.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/bayes/plot_01_bayes_rule_of_succession.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_probability_bayes_plot_01_bayes_rule_of_succession.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_probability_bayes_plot_01_bayes_rule_of_succession.py:


Bayes's problem: learning a probability from data
========================================================

Starting from Bayes's uniform prior, the posterior for an unknown
success probability :math:`p` after :math:`k` successes in :math:`n`
trials is :math:`\mathrm{Beta}(k+1, n-k+1)`. It narrows around the true
value as data accumulate, and its mean is Laplace's rule of succession
:math:`(k+1)/(n+2)`.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import beta_binomial_posterior, rule_of_succession








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Posteriors after more and more trials
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-39

.. code-block:: Python


    p_true = 0.3
    rng = np.random.default_rng(0)
    flips = rng.random(1000) < p_true
    x = np.linspace(0.0, 1.0, 500)

    fig, ax = plt.subplots()
    for n in (0, 5, 20, 100, 1000):
        k = int(flips[:n].sum())
        post = beta_binomial_posterior(k, n)
        lo, hi = post.ppf([0.025, 0.975])
        print(f"n={n:>4}, k={k:>3}: next-success probability {rule_of_succession(k, n):.4f}, 95% interval [{lo:.3f}, {hi:.3f}]")
        ax.plot(x, post.pdf(x), label=f"n = {n}")
    ax.axvline(p_true, color="0.3", ls="--", label="true p")
    ax.set_xlabel("p")
    ax.set_ylabel("posterior density")
    ax.legend()
    ax.set_title("Bayes (1763): Beta posteriors from a uniform prior")



.. image-sg:: /api/gallery/probability/bayes/images/sphx_glr_plot_01_bayes_rule_of_succession_001.png
   :alt: Bayes (1763): Beta posteriors from a uniform prior
   :srcset: /api/gallery/probability/bayes/images/sphx_glr_plot_01_bayes_rule_of_succession_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    n=   0, k=  0: next-success probability 0.5000, 95% interval [0.025, 0.975]
    n=   5, k=  3: next-success probability 0.5714, 95% interval [0.223, 0.882]
    n=  20, k=  7: next-success probability 0.3636, 95% interval [0.181, 0.570]
    n= 100, k= 25: next-success probability 0.2549, 95% interval [0.176, 0.343]
    n=1000, k=277: next-success probability 0.2774, 95% interval [0.250, 0.306]

    Text(0.5, 1.0, 'Bayes (1763): Beta posteriors from a uniform prior')




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.034 seconds)


.. _sphx_glr_download_api_gallery_probability_bayes_plot_01_bayes_rule_of_succession.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_bayes_rule_of_succession.ipynb <plot_01_bayes_rule_of_succession.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_bayes_rule_of_succession.py <plot_01_bayes_rule_of_succession.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_bayes_rule_of_succession.zip <plot_01_bayes_rule_of_succession.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_