Tutorial
=======================================

.. moduleauthor:: G. Peter Lepage <g.p.lepage@cornell.edu>

.. |Integrator| replace:: :class:`vegas.Integrator`
.. |PDFIntegrator| replace:: :class:`vegas.PDFIntegrator`
.. |AdaptiveMap| replace:: :class:`vegas.AdaptiveMap`
.. |vegas| replace:: :mod:`vegas`
.. |WAvg| replace:: :class:`vegas.RunningWAvg`
.. |chi2| replace:: :math:`\chi^2`
.. |x| replace:: x
.. |y| replace:: y
.. |~| unicode:: U+00A0
   :trim:

Introduction
-------------

Class :class:`vegas.Integrator` gives Monte Carlo estimates of arbitrary
multidimensional integrals using the *vegas* algorithm
(G. P. Lepage, J. Comput. Phys. 27 (1978) 192 and J. Comput. Phys. 439 (2021) 110386).
The algorithm has two components.
First an automatic transformation is applied to to the integration variables
in an attempt to flatten the integrand. Then a Monte Carlo estimate of the
integral is made using the  transformed variables. Flattening the integrand
makes the integral easier and improves the estimate.  The transformation
applied to the integration variables is optimized
over several iterations of the algorithm: information about the integrand that
is collected during one iteration is used to  improve the transformation used
in the next iteration.

Monte Carlo integration makes few assumptions about the
integrand --- it needn't be analytic nor even continuous. This
makes Monte Carlo integration unusually robust. It also makes it well suited
for adaptive integration. Adaptive strategies are essential for
multidimensional integration, especially in high dimensions, because
multidimensional space is large, with  lots of corners, making it
easy to lose important features in the integrand.

Monte Carlo integration also provides efficient and reliable methods for
estimating the
accuracy of its results. In particular, each Monte Carlo
estimate of an integral is a random number from a distribution
whose mean is the correct value of the integral. This distribution is
Gaussian or normal provided
the number of integrand samples is sufficiently large.
In practice we generate multiple
estimates of the integral
in order to verify that the distribution is indeed Gaussian.
Error analysis is straightforward if the
integral estimates are Gaussian.

The |vegas| algorithm has been in use for decades and implementations are
available in many programming languages, including Fortran (the original
version), C and C++. The algorithm used here is significantly improved over
the original implementation, and that used in most other implementations.
It uses two adaptive strategies: importance sampling, as in the original
implementation, and adaptive stratified sampling, which is new. The 
new algorithm is described in G. P. Lepage, arXiv_2009.05112_ 
(J. Comput. Phys. 439 (2021) 110386).

.. _arXiv_2009.05112: https://arxiv.org/abs/2009.05112

This module is written in Cython, so it is almost as fast as compiled Fortran or
C, particularly when the integrand is also coded in Cython (or some other
compiled language), as discussed below.

The following sections describe how to use |vegas|. Almost every
example shown is a complete code, which can be copied into a file
and run with Python. It is worthwhile playing with the parameters to see how
things change.

*About Printing:* The examples in this tutorial use the print function as it is
used in Python 3. Drop the outermost parenthesis in each print statement if
using Python 2, or add ::

    from __future__ import print_function

at the start of your file.


.. _basic_integrals:

Basic Integrals
----------------
Here we illustrate the use of |vegas| by estimating the integral

.. math::

    C\int_{-1}^1 dx_0 \int_0^1 dx_1 \int_0^1 dx_2 \int_0^1 dx_3
    \,\,\mathrm{e}^{- 100 \sum_{d}(x_d-0.5)^2}  ,

where constant :math:`C` is chosen so that the exact integral is 1.
The following code shows how this can be done::

    import vegas
    import math

    def f(x):
        dx2 = 0
        for d in range(4):
            dx2 += (x[d] - 0.5) ** 2
        return math.exp(-dx2 * 100.) * 1013.2118364296088

    integ = vegas.Integrator([[-1, 1], [0, 1], [0, 1], [0, 1]])

    result = integ(f, nitn=10, neval=1000)
    print(result.summary())
    print('result = %s    Q = %.2f' % (result, result.Q))

First we define the integrand ``f(x)`` where ``x[d]`` specifies a  point in the
4-dimensional space. We then create an  integrator, ``integ``, which is an
integration operator  that can be applied to any 4-dimensional function. It is
where we specify the integration volume.
Finally we apply ``integ`` to our integrand ``f(x)``,
telling the integrator to estimate  the integral using ``nitn=10`` iterations
of the |vegas| algorithm, each of which uses no more than ``neval=1000``
evaluations of the integrand. Each iteration produces an independent
estimate of the integral. The final estimate is the weighted average of
the results from all 10 iterations, and is returned by ``integ(f ...)``.
The call ``result.summary()`` returns
a summary of results from each iteration.

This code produces the following output:

.. literalinclude:: eg1a.out

There are several things to note here:

    **Adaptation:** Integration estimates are shown for
    each of the 10 iterations,
    giving both the estimate from just that iteration, and the weighted
    average of results from all iterations up to that point.  The
    estimates from the first two iterations are not accurate at
    all, with errors equal to 25--140% of the final result.
    |vegas| initially has no information about the integrand
    and so does a relatively poor job of estimating the integral.
    It uses information from the samples in one iteration, however,
    to remap the integration variables for subsequent iterations,
    concentrating samples where the function is largest and reducing
    errors.
    As a result, the per-iteration error
    is reduced to 3.4% by the fifth iteration, and almost to 1% by
    the end --- an improvement by a factor of more than 100 from the start. 
    Eventually the per-iteration error
    stops decreasing because |vegas| has found the optimal remapping,
    at which point
    it is fully adapted to the integrand.

    **Weighted Average:** The final result, 1.0057 ± 0.0064,
    is obtained from a weighted
    average of the separate results from each iteration:
    estimates are weighted by the inverse variance, thereby giving
    much less weight to the early iterations, where the errors are
    largest.
    The individual estimates are statistical: each
    is a random number drawn from a distribution whose mean
    equals the correct value of the integral, and the errors
    quoted are estimates of the standard deviations of those
    distributions. The distributions are Gaussian provided
    the number of integrand evaluations per iteration (``neval``)
    is sufficiently large, in which case the standard deviation
    is a reliable estimate of the error.
    The weighted average :math:`\overline I`  minimizes

    .. math::

      \chi^2 \,\equiv\, \sum_i \frac{(I_i - \overline I)^2}{\sigma_{i}^2}

    where :math:`I_i \pm \sigma_{i}` are the estimates from
    individual iterations. If the :math:`I_i` are Gaussian,
    :math:`\chi^2` should be of order the number of degrees of
    freedom (plus or minus the square root of double that number);
    here the number of degrees of freedom is the number of
    iterations minus 1.

    The distributions are likely non-Gaussian, and error estimates
    unreliable, if |chi2| is
    much larger than the number of iterations. This criterion is quantified
    by the *Q* or *p-value* of the :math:`\chi^2`,
    which is the probability that a
    larger :math:`\chi^2` could result from random (Gaussian)
    fluctuations. A very small *Q* (less than 0.05-0.1) indicates
    that the :math:`\chi^2` is too large to be accounted for by
    statistical fluctuations --- that is, the estimates of the integral
    from different iterations do not agree with each other to
    within errors. This means that ``neval`` is not sufficiently
    large to guarantee Gaussian behavior, and must be increased
    if the error estimates are to be trusted.


    ``integ(f...)`` returns a weighted-average object,
    of type :class:`vegas.RAvg`, that has the following
    attributes:

      ``result.mean`` --- weighted average of all estimates of the integral;

      ``result.sdev`` --- standard deviation of the weighted average;

      ``result.chi2`` --- :math:`\chi^2` of the weighted average;

      ``result.dof`` --- number of degrees of freedom;

      ``result.Q`` --- *Q* or *p-value* of the weighted average's |chi2|;

      ``result.itn_results`` --- list of the integral estimates from each iteration;

      ``result.sum_neval`` --- total number of integrand evaluations used.

      ``result.avg_neval`` --- average number of integrand evaluations per iteration

    In this example the final *Q* is 0.25, indicating that the
    :math:`\chi^2` for this average is not particularly unlikely and
    thus the error estimate is likely reliable.

