.. _bda-example-bayesian-analysis-of-a-bioassay-experiment-logistic-nonconjugate:

==============================================================================
Example: Bayesian analysis of a bioassay experiment (logistic, nonconjugate)
==============================================================================

**Part 1 · Stage 3 · 🧮 Multiparameter Models**  ·  Lesson 026 of 144  ·  *beginner*

:doc:`◀ Previous · Multivariate Normal with Unknown Mean and Variance <025-multivariate-normal-with-unknown-mean-and-variance>`   ·   :doc:`Next · Summary of Elementary Modeling and Computation ▶ <027-summary-of-elementary-modeling-and-computation>`   ·   :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   review, it may still contain mistakes, omissions, or out-of-date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.

Where the closed forms end
----------------------------

Every posterior so far has had a formula. This one does not, and that is the point. In a **bioassay**,
groups of animals receive increasing **doses** :math:`x_i` of a compound and the number of **deaths**
:math:`y_i` out of :math:`n_i` is recorded. A typical experiment is tiny — four dose groups, five
animals each.

The logistic dose–response model
----------------------------------

Deaths are binomial with a dose-dependent probability, modelled on the **log-odds** scale:

.. math::

   y_i \sim \mathrm{Binomial}(n_i,\, \theta_i), \qquad
   \mathrm{logit}(\theta_i) = \log\frac{\theta_i}{1 - \theta_i} = \alpha + \beta x_i .

Here :math:`\beta` is the **dose effect** — the scientific question is usually whether
:math:`\beta > 0`. A Beta prior cannot help: the parameters :math:`(\alpha, \beta)` enter through a
nonlinear link, so **no conjugate prior exists** and the posterior

.. math::

   p(\alpha, \beta \mid y) \;\propto\; p(\alpha, \beta)
   \prod_{i} \bigl[\mathrm{logit}^{-1}(\alpha + \beta x_i)\bigr]^{y_i}
             \bigl[1 - \mathrm{logit}^{-1}(\alpha + \beta x_i)\bigr]^{n_i - y_i}

has no closed form. This is the normal situation in applied work; Stage 2's tidy algebra was the
exception.

Compute it on a grid
----------------------

With only **two** parameters, you can simply **evaluate** the unnormalised posterior over a grid,
normalise by summing, and sample from the discrete approximation. It is brute force, and it is exact
enough to be a benchmark for the samplers of Part III.

.. code-block:: python

   import numpy as np
   from scipy.special import expit

   x = np.array([-0.86, -0.30, -0.05, 0.73])       # log dose
   n = np.array([5, 5, 5, 5]); y = np.array([0, 1, 3, 5])

   a, b = np.meshgrid(np.linspace(-5, 10, 400), np.linspace(-10, 40, 400))
   th = expit(a[..., None] + b[..., None] * x)     # broadcast over dose groups
   logpost = (y * np.log(th) + (n - y) * np.log1p(-th)).sum(-1)   # flat prior
   post = np.exp(logpost - logpost.max()); post /= post.sum()
   (post.sum(axis=0) @ np.linspace(-5, 10, 400))   # E[alpha | y], etc.

Reading the answer
--------------------

The posterior for :math:`(\alpha, \beta)` is **skewed and correlated** — no normal approximation would
capture its banana shape well — and :math:`\Pr(\beta > 0 \mid y)` is essentially 1: the dose kills.
Because any function of draws is itself a posterior draw, the **LD50** (the dose at which
:math:`\theta = 0.5`, namely :math:`-\alpha / \beta`) comes free, with a wide and asymmetric interval
that a plug-in estimate would badly misrepresent. Grids work in two dimensions; beyond three or four
they die of dimensionality — which is precisely why Part III exists.

.. hint::

   **Related lessons:** :doc:`Standard generalized linear model likelihoods <106-standard-generalized-linear-model-likelihoods>`  ·  :doc:`Numerical integration <062-numerical-integration>`  ·  :doc:`Normal Approximations to the Posterior Distribution <028-normal-approximations-to-the-posterior-distribution>`  ·  :doc:`Summary of Elementary Modeling and Computation <027-summary-of-elementary-modeling-and-computation>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/11/09/example-bayesian-analysis-of-a-bioassay-experiment-logistic-nonconjugate/ <https://insightful-data-lab.com/2025/11/09/example-bayesian-analysis-of-a-bioassay-experiment-logistic-nonconjugate/>`__ (insightful-data-lab.com).

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