Gaussian Markov Model

Formulation¶

Assumptions:

• Transition Function, $f$, and measurement function, $h$, are known.
• Gaussian system and measurement noise
• Gaussian distributions everywhere

Core Operations¶

• Posterior
• Filtering - Prediction + Correction
• Marginal Likelihood
• Posterior Samples

Bayesian¶

Joint Distribution¶

This represents how we decompose the time series. We use the Markov property that states that every subsequent prediction at time step $t+1$ is independent of any previous time steps, $t-\tau$.

We see that this factorizes.

Prediction Step¶

We can use a generic set of equations

Correction Step¶

For Gauss-Markov models, we can write a generic set of equations to calculate the analysis step.

This is expressed in terms of means and (cross)-covariances. This generic set of equations can be used to understand almost all filtering methods. For example:

• Linear + Conjugate -> Linear Kalman Filter
• Linearization via Taylor Approximations -> Extended Kalman Filter
• Sigma Points -> Unscented Kalman Filter
• Cubature Points -> Cubature Kalman Filter
• Moment Matching -> Assumed Density Filter

Marginal Likelihood¶

In the correction step, we see the normalization constant, $\boldsymbol{E}(\boldsymbol{\theta})$.

Smoothing¶

Exact Inference¶

There are very few circumstances when we can get exact inference.

Observation Operator Encoder¶

This can be seen from the [de Bézenac et al (2020)]

The joint distribution will be:

# initial state prior
mu_0: Array["Dy"] = param("mu_0", init_value=...)
Sigma_0: Array["Dy Dy"] = param("Sigma_0", init_value=..., constrains=positive)
z0: Array["Dy"] = sample("z0", Normal(mu_0, Sigma_0))

# transition prior
F: Array["Dy Dy"] = param("F", init_value=...)
b: Array["Dy"] = param("b", init_value=...)
mu_z = F @ z0 + b
Sigma_z: Array["Dy Dy"] = param("Sigma_z", init_value=..., constrains=positive)
z: Array["Dy"] = sample("z", Normal(mu_z, Sigma_z))

# transition prior
flow = NSF(features=y.shape, *args, **kwargs)
obs: Array["Dy"] = sample("obs", flow(z), obs=y)

Parameter Estimation¶

The assumptions can be any combination of the the following:

• We have an unknown dynamics model, $f$, and and unknown measurement model, $h$.
• We have an unknown dynamics and measurement model parameters, $\boldsymbol{\theta}$.
• We have an unknown initial distribution, $p(\boldsymbol{z}_0|\boldsymbol{\theta})$
• We have unknown Gaussian system and measurement noise, $\boldsymbol{\Sigma_z}$, $\boldsymbol{\Sigma_y}$

In the first case of parameter estimation, we assume that we do not know

We can decompose this expression further

Unfortunately, analytical expressions for the filtering distribution, $p(\boldsymbol{z}_t|\boldsymbol{y}_{1:t})$ and thus the data log-likelihood $\boldsymbol{L}(\boldsymbol{\theta})$ are only available for a small class of SSMs like the linear-Gaussian and discrete SSMs. Thus we need to use approximate filtering distributions to estimate the log-likelihood.

State Estimation¶

Assumptions:

• Transition Function, $f$, and measurement function, $h$, are known.
• Gaussian system and measurement noise

This is basically how we are able to fil

Deterministic Inference

• Approximate Model ($f$,$h$) - Linearization, Sigma Points
• Approximate Integral

Stochastic Inference:

• Ensembles
• Monte Carlo / Particle Filter