.. _bda-informative-prior-distributions:

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Informative Prior Distributions
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**Part 1 · Stage 2 · 📍 Single-Parameter Models & Priors**  ·  Lesson 014 of 144  ·  *beginner*

:doc:`◀ Previous · Summarizing Posterior Inference <013-summarizing-posterior-inference>`   ·   :doc:`Next · Normal Distribution with Known Variance ▶ <015-normal-distribution-with-known-variance>`   ·   :doc:`↑ Section <index>`


.. important::

   **✨ AI-generated content.** This page was written with the assistance of an
   AI language model and is provided as a learning aid. Despite careful
   review, it may still contain mistakes, omissions, or out-of-date
   information. Whether you are new to the topic, a team lead, or a senior
   practitioner, treat it as a starting point rather than an authoritative
   reference: read it critically and independently verify anything you act on
   (code, commands, figures, and factual claims) against official
   documentation and primary sources before relying on it.

Using what you know
---------------------

An **informative prior** deliberately encodes substantive knowledge about a parameter before the
current data arrive — from previous studies, physical constraints, or expert judgement. Its purpose is
not to bias the answer toward a preferred conclusion but to stop the analysis from pretending to know
nothing when, in fact, a great deal is known.

Priors as pseudo-data
-----------------------

Conjugate priors make "how much" precise: their hyperparameters read as **prior observations**. In the
Beta–Binomial, :math:`\mathrm{Beta}(\alpha, \beta)` acts like :math:`\alpha` prior successes and
:math:`\beta` prior failures, an effective prior sample size of :math:`\alpha + \beta`. In the
Poisson–Gamma, :math:`\mathrm{Gamma}(\alpha, \beta)` acts like :math:`\alpha` **pseudo-events**
observed in :math:`\beta` **pseudo-exposure**. This is the honest way to calibrate an informative
prior: ask *how many observations is my belief worth?*

.. code-block:: python

   from scipy import stats
   # "I've seen roughly 20 conversions in 100 visits before" -> Beta(20, 80)
   prior = stats.beta(20, 80)
   prior.mean(), prior.std()            # 0.20, ~0.04   (prior sample size 100)
   # 5 conversions in 40 new visits:
   post = stats.beta(20 + 5, 80 + 35)   # exact conjugate update
   post.mean()                          # 0.179 — pulled from 0.125 toward 0.20

Eliciting a prior
-------------------

Elicitation usually works backwards from summaries an expert can state: a plausible **central value**
and a range they would be surprised to see violated. Match those to a distribution's mean and spread,
then **check the implications** — draw from the prior, simulate data (a **prior predictive check**),
and ask whether the simulated datasets look like ones the world could produce. A prior that generates
absurd data is wrong, however reasonable it looked as a density.

The discipline
----------------

Informative priors carry real risk: with sparse data, they can dominate, and a confidently wrong prior
yields a confidently wrong posterior. Three habits keep them honest. **State** the prior and its
justification explicitly. **Report** its effective sample size, so readers see how much it contributes.
And run a **sensitivity analysis** — refit under reasonable alternatives, and say so when conclusions
move. Between the extremes of an informative prior and a deliberately vague one lies the
**weakly informative** prior of the coming lessons.

.. hint::

   **Related lessons:** :doc:`Posterior as a Compromise Between Data and Prior Information <012-posterior-as-a-compromise-between-data-and-prior-information>`  ·  :doc:`Noninformative Prior Distributions <018-noninformative-prior-distributions>`  ·  :doc:`Weakly Informative Prior Distributions <019-weakly-informative-prior-distributions>`  ·  :doc:`Informative Prior Distribution for Cancer Rates <017-informative-prior-distribution-for-cancer-rates>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/11/09/informative-prior-distributions/ <https://insightful-data-lab.com/2025/11/09/informative-prior-distributions/>`__ (insightful-data-lab.com).

.. tags:: purpose: reference, topic: data analysis, domain: bayesian, level: beginner