    **Precision:** The precision of |vegas| estimates is
    determined by ``nitn``, the number of iterations
    of the |vegas| algorithm,
    and by ``neval``, the maximum number of integrand evaluations
    made per iteration.
    The computing cost is typically proportional to the
    product of ``nitn`` and ``neval``.
    The number of integrand
    evaluations per iteration
    varies from iteration to iteration,
    here between 860 and 960. Typically |vegas| needs more
    integration points in early iterations, before it has fully
    adapted to the integrand.

    We can increase precision by increasing either ``nitn`` or ``neval``,
    but it is
    generally far better to increase ``neval``. For example,
    adding the following lines to the code above ::

      result = integ(f, nitn=100, neval=1000)
      print('larger nitn  => %s    Q = %.2f' % (result, result.Q))

      result = integ(f, nitn=10, neval=1e4)
      print('larger neval => %s    Q = %.2f' % (result, result.Q))

    generates the following results:

    .. literalinclude:: eg1b.out

    The total number of integrand evaluations, ``nitn * neval``, is
    about the same in both cases, but increasing ``neval`` is more
    than twice as accurate as increasing ``nitn``. Typically you
    want to use no more than 10 or 20 iterations beyond the
    point where |vegas| has fully adapted. You want some number of
    iterations so that you can verify Gaussian behavior by
    checking the |chi2| and *Q*, but not too many.

    It is also generally useful to compare two or more
    results from values of ``neval`` that differ by a
    significant factor (4--10, say). These should agree within
    errors. If they do not, it could be due to non-Gaussian
    artifacts caused by a small ``neval``. |vegas|
    estimates have two sources of error. One is the statistical
    error, which is what is quoted by |vegas|. The other is
    a systematic error due to residual non-Gaussian
    effects. The systematic error vanishes like
    ``1/neval`` or faster, and so becomes negligible compared with
    the statistical error as ``neval`` increases.
    The systematic error can bias the Monte Carlo estimate, however,
    if ``neval`` is insufficiently large. This usually
    results in a large |chi2| (and small *Q*), but a
    more reliable check is to compare
    results that use significantly different values of ``neval``.
    The systematic errors due to non-Gaussian behavior are
    likely negligible if the different estimates agree to
    within the statistical errors.

    The possibility of systematic biases
    is another reason for increasing ``neval``
    rather than ``nitn`` to obtain more precision.
    Making ``neval`` larger and larger is guaranteed
    to improve the Monte Carlo estimate, as the statistical
    error decreases and the
    systematic error decreases even more quickly.
    Making ``nitn`` larger and larger, on the other hand,
    is guaranteed eventually to give the wrong
    answer. This is because at some point the statistical error
    (which falls as ``sqrt(1/nitn)``) will no longer
    mask the systematic error (which is unaffected by ``nitn``).
    The systematic error for the integral
    above (with ``neval=1000``) is about -0.0008, which
    is negligible compared to the statistical error unless
    ``nitn`` is of order 1500 or larger --- so systematic errors
    aren't a problem with ``nitn=10``.

    **Early Iterations:** Integral estimates from early iterations,
    before |vegas| has adapted, can be quite
    crude. With very peaky integrands, these are often far from
    the correct answer with highly unreliable error  estimates. For
    example, the integral above becomes more 
    difficult if we double the length of each side of the
    integration volume by redefining ``integ`` as::

      integ = vegas.Integrator([[-2, 2], [0, 2], [0, 2], [0., 2]])

    The code above then gives:

    .. literalinclude:: eg1c.out

    |vegas| misses the peak completely in the first iteration,
    giving an estimate that is completely
    wrong (by 1000 standard deviations!).
    Some of its samples hit the peak's shoulders, so |vegas| is
    eventually able to find the peak (by iterations 5--6), but
    the integrand estimates are wildly non-Gaussian before that
    point. This results in a nonsensical final result, as
    indicated by the ``Q = 0.00``.

    It is common practice in using |vegas| to discard
    estimates from the first several iterations, before the
    algorithm has adapted, in order to avoid ruining the
    final result in this way. This is done by replacing the
    single call to ``integ(f...)`` in the original code
    with two calls::

      # step 1 -- adapt to f; discard results
      integ(f, nitn=10, neval=1000)

      # step 2 -- integ has adapted to f; keep results
      result = integ(f, nitn=10, neval=1000)
      print(result.summary())
      print('result = %s    Q = %.2f' % (result, result.Q))

    The integrator is trained in the first
    step, as it adapts to the integrand, and so is more or less
    fully adapted from the start in the second step, which yields:

    .. literalinclude:: eg1d.out

    The final result is now reliable.

    **Other Integrands:** Once ``integ`` has been trained on ``f(x)``,
    it can be usefully applied
    to other functions with similar structure. For example, adding
    the following at the end of the original code, ::

      def g(x):
          return x[0] * f(x)

      result = integ(g, nitn=10, neval=1000)
      print(result.summary())
      print('result = %s    Q = %.2f' % (result, result.Q))

    gives the following new output:

    .. literalinclude:: eg1e.out

    Again the grid is almost optimal for ``g(x)`` from the start,
    because ``g(x)`` peaks in the same region as ``f(x)``.
    The exact value for this integral is very close to 0.5.

    **Non-Rectangular Volumes:** |vegas| can integrate over volumes of
    non-rectangular shape. For example, we can replace integrand ``f(x)``
    above
    by the same Gaussian, but restricted to a 4-sphere of radius 0.2,
    centered on the Gaussian::

        import vegas
        import math

        def f_sph(x):
            dx2 = 0
            for d in range(4):
                dx2 += (x[d] - 0.5) ** 2
            if dx2 < 0.2 ** 2:
                return math.exp(-dx2 * 100.) * 1115.3539360527281318
            else:
                return 0.0

        integ = vegas.Integrator([[-1, 1], [0, 1], [0, 1], [0, 1]])

        integ(f_sph, nitn=10, neval=1000)           # adapt the grid
        result = integ(f_sph, nitn=10, neval=1000)  # estimate the integral
        print(result.summary())
        print('result = %s    Q = %.2f' % (result, result.Q))


    The normalization is adjusted to again make the
    exact integral equal 1. Integrating as before gives:

    .. literalinclude:: eg1f.out

    It is a good idea to make the actual integration volume as large a
    fraction as possible of the total volume used by |vegas| ---
    by choosing integration variables properly --- so
    |vegas| doesn't spend lots of effort on regions where the integrand
    is exactly 0. Also, it can be challenging for |vegas|
    to find the region of
    non-zero integrand in high dimensions: integrating ``f_sph(x)``
    in 20 dimensions instead of 4, for example,
    would require ``neval=1e16``
    integrand evaluations per iteration to have any chance of
    finding the region of non-zero integrand, because the volume of
    the 20-dimensional sphere is a tiny fraction of the total
    integration volume. The final error in the example above would have
    been cut in half had we used the integration volume
    ``4 * [[0.3, 0.7]]`` instead of ``[[-1, 1], [0, 1], [0, 1], [0, 1]]``.

    Note, finally, that integration to infinity is also possible:
    map the relevant variable into a different variable
    of finite range. For example,  an integral over :math:`x\equiv b z / (1-z)`
    from 0 to infinity is easily re-expressed as
    an integral over :math:`z` from 0 to 1, where the transformation
    emphasizes the region in :math:`x` of order free parameter :math:`b`.

    **Damping:** The result in the previous section
    can be improved somewhat by slowing down
    |vegas|’s adaptation::

        ...
        integ(f_sph, nitn=10, neval=1000, alpha=0.1)
        result = integ(f_sph, nitn=10, neval=1000, alpha=0.1)
        ...

    Parameter ``alpha`` controls the speed with which |vegas|
    adapts, with smaller ``alpha``\s giving slower adaptation.
    Here we reduce ``alpha`` to 0.1, from its default value of 0.5, and get
    the following output:

    .. literalinclude:: eg1h.out

    Notice how the errors fluctuate less from iteration to iteration
    with the smaller ``alpha`` in this case.
    Persistent, large fluctuations in the size
    of the per-iteration errors is often a signal that ``alpha`` should
    be reduced. With larger ``alpha``\s, |vegas| can over-react
    to random fluctuations it encounters as it samples the integrand.

