.. _bda-bayesian-inference-requires-a-model-for-data-collection:

========================================================================
Bayesian inference requires a model for data collection
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**Part 2 · Stage 7 · 🗳️ Data Collection & Decisions**  ·  Lesson 050 of 144  ·  *intermediate*

:doc:`◀ Previous · Implicit assumptions and model expansion: an example <049-implicit-assumptions-and-model-expansion-an-example>`   ·   :doc:`Next · Data-collection models and ignorability ▶ <051-data-collection-models-and-ignorability>`   ·   :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   review, it may still contain mistakes, omissions, or out-of-date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.

The data did not arrive by magic
----------------------------------

Every analysis so far quietly assumed that :math:`y` is simply "the data". But some process **chose**
which units you observe: a survey sampled households, an experiment assigned treatments, a clinic
recorded only patients who returned. That process is part of the probability model, and ignoring it can
invalidate everything downstream.

Model the inclusion indicator
-------------------------------

Write :math:`y = (y_{\text{obs}}, y_{\text{mis}})` for the complete data — everything that *could* have
been observed — and let :math:`I` be the **inclusion indicator**, :math:`I_i = 1` when unit :math:`i` is
observed. The honest joint model covers both:

.. math::

   p(y, I \mid \theta, \phi) = p(y \mid \theta) \; p(I \mid y, \phi),

where :math:`\phi` parameterises the **data-collection mechanism**. What you actually condition on is
:math:`(y_{\text{obs}}, I)`, so the correct posterior integrates the missing values out:

.. math::

   p(\theta, \phi \mid y_{\text{obs}}, I) \;\propto\;
   p(\theta, \phi) \int p(y_{\text{obs}}, y_{\text{mis}} \mid \theta)
   \; p(I \mid y_{\text{obs}}, y_{\text{mis}}, \phi) \; d y_{\text{mis}} .

Analysing :math:`p(\theta \mid y_{\text{obs}})` alone — as if the observed data were the whole story —
is a **modelling assumption**, not a neutral default.

When it bites
---------------

The mechanism matters exactly when :math:`I` depends on values you did **not** observe. Truncate a
sample at a threshold and the sample mean is biased for the population mean, however large :math:`n`
grows. Let sicker patients drop out and the surviving cohort looks healthier. Publish only significant
results and the literature's effect sizes are inflated. In each case the likelihood you wrote is not the
likelihood of the data you have.

.. code-block:: python

   import numpy as np
   # a truncated sample: only y > 0 recorded. Naive mean is biased upward.
   rng = np.random.default_rng(0)
   y_full = rng.normal(0.0, 1.0, 100_000)
   y_obs = y_full[y_full > 0]
   y_full.mean(), y_obs.mean()          # ≈ 0.00  vs  ≈ 0.80 — the mechanism is not ignorable

The good news
---------------

Under identifiable conditions — the subject of the next lesson — the mechanism **factors out** and can
be ignored entirely, which is why the first seven stages were not wrong, merely conditional. Knowing
those conditions is what separates an analysis that may safely omit the design from one that must model
it. Randomisation, it will turn out, is precisely a device for **making the mechanism ignorable by
construction**.

.. hint::

   **Related lessons:** :doc:`Data-collection models and ignorability <051-data-collection-models-and-ignorability>`  ·  :doc:`Censoring and truncation <056-censoring-and-truncation>`  ·  :doc:`Notation <118-notation>`  ·  :doc:`Sample surveys <052-sample-surveys>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/11/11/bayesian-inference-requires-a-model-for-data-collection/ <https://insightful-data-lab.com/2025/11/11/bayesian-inference-requires-a-model-for-data-collection/>`__ (insightful-data-lab.com).

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