.. _bda-effective-number-of-simulation-draws:

========================================================================
Effective number of simulation draws
========================================================================

**Part 3 · Stage 9 · ⛓️ MCMC: Gibbs, Metropolis & HMC**  ·  Lesson 073 of 144  ·  *intermediate*

:doc:`◀ Previous · Inference and assessing convergence <072-inference-and-assessing-convergence>`   ·   :doc:`Next · Example: hierarchical normal model ▶ <074-example-hierarchical-normal-model>`   ·   :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   review, it may still contain mistakes, omissions, or out-of-date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.

Correlated draws carry less information
-----------------------------------------

An MCMC chain's draws are **dependent** by construction: each is a perturbation of the last. A thousand
such draws therefore say less about the posterior than a thousand independent ones. The **effective
sample size** measures how much less.

The definition
----------------

If the chain's autocorrelation at lag :math:`t` is :math:`\rho_t`, then

.. math::

   S_{\text{eff}} = \frac{S}{1 + 2 \sum_{t=1}^{\infty} \rho_t} .

Independent draws (:math:`\rho_t = 0`) give :math:`S_{\text{eff}} = S`. Positively autocorrelated draws
— the usual case for random-walk Metropolis and Gibbs — give :math:`S_{\text{eff}} \ll S`. The estimate
truncates the sum once the estimated :math:`\rho_t` becomes noise, and in practice pools information
across chains.

Better than independent
-------------------------

A pleasing surprise: the sum can be **negative**. When successive draws are **anticorrelated**, the
effective sample size **exceeds** the number of iterations, :math:`S_{\text{eff}} > S`. Hamiltonian
Monte Carlo, and the NUTS sampler in particular, routinely achieves this for parameters whose posterior
is close to Gaussian with little dependence on the others — the chain deliberately overshoots, so
consecutive draws sit on opposite sides of the mode. For such problems MCMC beats independent sampling.

Bulk and tail
---------------

One number is not enough. **bulk-ESS** governs the precision of the mean and standard deviation;
**tail-ESS** governs the extreme quantiles, and a chain that mixes briskly through the centre can crawl
in the tails. Report both, and require both above roughly **400** before trusting any summary — including
:math:`\hat{R}` itself, whose reliability depends on adequate ESS.

.. code-block:: python

   import arviz as az
   az.summary(idata)                    # ess_bulk, ess_tail, mcse_mean, mcse_sd, r_hat
   az.ess(idata, method="tail")         # tail-ESS explicitly
   az.plot_autocorr(idata, var_names=["tau"])   # where does rho_t decay to zero?

Don't thin
------------

A persistent habit deserves retiring. **Thinning** — keeping every tenth draw — does not improve
inference. Discarding draws throws away information; the effective sample size of the thinned chain is
at best what the full chain already carried, and if the draws are anticorrelated, thinning actively
**destroys** the advantage. The only legitimate reason to thin is **memory**. If ESS is too low, the
answer is a better sampler, a reparameterisation, or a longer run — never a smaller one.

.. hint::

   **Related lessons:** :doc:`Inference and assessing convergence <072-inference-and-assessing-convergence>`  ·  :doc:`How many simulation draws are needed? <066-how-many-simulation-draws-are-needed>`  ·  :doc:`Efficient Metropolis jumping rules <076-efficient-metropolis-jumping-rules>`  ·  :doc:`Hamiltonian Monte Carlo <078-hamiltonian-monte-carlo>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/11/12/e%ef%ac%80ective-number-of-simulation-draws/ <https://insightful-data-lab.com/2025/11/12/e%ef%ac%80ective-number-of-simulation-draws/>`__ (insightful-data-lab.com).

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