    In general, we want ``alpha`` to be large enough so that |vegas| adapts
    quickly to the integrand, but not so large that it has difficulty
    holding on to the optimal tuning once it has found it. The best value
    depends upon the integrand.

    **adapt=False:** Adaptation can be turned off completely
    by setting parameter ``adapt=False``. There are three reasons one
    might do this. The first is if |vegas| is exhibiting the
    kind of instability discussed in the previous section --- one might
    use the following code, instead of that presented there::

        ...
        integ(f_sph, nitn=10, neval=1000, alpha=0.1)
        result = integ(f_sph, nitn=10, neval=1000, adapt=False)
        ...

    The second reason is that |vegas| runs slightly faster when it is
    no longer adapting to the integrand. The difference is not signficant
    for complicated integrands, but is noticable in simpler cases.

    The third reason for turning off adaptation is that |vegas| uses
    unweighted averages, rather than weighted averages, to combine
    results from different iterations when ``adapt=False``.
    Unweighted averages are not biased. They have no systematic error
    of the sort discussed above, and so give correct results even
    for very large numbers of iterations, ``nitn``.

    The lack of systematic biases is *not* a strong reason for turning
    off adaptation, however, since the biases are
    usually negligible (see above). The most important reason is the
    first: stability. 
    
    ``adapt=False`` is particularly useful when the number of
    integrand evaluations ``neval`` is small for the integrand, leading
    to large fluctuations in the errors from iteration to 
    iteration. For example, 
    the following output is from an estimate (with ``neval=2.5e4``)
    of an eight-dimensional
    integral with three sharp peaks along the diagonal (Eq. |~| (45) in 
    arXiv_2009.05112_, normalized so that the correct answer equals 1)::

        itn   integral        wgt average     chi2/dof        Q
        -------------------------------------------------------
         1   0.75(43)        0.75(43)            0.00     1.00
         2   0.506(58)       0.510(58)           0.32     0.57
         3   0.80(21)        0.530(56)           1.02     0.36
         4   0.76(11)        0.576(50)           1.81     0.14
         5   1.27(29)        0.596(49)           2.74     0.03
         6   1.10(19)        0.629(48)           3.56     0.00
         7   0.802(73)       0.681(40)           3.63     0.00
         8   2.8(2.0)        0.681(40)           3.27     0.00
         9   0.907(90)       0.719(36)           3.52     0.00
        10   1.07(16)        0.736(35)           3.65     0.00

        itn   integral        average         chi2/dof        Q
        -------------------------------------------------------
         1   1.13(14)        1.13(14)            0.00     1.00
         2   1.064(96)       1.095(86)           0.13     0.72
         3   1.03(10)        1.072(67)           0.19     0.83
         4   0.924(94)       1.035(55)           0.58     0.63
         5   0.858(71)       1.000(46)           1.08     0.37
         6   0.97(11)        0.995(43)           0.84     0.52
         7   0.924(69)       0.985(38)           0.84     0.54
         8   1.19(16)        1.010(39)           1.01     0.42
         9   1.74(73)        1.092(88)           1.01     0.42
        10   0.942(89)       1.077(80)           1.02     0.42
            
    The first 10 iterations are used to train the |vegas| map; their
    results are discarded. The 
    next 10 iterations, with ``adapt=False``, have uncertainties that
    fluctuate in size by an order of magnitude, but still give a reliable 
    estimate for the integral (1.08(8)). Allowing |vegas|
    to continue adapting in the the second set of iterations gives
    results like 0.887(25), which is 4.5 standard deviations too low;
    the real uncertainty is larger than |~| ±0.025.

    Training the integrator and then setting ``adapt=False`` for the
    final results works best if the number of evaluations per iteration
    (``neval``) is the same in both steps. This is because the second
    of |vegas|'s adaptation strategies (:ref:`adaptive-stratified-sampling`) is
    usually reinitialized when ``neval`` changes, and so is not
    used at all when ``neval`` is changed at the same time ``adapt=False``
    is set.

Multiple Integrands Simultaneously
-----------------------------------
|vegas| can be used to integrate multiple integrands simultaneously, using
the same integration points for each of the integrands. This is useful
in situations where the integrands have similar structure, with peaks in
the same locations. There can be  signficant advantages in sampling
different integrands at precisely the same points in |x| space, because
then Monte Carlo estimates for the different integrals are correlated.
If the integrands are very similar to each other, the correlations can be
very strong. This leads to greatly reduced errors in ratios or differences
of the resulting integrals as the fluctuations cancel.

Consider a simple example. We want to compute
the normalization and first two moments of a
sharply peaked probability distribution:

.. math::
    I_0  &\equiv  \int_0^1 d^4x\;
        \mathrm{e}^{- 200 \sum_{d}(x_d-0.5)^2}\\
    I_1  &\equiv \int_0^1 d^4x\; x_0 \;
        \mathrm{e}^{- 200 \sum_{d}(x_d-0.5)^2} \\
    I_2 &\equiv \int_0^1 d^4x\; x_0^2 \;
        \mathrm{e}^{- 200 \sum_{d}(x_d-0.5)^2}

From these integrals we determine the mean and width of the  distribution
projected onto one of the axes:

.. math::
    \langle x \rangle  &\equiv I_1 / I_0 \\[1ex]
    \sigma_x^2  &\equiv \langle x^2 \rangle - \langle x \rangle^2 \\
               &= I_2 / I_0  - (I_1 / I_0)^2

This can be done using the following code::

    import vegas
    import math
    import gvar as gv

    def f(x):
        dx2 = 0.0
        for d in range(4):
            dx2 += (x[d] - 0.5) ** 2
        f = math.exp(-200 * dx2)
        return [f, f * x[0], f * x[0] ** 2]

    integ = vegas.Integrator(4 * [[0, 1]])

    # adapt grid
    training = integ(f, nitn=10, neval=2000)

    # final analysis
    result = integ(f, nitn=10, neval=10000)
    print('I[0] =', result[0], '  I[1] =', result[1], '  I[2] =', result[2])
    print('Q = %.2f\n' % result.Q)
    print('<x> =', result[1] / result[0])
    print(
        'sigma_x**2 = <x**2> - <x>**2 =',
        result[2] / result[0] - (result[1] / result[0]) ** 2
        )
    print('\ncorrelation matrix:\n', gv.evalcorr(result))

The code is very similar to that used in the previous section. The
main difference is that the integrand function and |vegas|
return arrays of results --- in
both cases, one result for each of the three integrals. |vegas| always adapts to
the first integrand in the array. The ``Q`` value is for all three
of the integrals, taken together.

The code produces the following output:

.. literalinclude:: eg3a.out

The estimates for the individual integrals are separately accurate to
about ±0.05%,
but the estimate for :math:`\langle x \rangle = I_1/I_0`
is accurate to ±0.01%.
This is almost an order
of magnitude (8x) more accurate than we would obtain absent correlations.
The correlation matrix shows that there is 98% correlation between the
statistical fluctuations in estimates for :math:`I_0` and :math:`I_1`,
and so the bulk of these fluctuations cancel in the ratio.
The estimate for the variance :math:`\sigma^2_x`
is 48x more accurate than we would
have obtained had the integrals been evaluated separately. Both estimates
are correct to within the quoted errors.

The individual results are objects of type :class:`gvar.GVar`, which
represent Gaussian random variables. Such objects have means
(``result[i].mean``) and standard deviations (``result[i].sdev``), but
also can be statistically correlated with other :class:`gvar.GVar`\s.
Such correlations are handled automatically by :mod:`gvar` when
:class:`gvar.GVar`\s are combined with each other or with numbers in
arithmetical expressions. (Documentation for :mod:`gvar` can be found
at https://gvar.readthedocs.io or with the source code
at https://github.com/gplepage/gvar.git.)

