.. _bda-other-standard-single-parameter-models:

========================================================================
Other Standard Single-Parameter Models
========================================================================

**Part 1 · Stage 2 · 📍 Single-Parameter Models & Priors**  ·  Lesson 016 of 144  ·  *beginner*

:doc:`◀ Previous · Normal Distribution with Known Variance <015-normal-distribution-with-known-variance>`   ·   :doc:`Next · Informative Prior Distribution for Cancer Rates ▶ <017-informative-prior-distribution-for-cancer-rates>`   ·   :doc:`↑ Section <index>`


.. important::

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   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   review, it may still contain mistakes, omissions, or out-of-date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
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The same pattern, different data
----------------------------------

Binomial and normal are two instances of one recipe. Whenever the likelihood belongs to the
**exponential family**, a conjugate prior exists, the posterior stays in the prior's family, and the
update amounts to **adding sufficient statistics** to the hyperparameters. Three more cases cover most
of applied practice.

Poisson–Gamma (counts)
------------------------

For counts, :math:`y_i \sim \mathrm{Poisson}(\theta)` with a :math:`\mathrm{Gamma}(\alpha, \beta)`
prior on the rate:

.. math::

   \theta \mid y \;\sim\; \mathrm{Gamma}\Bigl(\alpha + \textstyle\sum_i y_i,\; \beta + n\Bigr),
   \qquad
   \mathrm{E}[\theta \mid y] = \frac{\alpha + \sum_i y_i}{\beta + n}.

The hyperparameters read as **pseudo-events** :math:`\alpha` observed in **pseudo-exposure**
:math:`\beta`, and the posterior mean is again a weighted average of the prior mean
:math:`\alpha/\beta` and the sample rate :math:`\bar{y}`. With exposures :math:`x_i` (person-years,
say), the update uses :math:`\beta + \sum_i x_i`.

Exponential and normal-variance
---------------------------------

For waiting times, :math:`y_i \sim \mathrm{Exponential}(\theta)` also takes a **Gamma** conjugate
prior, updating to :math:`\mathrm{Gamma}(\alpha + n,\ \beta + \sum_i y_i)`. And for a normal with
**known mean and unknown variance**, the conjugate prior on :math:`\sigma^2` is the
**inverse-gamma** (equivalently, scaled inverse-:math:`\chi^2`) — the piece needed to complete the
normal model in the next stage.

.. code-block:: python

   from scipy import stats
   y = [3, 5, 2, 4]                       # counts over n = 4 periods
   a, b = 2, 1                            # Gamma prior: 2 pseudo-events / 1 pseudo-period
   post = stats.gamma(a + sum(y), scale=1 / (b + len(y)))
   post.mean(), post.interval(0.95)       # posterior rate and 95% CI

Convenient, not sacred
------------------------

Conjugacy is prized for being **exact, instant and interpretable** — ideal for teaching, for sequential
updating, and as a building block inside larger samplers (Gibbs, in Part III). But conjugate families
exist for **algebraic convenience**, not because they encode anyone's real prior beliefs, and most
useful models have no conjugate form at all. Modern practice keeps the intuition — priors as pseudo-data,
posteriors as compromises — and reaches for **MCMC** whenever the model demands it.

.. hint::

   **Related lessons:** :doc:`Normal Distribution with Known Variance <015-normal-distribution-with-known-variance>`  ·  :doc:`Estimating a Probability from Binomial Data <011-estimating-a-probability-from-binomial-data>`  ·  :doc:`Informative Prior Distributions <014-informative-prior-distributions>`  ·  :doc:`Standard generalized linear model likelihoods <106-standard-generalized-linear-model-likelihoods>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/11/09/other-standard-single-parameter-models/ <https://insightful-data-lab.com/2025/11/09/other-standard-single-parameter-models/>`__ (insightful-data-lab.com).

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