.. _bda-latent-gaussian-process-models:

========================================================================
Latent Gaussian process models
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**Part 5 · Stage 15 · 🌊 Basis Functions & Gaussian Processes**  ·  Lesson 131 of 144  ·  *advanced*

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.. important::

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   review, it may still contain mistakes, omissions, or out-of-date
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   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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Gaussian processes for non-Gaussian data
------------------------------------------

Gaussian-process regression as introduced assumes a **Gaussian** likelihood — continuous, normally-noised
observations. Most GP applications are not like that: the outcome is binary, a count, a category. A
**latent** Gaussian process handles them by putting the GP **inside** the model, on an unobserved
function that is then passed through a link and a non-Gaussian likelihood — exactly the GLM move, with a
GP where the linear predictor used to be.

The construction
------------------

Let a latent function :math:`f` have a GP prior, and let the observations depend on it through a link:

.. math::

   f \sim \mathcal{GP}(0, k), \qquad
   g\bigl(\mathrm{E}[y_i]\bigr) = f(x_i), \qquad
   y_i \sim \text{ExponentialFamily}.

For binary data this is **GP classification** — a logistic link on a smoothly varying latent function,
giving a probability surface that bends with the inputs. For counts it is a **log-Gaussian Cox process** —
a Poisson whose log-rate is a GP, the natural model for events whose intensity varies smoothly over space
or time.

.. code-block:: python

   import pymc as pm
   with pm.Model():
       ell = pm.Gamma("ell", 2, 1); eta = pm.HalfNormal("eta", 1)
       cov = eta**2 * pm.gp.cov.ExpQuad(1, ls=ell)
       gp = pm.gp.Latent(cov_func=cov)                      # explicit latent function
       f = gp.prior("f", X=X)                               # the latent GP values
       pm.Bernoulli("y", p=pm.math.sigmoid(f), observed=y)  # non-Gaussian likelihood

The computational cost
------------------------

The price is real. With a Gaussian likelihood the latent function integrates out analytically; with a
non-Gaussian one it **cannot**, so :math:`f` must be sampled or approximated. This is precisely the
**latent Gaussian model** class that motivated **INLA** back in the computation stage — nested Laplace
approximation was built for exactly this structure, and is far faster than MCMC here. Sampling the latent
values with HMC also invites the funnel geometry of hierarchical models, so non-centred parameterisations
and careful diagnostics apply.

Why it matters
----------------

Latent GPs are the general-purpose tool for **flexible, non-Gaussian regression**: spatial disease
mapping (a Cox process over geography), smoothly-varying classification boundaries, time-varying rates.
They unify the threads of Part V — the GP supplies the flexible function, the link and likelihood handle
the data type, and the computation connects back to the approximation methods built earlier. Wherever a
smooth latent surface drives non-normal observations, a latent Gaussian process is the model.

.. hint::

   **Related lessons:** :doc:`Gaussian process regression <129-gaussian-process-regression>`  ·  :doc:`Other approximations <089-other-approximations>`  ·  :doc:`Standard generalized linear model likelihoods <106-standard-generalized-linear-model-likelihoods>`  ·  :doc:`Functional data analysis <132-functional-data-analysis>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/12/09/latent-gaussian-process-models/ <https://insightful-data-lab.com/2025/12/09/latent-gaussian-process-models/>`__ (insightful-data-lab.com).

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