Integrands can return dictionaries instead of arrays. The example above,
for example, can be rewritten as ::

    import vegas
    import math
    import gvar as gv

    def f(x):
        dx2 = 0.0
        for d in range(4):
            dx2 += (x[d] - 0.5) ** 2
        f = math.exp(-200 * dx2)
        return {'1':f, 'x':f * x[0], 'x**2':f * x[0] ** 2}

    integ = vegas.Integrator(4 * [[0, 1]])

    # adapt grid
    training = integ(f, nitn=10, neval=2000)

    # final analysis
    result = integ(f, nitn=10, neval=10000)
    print(result)
    print('Q = %.2f\n' % result.Q)
    print('<x> =', result['x'] / result['1'])
    print(
        'sigma_x**2 = <x**2> - <x>**2 =',
        result['x**2'] / result['1'] - (result['x'] / result['1']) ** 2
        )

which returns the following output:

.. literalinclude:: eg3b.out

The result returned by |vegas| is a dictionary using the same keys as the
dictionary returned by the integrand. Using a dictionary with descriptive
keys, instead of an array, can often make code more intelligible, and,
therefore, easier to write  and maintain. Here the values in the integrand's
dictionary are all numbers; in general, values can be  either numbers or
arrays (of any shape).

Calculating Distributions
----------------------------
|vegas| is often used to calculate distributions. The following
code, for example, evaluates both an integral ``I`` and the contributions
``dI`` to the integral coming from each of five different intervals ``dr``
in the radius measured
from the center of the integration volume. The normalized contributions
``dI/I`` are then tabulated::

    import vegas
    import numpy as np

    RMAX = (2 * 0.5**2) ** 0.5

    def fcn(x):
        dx2 = 0.0
        for d in range(2):
            dx2 += (x[d] - 0.5) ** 2
        I = np.exp(-dx2)
        # add I to appropriate bin in dI
        dI = np.zeros(5, dtype=float)
        dr = RMAX / len(dI)
        j = int(dx2 ** 0.5 / dr)
        dI[j] = I
        return dict(I=I, dI=dI)

    integ = vegas.Integrator(2 * [(0,1)])

    # results returned in a dictionary
    result = integ(fcn)
    print(result.summary())
    print('   I =', result['I'])
    print('dI/I =', result['dI'] / result['I'])
    print('sum(dI/I) =', sum(result['dI']) / result['I'])

Note the check at the end, to verify that the sum of the
``dI[i]``\s equals the original integral. Running this script gives
the following output:

.. literalinclude:: eg4.out

The integrator adapts to the full integral ``I`` but also gives
accurate results for the distribution ``dI`` (though not quite as
accurate). Note that ``sum(dI/I)`` is much more accurate than
any individual ``dI/I``, because of correlations between
the different ``dI/I`` values. (The uncertainty on ``sum(dI/I)`` would
be exactly zero absent roundoff errors.) Typically one has
more than five bins.

Bayesian Integrals
---------------------
The |vegas| module has a special-purpose integrator for 
evaluating averages over probability distributions. Given a 
multi-dimensional Gaussian probability distribution 
``g`` (specified by an array of ``gvar.GVar``\s or a dictionary whose 
values are ``gvar.GVar``\s or arrays of ``gvar.GVar``\s),
``vegas.PDFIntegrator(g)`` creates an integrator that 
is optimized to evaluate integrals of ``f(p) * pdf(p)``
where ``f(p)`` is an arbitrary function and ``pdf(p)``
is the probability density function (PDF) corresponding to ``g``.
Here ``p`` is a point in the parameter space of the 
distribution — an array or dictionary having the 
same layout as ``g``.

|PDFIntegrator| integrates over the entire 
parameter space of the
distribution but re-expresses integrals in terms of variables
that diagonalize ``g``'s covariance matrix and are centered at
its mean value. This greatly facilitates integration over these
variables using |vegas|, making integrals over
many parameters feasible, even when the parameters are highly
correlated. |PDFIntegrator| also pre-adapts the integrator 
to ``g``'s PDF so it is usually unnecessary to discard early iterations.

The PDF corresponding to ``g`` can be replaced by an arbitrary
PDF function ``pdf(p)``. The integration variables are still
specified by and optimized for ``g``, but the PDF used in the 
integrals is ``pdf(p)``.
|PDFIntegrator| normally evaluates the 
integrals of both ``pdf(p) * f(p)`` and ``pdf(p)``. The expectation 
value is the ratio of the two integrals, so the PDF need not be 
normalized. Note also that Monte Carlo uncertainties in 
the two integrals can be highly correlated, in which case 
the uncertainties are significantly reduced in the ratio.  

A simple illustration of |PDFIntegrator| is given by the following
code::

    import vegas
    import gvar as gv

    # multi-dimensional Gaussian distribution
    g = gv.BufferDict()
    g['a'] = gv.gvar([0., 1.], [[1., 0.99], [0.99, 1.]])
    g['fb(b)'] = gv.BufferDict.uniform('fb', 0.0, 2.0)

    # integrator for expectation values in distribution g
    g_expval = vegas.PDFIntegrator(g)

    # want expectation value of [fp, fp**2]
    def f_f2(p):
        a = p['a']
        b = p['b']
        fp = a[0] * a[1] + b
        return [fp, fp ** 2]

    # <f_f2> in distribution g
    results = g_expval(f_f2, neval=2000, nitn=5)
    print(results.summary())
    print('results =', results, '\n')

    # mean and standard deviation of fp's distribution
    fmean = results[0]
    fsdev = gv.sqrt(results[1] - results[0] ** 2)
    print ('fp.mean =', fmean, '   fp.sdev =', fsdev)
    print ("Gaussian approx'n for fp =", f_f2(g)[0], '\n')

    # g's pdf norm
    print('PDF norm =', results.pdfnorm)

Here the distribution ``g`` describes two highly correlated Gaussian
variables, ``a[0]`` and ``a[1]``, and a third uncorrelated variable ``b``
that is uniformly distributed on the interval [0,2] (see the :mod:`gvar` 
documentation for more information).
We use the integrator to calculated  the expectation value of 
``fp = a[0] * a[1] + b`` and ``fp**2``, so we can compute the 
mean and standard 
deviation of the ``fp`` |~| distribution. The output from this code 
shows that the Gaussian approximation 1.0(1.3) for the mean and 
standard deviation is not particularly 
close to the correct value |~| 2.0(1.8):

.. literalinclude:: eg7.out

In general function ``f(p)`` can return a number, or an array of
numbers, or a dictionary whose values are numbers or arrays of numbers.
This allows multiple expectation values to be evaluated simultaneously.

The discussion in :ref:`case_curve_fitting` illustrates how 
|PDFIntegrator| can be used with a non-Gaussian PDF in two
examples, one with 4 |~| parameters 
and the other with 22 |~| parameters. 


Faster Integrands
-------------------------
The computational cost of a realistic multidimensional integral
comes mostly from
the cost of evaluating the integrand at the Monte Carlo sample
points. Integrands written in pure Python are probably fast
enough for problems where ``neval=1e3`` or ``neval=1e4`` gives
enough precision. Some problems, however, require
hundreds of thousands or millions of function evaluations, or more.

We can significantly reduce the cost of evaluating the integrand
by using |vegas|'s batch mode. For example, replacing ::

    import vegas
    import math

    def f(x):
        dim = len(x)
        norm = 1013.2118364296088 ** (dim / 4.)
        dx2 = 0.0
        for d in range(dim):
            dx2 += (x[d] - 0.5) ** 2
        return math.exp(-100. * dx2) * norm

    integ = vegas.Integrator(4 * [[0, 1]])

    integ(f, nitn=10, neval=2e5)
    result = integ(f, nitn=10, neval=2e5)
    print('result = %s   Q = %.2f' % (result, result.Q))



by ::

    import vegas
    import numpy as np

    @vegas.batchintegrand
    def f_batch(x):
        # evaluate integrand at multiple points simultaneously
        dim = x.shape[1]
        norm = 1013.2118364296088 ** (dim / 4.)
        dx2 = 0.0
        for d in range(dim):
            dx2 += (x[:, d] - 0.5) ** 2
        return np.exp(-100. * dx2) * norm

    integ = vegas.Integrator(4 * [[0, 1]])

    integ(f_batch, nitn=10, neval=2e5)
    result = integ(f_batch, nitn=10, neval=2e5)
    print('result = %s   Q = %.2f' % (result, result.Q))

reduces the cost of the integral by an order of magnitude. Internally |vegas|
processes integration points in batches. (|vegas| parameter ``nhcube_batch``
determines the number of integration
points per batch (typically 1000s).) In batch mode,
|vegas| presents integration points to the integrand in batches
rather than offering them one at a
time. Here, for example, function ``f_batch(x)`` accepts  an array of integration
points --- ``x[i, d]`` where ``i=0...`` labels the integration point and
``d=0...`` the direction --- and returns an array of integrand values
corresponding  to those points. The decorator
:func:`vegas.batchintegrand` tells |vegas| that it should send
integration points to ``f(x)`` in batches.

