Seminar talk: How can we learn multidimensional joint-probability distributions?
==================================================================================

:Date: 2026-03-19
:Tags: copulas, dependence, joint distributions, seminar
:download:`Original slides <../_static/joint-probability.pptx>`

A seminar talk I gave as part of the **copulas / dependence-modelling** track
at AI4Value — sitting alongside `sklar-engine <https://github.com/PlayerPlanet>`_,
the dependence-modelling tooling I've been building at work. The motivating
question is one I keep coming back to: given a handful of marginals, how do we
learn the **joint** distribution without assuming a parametric form for it?

The slides walk through three threads:

- **Marginals and copulas.** Sklar's theorem as the bridge: a joint distribution
  is its marginals plus a copula, and the copula is the part that actually
  encodes dependence.
- **Parametric vs. non-parametric copulas.** Gaussian- and t-copulas are the
  classical defaults; empirical copulas and kernel estimators trade assumptions
  for variance. The "elliptical" stretch example (the diagonal density plot)
  is the canonical case where a Gaussian copula fits, and the canonical case
  where tail dependence makes it fail.
- **Inference on the joint.** Maximum-likelihood fits under parametric copulas,
  pseudo-likelihood and IFM for semi-parametric fits, and the simulation-based
  route when the joint is implicit.

The honest framing in the talk — and the one I now use as the through-line —
is that **what you can learn about a joint distribution is bounded by what you
already believe about its marginals**. Pick the marginals first; the copula
fills in the dependence.

Slides
------

.. figure:: ../_static/slides/joint-probability/jp-image8.png
   :alt: Slide 3
   :width: 100%

   Slide 3 — Gaussian marginals (the building blocks).

.. figure:: ../_static/slides/joint-probability/jp-image9.png
   :alt: Slide 5
   :width: 100%

   Slide 5 — an elliptical Gaussian joint, centred on (0.5, 0.5) and stretched
   along the diagonal. The copula is Gaussian; the dependence is +1 along u=v.

.. figure:: ../_static/slides/joint-probability/jp-image10.png
   :alt: Slide 6
   :width: 100%

   Slide 6 — the same joint as a heatmap. A copula plot, not a scatter plot:
   this is the **density** under the copula transform.

.. figure:: ../_static/slides/joint-probability/jp-image11.png
   :alt: Slide 7
   :width: 100%

   Slide 7.

.. figure:: ../_static/slides/joint-probability/jp-image12.png
   :alt: Slide 8
   :width: 100%

   Slide 8.

.. figure:: ../_static/slides/joint-probability/jp-image13.png
   :alt: Slide 10
   :width: 100%

   Slide 10.

.. figure:: ../_static/slides/joint-probability/jp-image14.png
   :alt: Slide 12
   :width: 100%

   Slide 12 — comparison panels (slides 12–14 share this figure in different
   contexts).

.. figure:: ../_static/slides/joint-probability/jp-image15.png
   :alt: Slide 15
   :width: 100%

   Slide 15 — repeated across slides 15–17.

.. figure:: ../_static/slides/joint-probability/jp-image16.png
   :alt: Slide 18
   :width: 100%

   Slide 18.

.. figure:: ../_static/slides/joint-probability/jp-image17.png
   :alt: Slide 19
   :width: 100%

   Slide 19 — tail panels.

.. figure:: ../_static/slides/joint-probability/jp-image18.png
   :alt: Slide 19
   :width: 100%

   Slide 19 — tail panels.

.. figure:: ../_static/slides/joint-probability/jp-image19.png
   :alt: Slide 19
   :width: 100%

   Slide 19 — tail panels.

.. figure:: ../_static/slides/joint-probability/jp-image20.png
   :alt: Slide 20
   :width: 100%

   Slide 20.

.. figure:: ../_static/slides/joint-probability/jp-image21.png
   :alt: Slide 21
   :width: 100%

   Slide 21.

Links
-----

- :download:`Download the original deck (PPTX) <../_static/joint-probability.pptx>`
- Source repo for `sklar-engine` (work-in-progress): https://github.com/PlayerPlanet