The six-step cycle, written in probabilities

Abstract¶

The six-step figure is usually read with functions: a model $f$ , an observation operator $h$ and an emulator $f(\cdot; \gvect{\theta}_\text{emu})$ . In this page, we rewrite every box as a statement about distributions, which means a prior, a likelihood, a posterior, and learned densities that approximate them. So a deterministic model becomes one special case instead of the starting point. Nothing here assumes linearity. Linear and nonlinear instances only appear as worked examples underneath each step, to show where the closed forms come from and why they stop.

Keywords:six-step cycleprobabilistic modelposterioremulatoramortized inferencesimulation-based inference¶

Open the full page

Where it sits. Start here. Every later page is one box of this cycle, written for the ocean. The full page is a self-contained document with operator boxes (inputs, outputs, shapes and units), figures and code sketches, which opens on its own and links back here.

TL;DR: Six boxes, six distributions¶

Here $\vect{x} \in \mathbb{R}^{D_x}$ is the state, $\vect{y} \in \mathbb{R}^{D_y}$ the observations and $\gvect{\theta} \in \mathbb{R}^{D_\theta}$ the physical parameters.

Step 1 is a joint distribution $p(\gvect{\theta}) p(\vect{x} \mid \gvect{\theta}) p(\vect{y} \mid \vect{x})$ that you can sample. A deterministic simulator is the special case where $p(\vect{x} \mid \gvect{\theta})$ is a point mass.
Step 2 is the posterior $p(\gvect{\theta}, \vect{x} \mid \vect{y})$ of that joint. It is exact by definition, and every algorithm (Kalman, 4D-Var, EnKF, particles, MCMC) is an approximation of it, which is slow but trusted.
Steps 3 to 5 replace densities with learned ones: $q(\vect{x}_k \mid \vect{x}_{k-1}, \gvect{\theta}; \gvect{\theta}_\text{emu})$ for the simulator’s transition (the emulator), the same posterior computed with that emulator (step 4), and $q(\gvect{\theta}, \vect{x} \mid \vect{y}; \gvect{\nu})$ for the posterior itself (amortized). Each one is trained by a KL divergence to the distribution it replaces.
Training and use are separate phases. Steps 3 and 5 are fitted once, offline, on simulations. Then every route (2, 4 or 5) returns posterior draws, and the draws are used: estimates, forecasts, predictions of new data and probabilities of events.
Step 6 compares distributions, which means each learned one against the one it approximates, and the whole model against data.
Linear-Gaussian is the one case where every integral is matrix algebra. Outside of it, posteriors skew and split, their shape depends on the data, and every shortcut that assumes one Gaussian can fail.

On the page¶

The pipeline: train once, use per window
Why distributions instead of functions
Notation
Step 1 · The simple model is a joint distribution
Step 2 · Model-based inference is the posterior of that joint
Step 3 · The emulator is a learned conditional density
Step 4 · Emulator-based inference: the same posterior, one density swapped
Step 5 · Amortized inference: learn the posterior as a function of the data
Using a posterior: estimate, forecast, predict
Step 6 · Improve: compare distributions, then change the structure
Linear versus nonlinear, side by side
Bridge to ocean state estimation