An alternative to a function decorated with :func:`vegas.batchintegrand` is
a class derived from :class:`vegas.BatchIntegrand` that
behaves like a batch integrand::

    import vegas
    import numpy as np

    @vegas.batchintegrand
    class f_batch:
        def __init__(self, dim):         
            self.dim = dim
            self.norm = 1013.2118364296088 ** (dim / 4.)

        def __call__(self, x):
            # evaluate integrand at multiple points simultaneously
            dx2 = 0.0
            for d in range(self.dim):
                dx2 += (x[:, d] - 0.5) ** 2
            return np.exp(-100. * dx2) * self.norm

    f = f_batch(dim=4)
    integ = vegas.Integrator(f.dim * [[0, 1]])

    integ(f, nitn=10, neval=2e5)
    result = integ(f, nitn=10, neval=2e5)
    print('result = %s   Q = %.2f' % (result, result.Q))

This version is as fast as the previous batch integrand, but is
potentially more flexible because it is built around a class rather
than a function.  (Some classes won't allow decorators. An alternative 
to the decorator is to derive the class from ``vegas.BatchIntegrand``.)

The batch integrands here are fast because they are expressed in terms
:mod:`numpy` operators that act on entire arrays --- they evaluate the
integrand for all integration points in a batch at the same time.
That optimization is not always possible or simple.
It is unnecessary if we write the integrand in Cython, which
is a compiled hybrid of Python and C. The Cython version
of the (batch) integrand is::

    # file: cython_integrand.pyx

    import numpy as np

    # use exp from C
    from libc.math cimport exp

    def f_batch(double[:, ::1] x):
        cdef int i          # labels integration point
        cdef int d          # labels direction
        cdef int dim = x.shape[1]
        cdef double norm = 1013.2118364296088 ** (dim / 4.)
        cdef double dx2
        cdef double[::1] ans = np.empty(x.shape[0], float)
        for i in range(x.shape[0]):
            # integrand for integration point x[i]
            dx2 = 0.0
            for d in range(dim):
                dx2 += (x[i, d] - 0.5) ** 2
            ans[i] = exp(-100. * dx2) * norm
        return ans

We put this in a separate file called, say,
``cython_integrand.pyx``, and rewrite the main code as::

    import numpy as np
    import pyximport
    pyximport.install(inplace=True)

    import vegas
    from cython_integrand import f_batch
    f = vegas.batchintegrand(f_batch)

    integ = vegas.Integrator(4 * [[0, 1]])

    integ(f, nitn=10, neval=2e5)
    result = integ(f, nitn=10, neval=2e5)
    print('result = %s   Q = %.2f' % (result, result.Q))

Module :mod:`pyximport` is used here to cause the Cython
module ``cython_integrand.pyx`` to be compiled the first time
it is imported. The compiled code is used in subsequent
imports, so compilation occurs only once.

Batch mode is also useful for array-valued integrands.
The code from the previous section could have been written as::

    import vegas
    import gvar as gv
    import numpy as np

    dim = 4

    @vegas.batchintegrand
    def f(x):
        ans = np.empty((x.shape[0], 3), float)
        dx2 = 0.0
        for d in range(dim):
            dx2 += (x[:, d] - 0.5) ** 2
        ans[:, 0] = np.exp(-200 * dx2)
        ans[:, 1] = x[:, 0] * ans[:, 0]
        ans[:, 2] = x[:, 0] ** 2 * ans[:, 0]
        return ans

    integ = vegas.Integrator(4 * [[0, 1]])

    # adapt grid
    training = integ(f, nitn=10, neval=2000)

    # final analysis
    result = integ(f, nitn=10, neval=10000)
    print('I[0] =', result[0], '  I[1] =', result[1], '  I[2] =', result[2])
    print('Q = %.2f\n' % result.Q)
    print('<x> =', result[1] / result[0])
    print(
        'sigma_x**2 = <x**2> - <x>**2 =',
        result[2] / result[0] - (result[1] / result[0]) ** 2
        )
    print('\ncorrelation matrix:\n', gv.evalcorr(result))

Note that the batch index (here ``:``) always comes first. An extra
(first) index is also added to each value in the dictionary returned
by a dictionary-valued batch integrand: e.g., ::

    dim = 4

    @vegas.batchintegrand
    def f(x):
        ans = {}
        dx2 = 0.0
        for d in range(dim):
            dx2 += (x[:, d] - 0.5) ** 2
        ans['1'] = np.exp(-200 * dx2)
        ans['x'] = x[:, 0] * ans['1']
        ans['x**2'] = x[:, 0] ** 2 * ans['1']
        return ans

It is sometimes more convenient to have the batch index be the last index 
(the rightmost) rather than the first. Then ``@vegas.batchintegrand`` is
replaced by ``@vegas.rbatchintegrand``, and ``vegas.BatchIntegrand`` by
``vegas.RBatchIntegrand``. (Note that ``@vegas.batchintegrand``
and ``@vegas.lbatchintegrand`` are the same, as are ``vegas.BatchIntegrand``
and ``vegas.LBatchIntegrand``.)


Multiple Processors
---------------------------
|vegas| supports parallel evaluation of integrands 
on multiple processors. This
can shorten execution time substantially when the integrand is 
costly to evaluate. The following code, for example,
runs more than five times faster 
when using ``nproc=8`` processors instead of the 
default ``nproc=1`` (on a 2019 laptop)::

    import vegas
    import numpy as np

    # Integrand: ridge of N Gaussians spread along part of the diagonal
    def ridge(x):
        N = 10000
        x0 = np.linspace(0.4, 0.6, N)
        dx2 = 0.0
        for xd in x:
            dx2 += (xd - x0) ** 2
        return np.average(np.exp(-100. * dx2)) *  (100. / np.pi) ** (len(x) / 2.)

    def main():
        integ = vegas.Integrator(4 * [[0, 1]], nproc=8)  # 8 processors
        # adapt
        integ(ridge, nitn=10, neval=1e4)
        # final results
        result = integ(ridge, nitn=10, neval=1e4)
        print('result = %s    Q = %.2f' % (result, result.Q))

    if __name__ == '__main__':
        main()

The code doesn't run eight times faster because it takes time to 
initiate the ``nproc`` processes, and to feed data and 
collect results from them. 
Parallel processing only becomes useful when integrands are 
sufficiently costly that such overheads become negligible.

Parallel processing is managed by Python's 
:mod:`multiprocessing` module. 
The ``if __name__ == '__main__'`` construct at the end of
this code is essential when running on Windows or MacOS (in its 
default mode) as it prevents additional processes being launched
when the main module is imported as part of spawning the ``nproc`` 
processes; see the :mod:`multiprocessing` documentation for more 
details. This is not an issue for Linux/Unix. It is also important 
that the integrand and its return values can be pickled using 
Python's :mod:`pickle` module. This is the case for most pure 
Python integrands.

