be useful to examine the normalized residuals from a fit (in array
``fit.residuals``). These are the differences between
the data and the corresponding values from the fit function using the best-fit
values for the fit parameters. The differences are projected onto the 
eigenvectors of the correlation matrix and normalized by dividing by the 
square root of the corresponding eigenvalues. 
The statistical assumptions underlying |nonlinear_fit| imply that the 
normalized fit residuals should be uncorrelated and distributed 
randomly about zero in 
a Gaussian distribution. 

One way to test whether residuals from a fit have a Gaussian distribution
is to make a *Q-Q plot.* Plots for the two fits from the previous section
(one without extra noise on the left, and the other with noise) are:

.. image:: eg10e1.png
    :width: 49%

.. image:: eg10e2.png
    :width: 49%

These plots were made using 

    :meth:`lsqfit.nonlinear_fit.qqplot_residuals`, 

by adding the following lines at the end 
of the ``main()`` method::

    fit.qqplot_residuals().show()
    noisyfit.qqplot_residuals().show()

In each case the residuals are first ordered, from smallest to largest. 
They are then plotted against the value
expected for a residual at that position in the list if 
the list elements were drawn at 
random from a Gaussian distribution of unit width and zero mean. (For 
example, given 100 samples from the Gaussian distribution, the sample 
in position 16 of the ordered list should have a value around -1 since 
16% of the values should be more than one standard deviation below the 
mean.) The residuals are consistent with a Gaussian distribution if they
fall on or near a straight line in such a plot.

The plots show fits of the residuals to straight lines (red solid lines).
The residuals in the left plot (without additional noise) are reasonably
consistent with a straight line (and, therefore, a Gaussian distribution), 
but the slope |~| (0.55) is much less than 
one. This is because ``chi2/dof = 0.38`` for this fit 
is much smaller than one. Typically
we expect the slope to be roughly the square root of |~| ``chi2/dof``
(since ``chi2`` equals the sum of the residuals squared).

The residuals in the right plot are also quite linear in the Q-Q plot. 
In this case the fit residuals include extra noise associated with 
the priors and with the SVD cut, as discussed in the previous section.
As a result ``chi2/dof = 0.81`` is much closer to one, as is the 
resulting slope |~| (0.90) in the Q-Q plot. The fit line through 
the residuals is much closer here to the dashed line in the plot,
which is what would result if the residuals had unit standard 
deviation and zero mean.

The fits pictured above have relatively few residuals. 
Q-Q plots become increasingly 
compelling as the number of residuals increases. 
The following plots, from fits without (left) 
and with (right) prior/SVD noise,
come from a lattice QCD analysis with 383 |~| residuals:

.. image:: bmixing1.png 
    :width: 49%

.. image:: bmixing2.png 
    :width: 49%



