.. _bda-summary-of-elementary-modeling-and-computation:

========================================================================
Summary of Elementary Modeling and Computation
========================================================================

**Part 1 · Stage 3 · 🧮 Multiparameter Models**  ·  Lesson 027 of 144  ·  *beginner*

:doc:`◀ Previous · Example: Bayesian analysis of a bioassay experiment (logistic, nonconjugate) <026-example-bayesian-analysis-of-a-bioassay-experiment-logistic-nonconjugate>`   ·   :doc:`Next · Normal Approximations to the Posterior Distribution ▶ <028-normal-approximations-to-the-posterior-distribution>`   ·   :doc:`↑ Section <index>`


.. important::

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What Part I established
-------------------------

Everything so far rests on a single move: write a **full probability model**
:math:`p(\theta, y) = p(\theta) p(y \mid \theta)`, condition on the data, and summarise. The stages
built this out — one parameter, then several — while the arithmetic stayed the same. Three ideas do
most of the work.

The recurring three
---------------------

* **The posterior is a compromise.** Beta–Binomial, Normal–Normal, Poisson–Gamma, the multivariate
  normal: in each, the posterior mean is a **weighted average** of prior and data, with weights given
  by **effective sample size** or **precision**. Data eventually win; the prior matters most when they
  are scarce.
* **Nuisance parameters are integrated, not fixed.** Marginalising honestly is what turns a normal into
  a :math:`t`. Plug-in estimates discard the second term of the variance decomposition and understate
  uncertainty.
* **Priors are choices, and they are checkable.** Informative (pseudo-data), noninformative (invariant,
  possibly improper), weakly informative (regularising) — each states something, and prior predictive
  simulation reveals what.

Two computational lessons
---------------------------

**Conjugacy is a convenience.** It gives exact, instant, interpretable updates, and it will return in
Part III as the engine inside Gibbs samplers. But it exists for algebraic reasons, not scientific ones,
and the bioassay showed how quickly a realistic model escapes it.

**Draws are enough.** Every quantity of interest — a mean, an interval, :math:`\Pr(\beta > 0 \mid y)`,
the LD50, a contrast — is a summary of posterior draws. Functions of draws are draws from the function's
posterior; **dropping a column is marginalisation**. This is why the simulation-based workflow scales
where the algebra cannot.

.. code-block:: python

   # the entire Part I workflow, in five lines
   import pymc as pm, arviz as az
   with pm.Model():
       theta = pm.Beta("theta", 1, 1)                 # prior
       pm.Binomial("y", n=10, p=theta, observed=8)    # likelihood
       idata = pm.sample()                            # condition on data
   az.summary(idata)                                  # summarise; then check the fit

What comes next
-----------------

Part I answered *how to compute a posterior* when the model is given. Three questions remain, and they
organise the rest of the course: **is the model any good?** (checking and comparison, Part II);
**what if the posterior has no formula?** (approximation and MCMC, Part III); and **how do we build
models with structure** — groups, predictors, nonlinearity? (hierarchies, regression, nonparametrics,
Parts IV–V). The next stage begins the answer by asking what happens to a posterior as data
accumulate.

.. hint::

   **Related lessons:** :doc:`The three steps of Bayesian data analysis <001-the-three-steps-of-bayesian-data-analysis>`  ·  :doc:`Example: Bayesian analysis of a bioassay experiment (logistic, nonconjugate) <026-example-bayesian-analysis-of-a-bioassay-experiment-logistic-nonconjugate>`  ·  :doc:`Posterior as a Compromise Between Data and Prior Information <012-posterior-as-a-compromise-between-data-and-prior-information>`  ·  :doc:`The Place of Model Checking in Applied Bayesian Statistics <040-the-place-of-model-checking-in-applied-bayesian-statistics>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/11/09/summary-of-elementary-modeling-and-computation/ <https://insightful-data-lab.com/2025/11/09/summary-of-elementary-modeling-and-computation/>`__ (insightful-data-lab.com).

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