The code above will generate an ``AttributeError`` when run in some 
interactive environments (as opposed to running from the command line)
on some platforms. This can usually be fixed by putting the integrand 
function ``ridge(x)`` into a file and importing it into the script.

|vegas| also supports multi-processor evaluation of integrands using MPI
(via the Python module :mod:`mpi4py` which must be installed separately).
Placing the code above in a file ``ridge.py``, with ``nproc=1`` (or 
omitting ``nproc``), it can be run on 8 processors using ::

    mpirun -np 8 python ridge.py

The speedup is similar to that from using the 
:mod:`multiprocessing` module, above. 
Note that the random number generator used by |vegas| must be
synchronized so that it
produces the same random numbers on the different processors. This
happens automatically for the default random-number generator.

|vegas|'s batch mode makes it possible to implement other strategies
for distributing integrand evaluations across multiple processors.
For example, we can create a class ``parallelintegrand``
whose function is similar to decorator :func:`vegas.batchintegrand`,
but where Python's
:mod:`multiprocessing` module provides parallel processing::

    import multiprocessing
    import numpy as np
    import vegas

    class parallelintegrand(vegas.BatchIntegrand):
        """ Convert (batch) integrand into multiprocessor integrand.

        Integrand should return a numpy array.
        """
        def __init__(self, fcn, nproc=4):
            " Save integrand; create pool of nproc processes. "
            super().__init__()
            self.fcn = fcn
            self.nproc = nproc
            self.pool = multiprocessing.Pool(processes=nproc)
        def __del__(self):
            " Standard cleanup. "
            self.pool.close()
            self.pool.join()
        def __call__(self, x):
            " Divide x into self.nproc chunks, feeding one to each process. "
            nx = x.shape[0] // self.nproc + 1
            # launch evaluation of self.fcn for each chunk, in parallel
            results = self.pool.map(
                self.fcn,
                [x[i*nx : (i+1)*nx] for i in range(self.nproc)],
                1,
                )
            # convert list of results into a single numpy array
            return np.concatenate(results)

Then ``fparallel = parallelintegrand(f, 4)``, for example, will create a
new integrand ``fparallel(x)`` that uses 4 CPU cores.

Saving Results Automatically
-----------------------------
Results returned by a |vegas| integrator can be pickled for later use using ``pickle.dump/load`` (or 
``gvar.dump/load``) in the usual way. Results can also be saved automatically using the 
``save`` keyword to specify a file name for the pickled result: for example, running ::

    import vegas
    import math

    def f(x):
        dx2 = 0
        for d in range(4):
            dx2 += (x[d] - 0.5) ** 2
        return math.exp(-dx2 * 100.) * 1013.2118364296088

    integ = vegas.Integrator([[-1, 1], [0, 1], [0, 1], [0, 1]])

    result = integ(f, nitn=5, neval=1000, save='save.pkl')
    print(result.summary())
    print('result = %s    Q = %.2f' % (result, result.Q))

prints out

.. literalinclude:: eg9a.out

but also store ``result`` in file ``save.pkl``. The result can be retrieved 
later using, for example, ::

    import pickle

    with open('save.pkl', 'rb') as ifile:
        result = pickle.load(ifile)

    print(result.summary())
    print('result = %s    Q = %.2f' % (result, result.Q))

which gives exactly the same output.

This feature is most useful for expensive integrations, ones taking minutes or hours
to complete. This is because the pickled file is updated after every |vegas| 
iteration. This means that a short script like the one above can be used to 
monitor progress before the integration is finished. It also means that results 
up through the most recent iteration are saved even if the integration script is 
terminated early or crashes.

It is also possible to save an adapted integrator using ``pickle.dump/load`` 
(or ``gvar.dump/load``). This can also be done automatically, by 
replacing, for example, ``save='save.pkl'`` with ``saveall='saveall.pkl'``
in the script above. The pickled file then returns a tuple containing
the most recent ``result`` and ``integ``. Having the (adapted) integrator,
it is possible to further refine a result later: for example, running ::

    import pickle

    def f(x):
        dx2 = 0
        for d in range(4):
            dx2 += (x[d] - 0.5) ** 2
        return math.exp(-dx2 * 100.) * 1013.2118364296088

    with open('saveall.pkl', 'rb') as ifile:
        result, integ = pickle.load(ifile)
    new_result = integ(f, nitn=5)

    print('\nNew results:')
    print(new_result.summary())

    print('\nCombined results:')
    result.extend(new_result)
    print(result.summary())
    print('Combined result = %s    Q = %.2f' % (result, result.Q))

greatly improves the final result by adding 5 |~| additional iterations 
to what was done  earlier. The new iterations are in ``new_result``
and tabulated under "New Results" in the output:

.. literalinclude:: eg9b.out

The new results are merged onto the end of the original results using 
``result.extend(new_result)`` to obtain the combined results from all 
10 |~| iterations (old plus new).

Saving integrators is again useful for costly
integrations that might need to be restarted later since the saved integrator
remembers the variable transformations made to minimize errors, and
so need not be readapted to the integrand when used again. The resulting 
pickle file can be large, however, particularly if ``neval`` is large. 
The (adapted) :class:`vegas.AdaptiveMap` ``integ.map`` can also 
be pickled by itself and results in a smaller file.



Sums with |vegas|
-------------------
The code in the previous sections is inefficient in the way it
handles the sum over 10,000 Gaussians. It is not necessary to include every
term in the sum for every integration point. Rather we can sample the sum,
using |vegas| to do the sampling. The trick is to replace the sum with
an equivalent integral:

.. math::

    \sum_{i=0}^{N-1} f(i) = N \int_0^1 dx \; f(\mathrm{floor}(x N))

where :math:`\mathrm{floor}(x)` is the largest
integer smaller than :math:`x`. The
resulting integral can then be handed to |vegas|. Using this trick,
the integral in the previous section can be re-cast as a 5-dimensional
integral::

    import vegas
    import numpy as np

    # Integrand: ridge of N Gaussians spread evenly along the diagonal
    def ridge(x):
        N = 10000
        dim = 4
        x0 = 0.4 + 0.2 * np.floor(x[-1] * N) / (N - 1.)
        dx2 = 0.0
        for xd in x[:-1]:
            dx2 += (xd - x0) ** 2
        return np.exp(-100. * dx2) *  (100. / np.pi) ** (dim / 2.)

    def main():
        integ = vegas.Integrator(5 * [[0, 1]])
        # adapt
        integ(ridge, nitn=10, neval=5e4)
        # final results
        result = integ(ridge, nitn=10, neval=5e4)
        print('result = %s    Q = %.2f' % (result, result.Q))

    if __name__ == '__main__':
        main()

This code gives a result with the same precision, but is 5x |~| faster
than the code in the previous section (with ``nproc=1``; it is another 
3x |~| faster when ``nproc=8``).

The same trick can be generalized to sums over multiple indices, including sums
to infinity. |vegas| will provide Monte Carlo estimates of the sums, emphasizing
the more important terms.


|vegas| Maps and Preconditioning |vegas| 
-------------------------------------------
|vegas| adapts by remapping the integration variables (see :ref:`importance_sampling`). It 
is possible to precondition this map, before creating a :class:`vegas.Integrator`.
Preconditioned maps can improve |vegas| results when 
much is known about the integrand ahead of time. Consider,
for example, the integral

.. math:: 
    C\int_0^1 \!d^5x\,
    \,\sum_{i=1}^2 \mathrm{e}^{-50 |\mathbf{x} - \mathbf{r}_i|},

which has high, narrow peaks at 

.. math::
    \mathbf{r}_1 = (0.45, 0.45, 0.45, 0.45, 0.45),
    
.. math::
    \mathbf{r}_2 = (0.7, 0.7, 0.7, 0.7, 0.7).

Given the locations of the peaks we can create a |vegas| map before integrating
that emphasizes the regions around them::

    import vegas 
    import numpy as np 

    @vegas.batchintegrand
    def f(x):
        ans = 0
        for c in [0.45, 0.7]:
            dx2 = np.sum((x - c) ** 2, axis=1)
            ans += np.exp(-50 * np.sqrt(dx2))
        return ans * 247366.171

    dim = 5 
    map = vegas.AdaptiveMap(dim * [[0, 1]])     # uniform map
    x = np.concatenate([                        # 2000 points near peaks
        np.random.normal(loc=0.45, scale=3/50, size=(1000, dim)),
        np.random.normal(loc=0.7, scale=3/50, size=(1000, dim)),
        ])
    map.adapt_to_samples(x, f(x), nitn=5)       # precondition map
    integ = vegas.Integrator(map, alpha=0.1)
    r = integ(f, neval=1e4, nitn=5)
    print(r.summary())

|vegas| maps are objects of type :class:`vegas.AdaptiveMap`. Here we
create a uniform map object ``map``. We then 
generate 2000 random points ``x`` from normal distributions centered around 
the peak locations. ``map.adapt_to_samples(x, f(x), nitn=5)`` optimizes 
the map for integrating ``f(x)`` based on information about the integrand
at the random 
points ``x``. As a result, |vegas| is almost fully adapted to the 
integrand already in its first iteration: 

.. literalinclude:: eg5b.out

This can be contrasted with what happens without preconditioning, where 
the integrator is still far from converged by the fifth iteration:

.. literalinclude:: eg5a.out

The exact distribution of random points ``x`` isn't important; what 
matters is that they cover and are concentrated in 
the dominant regions contributing to the integral. Should it be needed,
a sample ``xsample=(x, f(x))`` 
in ``x``-space is easily converted to the equivalent sample in ``y``-space using::

    x, fx = xsample
    y = np.empty(x.shape, float)
    jac = np.empty(x.shape[0], float)
    map.invmap(x, y, jac)
    ysample = (y, jac * fx)

where ``map.invmap(x, y, jac)`` fills array ``y`` with the ``y``-space 
points corresponding to ``x``, and array ``jac`` with the transformation's
Jacobian.

Note that |vegas| maps can be used with integrators other than |vegas|.
The |vegas| map maps the integration variables ``x[d]`` into 
new variables ``y[d]`` (where ``0 < y[d] < 1``) that make the integrand 
easier to integrate. The integrator is used 
to evaluate the integral in ``y``-space. To illustrate how this 
works, we replace the last three lines of the code at the start of
this section with 
the following::

    def smc(f, neval, dim):
        " integrates f(y) over dim-dimensional unit hypercube "
        y = np.random.uniform(0,1, (neval, dim))
        fy = f(y)
        return (np.average(fy), np.std(fy) / neval ** 0.5)

    def g(y):
        jac = np.empty(y.shape[0], float)
        x = np.empty(y.shape, float)
        map.map(y, x, jac)
        return jac * f(x)

    # with map
    r = smc(g, 50_000, dim)
    print('   SMC + map:', f'{r[0]:.3f} +- {r[1]:.3f}')

    # without map
    r = smc(f, 50_000, dim)
    print('SMC (no map):', f'{r[0]:.3f} +- {r[1]:.3f}')

Here ``smc`` is a Simple Monte Carlo (SMC) integrator and ``g(y)`` is the 
integrand in ``y``-space that corresponds to ``f(x)`` in ``x``-space.
The mapping is done by ``map.map(y, x, jac)`` which fills the arrays
``x`` and ``jac`` with the ``x`` values and Jacobian corresponding 
to integration points ``y``. The result is 

.. literalinclude:: eg5c.out

where we give results from SMC with and without using 
the |vegas| map. The |vegas| map greatly improves 
the SMC result, which, not surprisingly, is significantly
less accurate the the |vegas| result above.
Maps for use with other integrators can be built 
directly, as above, or they can be built for a particular 
integrand by running several iterations of |vegas| with 
the integrand and using the |vegas| integrator's map: ``integ.map``.


|vegas| Stratifications
-------------------------
Having mapped the integration variables ``x[d]`` to new variables ``y[d]``,
|vegas| evaluates the integral in ``y``-space using stratified Monte 
Carlo sampling (see :ref:`adaptive-stratified-sampling`). This is 
done by dividing each axis ``d`` into ``nstrat[d]`` equal stratifications, 
which divide the ``D``-dimensional integration volume into 
``prod(nstrat)`` sub-volumes or (rectangular) hypercubes. |vegas| does a 
separate integral 
in each hypercube, adjusting the number of integrand evaluations 
used in each one to minimize errors. By default, the number of
stratifications is set automatically based on the number ``neval`` 
of integrand evaluations per iteration: ``nstrat[d]`` is set 
equal to :math:`M_\mathrm{st}+1` for the first :math:`D_0` directions 
and :math:`M_\mathrm{st}` for 
the remaining directions, where :math:`M_\mathrm{st}` 
and :math:`D_0` are chosen 
to maximize the number stratifications consistent with ``neval``.
It is also possible, however, to specify the number of 
stratifications ``nstrat[d]`` for each direction separately. 

Requiring (approximately) the same number of stratifications in each direction 
greatly limits the number of stratifications in very high 
dimensions, since the product of the ``nstrat[d]`` must be 
less than ``neval/2`` (so there are at least two integrand 
samples in each hypercube). This restricts |vegas|'s ability to 
adapt. Often there is a subset of integration directions 
that are more challenging than the others. In high 
dimensions (and possibly lower dimensions) 
it is worthwhile using larger values 
for ``nstrat[d]`` for those directions.

An example is the 20-dimensional integral 

.. math:: 
    C\int_0^1 \!d^{20}x\,
    \,\sum_{i=1}^3 \mathrm{e}^{-100 (\mathbf{x} - \mathbf{r}_i)^2},

which has high, narrow peaks at 

.. math::
    \mathbf{r}_1 = (0.23, 0.23, 0.23, 0.23, 0.23, 0.45, \ldots, 0.45),
    
.. math::
    \mathbf{r}_2 = (0.39, 0.39, 0.39, 0.39, 0.39, 0.45, \ldots, 0.45),
    
.. math::
    \mathbf{r}_3 = (0.74, 0.74, 0.74, 0.74, 0.74, 0.45, \ldots, 0.45).

These peaks are aligned along the diagonal of the integration volume for the 
first five directions, but are on top of each other in the remaining directions.
This makes the integrals over the first five directions much more challenging
than the remaining integrals.

To integrate this function, we specify ``nstrat[d]`` rather than ``neval``. 
We set ``nstrat[d]=10`` for the first five directions, and ``nstrat[d]=1`` 
for all the rest. This fixes the number of function evaluations to be 
approximately eight times the number of hypercubes created by 
the stratifications (i.e., eight times 100,000), which provides enough
integrand samples for adaptive stratified sampling while also 
guaranteeing at least two samples per hypercube.
The following code ::

    import vegas 
    import numpy as np 

    dim = 20
    r = np.array([
        5 * [0.23] + (dim - 5) * [0.45],
        5 * [0.39] + (dim - 5) * [0.45],
        5 * [0.74] + (dim - 5) * [0.45],
        ])

    @vegas.batchintegrand
    def f(x):
        ans = 0
        for ri in r:
            dx2 = np.sum((x - ri[None, :]) ** 2, axis=1)
            ans += np.exp(-100 * dx2)
        return ans * 356047712484621.56

    integ = vegas.Integrator(dim * [[0, 1]], alpha=0.25)
    nstrat = 5 * [10] + (dim - 5) * [1]
    integ(f, nitn=15, nstrat=nstrat)            # warmup
    r = integ(f, nitn=5, nstrat=nstrat)
    print(r.summary())
    print('nstrat =', np.array(integ.nstrat))

gives excellent results:

.. literalinclude:: eg6a.out

|vegas| struggles to converge, however,
when in its default mode with the same number of 
integrand evaluations (about 800,000 per iteration):

.. literalinclude:: eg6b.out 

.. _vegas_Jacobian:

|vegas| Jacobian
-------------------
The |vegas| Jacobian is useful when integrating multiple integrands simultaneously when some of the integrands depend only on a subset of the integration variables.

Consider, for example, the integral

.. math::

     \int^{\pi/2}_0 dx_0 \, \frac{1}{(x_0-\delta)^2 + 0.01}.

We want to compare this integral for two different cases: a) 
where :math:`\delta=x_1` is a random number near zero drawn from a distribution
proportional to :math:`\mathrm{exp}(-100\, \mathrm{sin} (x_1))` 
with :math:`0\le x_1 \le \pi/2`; 
and b) where :math:`\delta=0`. The result for the first case is 
:math:`I_a = I_a^\mathrm{num}/I_a^\mathrm{den}` where

.. math::

    I_a^\mathrm{num} &= \int^{\pi/2}_0 dx_0 \int^{\pi/2}_0 dx_1
    \, \mathrm{e}^{-100\,\mathrm{sin}(x_1)}
     \, \frac{1}{(x_0-x_1)^2 + 0.01} \\
    I_a^\mathrm{den} &= \int^{\pi/2}_0 dx_1 
    \, \mathrm{e}^{-100 \,\mathrm{sin}(x_1)}.

The result for the second case is

.. math::

    I_b = \int^{\pi/2}_0 dx_0 
    \, \frac{1}{(x_0)^2 + 0.01}.

The first two integrals should have very similar dependence on :math:`x_1`, while 
the first and third integrals should have similar dependence on :math:`x_0`.
Given these similarities, we would like to do all three integrals simultaneously;
but one integral is two-dimensional and the other two are one-dimensional.

One solution is to turn the one-dimensional integrals into two-dimensional 
integrals by rewriting :math:`I_a^\mathrm{den}` as

.. math::

    I_a^\mathrm{den} &= 
    \int^{\pi/2}_0 \frac{dx_0}{\pi/2} 
    \int^{\pi/2}_0 dx_1 
    \, \mathrm{e}^{-100 \,\mathrm{sin}(x_1)}.

and similarly for :math:`I_b`.
These are then easily integrated together --- ::

    import vegas
    import numpy as np

    integ = vegas.Integrator(2 * [(0., np.pi/2.)])

    @vegas.rbatchintegrand
    def f(x):
        Ia_num = np.exp(-1e2 * np.sin(x[1])) / ((x[0] - x[1])**2 + 0.01)
        Ia_den = np.exp(-1e2 * np.sin(x[1])) / (np.pi/2)
        Ib = 1 /  (x[0]**2 + 0.01) / (np.pi/2)
        return dict(Ia_num=Ia_num, Ia_den=Ia_den, Ib=Ib)

    w = integ(f, neval=20000, nitn=10)
    r = integ(f, neval=20000, nitn=5)
    print(r.summary(True))
    print('Ia =', r['Ia_num'] / r['Ia_den'], 'Ia - Ib =', r['Ia_num'] / r['Ia_den'] - r['Ib'])

--- but the result is useless:

.. literalinclude:: eg8a.out

:math:`I_b` is almost 100x less accurate than :math:`I_a`, despite 
having an integrand with no ``x[1]`` dependence. The problem is that the 
|vegas| map for ``x[1]`` is tailored to :math:`I_a^\mathrm{num}`, which has 
a strong peak at ``x[1]=0``. That map is highly sub-optimal for integrating
a function, like that in :math:`I_b`, that is independent of ``x[1]``.

A much better approach is to rewrite the one-dimensional integrals
as two-dimensional integrals using the |vegas|
map's Jacobian :math:`dx_d/dy_d` in the extra direction |~| :math:`d`.
For example,

.. math::

    I_a^\mathrm{den} &= 
    \int^{\pi/2}_0 \frac{dx_0}{dx_0/dy_0} 
    \int^{\pi/2}_0 dx_1 
    \, \mathrm{e}^{-100 \,\mathrm{sin}(x_1)}.

This turns the :math:`x_0` integral into an integral over :math:`0\le y_0 \le 1`` 
with an integrand equal to |~| 1. The Monte Carlo integral over :math:`y_0` 
in |vegas| is then exact; it is unaffected by the map for that direction. 
This is easily implemented for both one-dimensional integrals by setting 
keyword ``uses_jac=True`` in the integrator and modifying the integrand::

    @vegas.rbatchintegrand
    def f(x, jac):
        Ia_num = np.exp(-1e2 * np.sin(x[1])) / ((x[0] - x[1])**2 + 0.01)
        Ia_den = np.exp(-1e2 * np.sin(x[1])) / jac[0]
        Ib = 1 /  (x[0]**2 + 0.01) / jac[1]
        return dict(Ia_num=Ia_num, Ia_den=Ia_den, Ib=Ib)

    w = integ(f, uses_jac=True, neval=20000, nitn=10)
    r = integ(f, uses_jac=True, neval=20000, nitn=5)

Setting ``uses_jac=True`` causes |vegas| to call the integrand with 
two arguments instead of one: ``f(x, jac=jac)`` where ``jac[d]`` is 
the Jacobian for direction |~| ``d`` (array ``jac`` has the same shape
as ``x``). This gives much better results --- 

.. literalinclude:: eg8b.out

--- where now :math:`I_a` and :math:`I_b` are comparable in precision and the difference is 
quite accurate. Note that the error on the difference is smaller than either of 
the separate errors, because of correlations between the two results.


|vegas| as a Random Number Generator
-------------------------------------
A |vegas| integrator generates random points in its integration volume from a
distribution that is optimized for integrals of whatever function it
was trained on. The integrator
provides low-level access to the random-point generator
through the iterators :meth:`vegas.Integrator.random` and
:meth:`vegas.Integrator.random_batch`.

To illustrate, the following code snippet estimates the integral of function
``f(x)`` using integrator ``integ``::

    integral = 0.0
    for x, wgt in integ.random():
        integral += wgt * f(x)

Here ``x[d]`` is a random point in the integration volume and ``wgt`` is the
weight |vegas| assigns to that point in an integration. The iterator generates
integration points and weights corresponding to a single iteration of the
|vegas| algorithm. In practice, we would train ``integ`` on a function whose
shape is similar to that of ``f(x)`` before using it to estimate the integral
of ``f(x)``.

It is usually more efficient to generate and use integration points in
batches. The :meth:`vegas.Integrator.random_batch` iterator does just
this::

    integral = 0.0
    for x, wgt in integ.random_batch():
        integral += wgt.dot(batch_f(x))

Here ``x[i, d]`` is an array of integration points, ``wgt[i]`` contains the
corresponding weights, and ``batch_f(x)`` returns an array containing the
corresponding integrand values.

The random points generated by |vegas| are stratified into hypercubes: |vegas|
uses transformed integration variables to improve its Monte Carlo
estimates. It further improves those estimates by subdividing the
integration volume in the transformed variables into a large number of
hypercubes, and doing a Monte Carlo integral in each hypercube separately
(see previous section).
The final result is the sum of the results from all the hypercubes.
To mimic a full |vegas| integral estimate using the iterators above, we need
to know which points belong to which hypercubes. The following code
shows how this is done::

    integral = 0.0
    variance = 0.0
    for x, wgt, hcube in integ.random_batch(yield_hcube=True):
        wgt_fx = wgt * batch_f(x)
        # iterate over hypercubes: compute variance for each,
        #                          and accumulate for final result
        for i in range(hcube[0], hcube[-1] + 1):
            idx = (hcube == i)          # select array items for h-cube i
            nwf = np.sum(idx)           # number of points in h-cube i
            wf = wgt_fx[idx]
            sum_wf = np.sum(wf)         # sum of wgt * f(x) for h-cube i
            sum_wf2 = np.sum(wf ** 2)   # sum of (wgt * f(x)) ** 2
            integral += sum_wf
            variance += (sum_wf2 * nwf - sum_wf ** 2) / (nwf - 1.)
    # answer = integral;   standard deviation = variance ** 0.5
    result = gvar.gvar(integral, variance ** 0.5)

Here ``hcube[i]`` identifies the hypercube containing ``x[i, d]``. This example is 
easily modified to provide information about the |vegas| Jacobian to the integrand, if needed 
(see the section :ref:`vegas_Jacobian`)::

    integral = 0.0
    variance = 0.0
    for x, y, wgt, hcube in integ.random_batch(yield_hcube=True, yield_y=True):
        wgt_fx = wgt * batch_f(x, jac=integ.map.jac1d(y))
        ...

Implementation Notes
---------------------
This implementation relies upon Cython for its speed and
numpy for array processing. It also uses :mod:`matplotlib`
for graphics and :mod:`mpi4py` for MPI support, but graphics
and MPI are optional.

|vegas| also uses the :mod:`gvar` module (``pip install gvar``).
Integration results are returned as objects of type
:class:`gvar.GVar`, which is a class representing Gaussian
random variables (i.e., something with a mean and standard
deviation). These objects can be combined with numbers and
with each other in arbitrary arithmetic expressions to
get new :class:`gvar.GVar`\s with the correct standard
deviations, and properly correlated with other
:class:`gvar.GVar`\s --- that is the tricky part.




