Kalman model

What is Kalman Model/Filter

When the hidden variables of a Markov Chain are continuous and probability distributions are Gaussians, then the model is usually called a Kalman model.

When these models are used to make estimates about a given point of time, given only the past, they are generally called “filters” (Kalman filter)
when they make estimates given both past and future, they are called “smoothers.”

How does it work

Source: Neuromatch - The Kalman Filter

Kalman filters lie in the linear dynamical system

s_{t} = D s_{t - 1} + w_{t - 1}

where $D$ is a scalar that models how the estimates changes over time, and $w_{t} \sim N (0, σ_{p}^{2})$ is white Gaussian noise.

Thus, a Kalman filter estimates a posterior probability distribution recursively over time using a mathematical model of the process and incoming measurements.

The posterior mean can be expressed as:

{\bar{μ}}_{t} = (1 - K) D {\bar{μ}}_{t - 1} + K m_{t} = D {\bar{μ}}_{t - 1} + K (m_{t} - D {\bar{μ}}_{t - 1})

where $K$ is known as the Kalman gain and it is a function of $t$ . $m$ denotes measurements.

Implement Kalman filter with the sum rule and product rule for Gaussians

Step 1: Change yesterday's posterior into today's prior
Use the mathematical model to calculate how deterministic changes in the process shift yesterday's posterior, $N (μ_{s_{t - 1}}, σ_{s_{t - 1}}^{2})$ , and how random changes in the process broaden the shifted distribution:

p (s_{t} | m_{1 : t - 1}) = p (D s_{t - 1} + w_{t - 1} | m_{1 : t - 1}) = N (D μ_{s_{t - 1}} + 0, D^{2} σ_{s_{t - 1}}^{2} + σ_{p}^{2})

where $σ_{p}$ denotes process noise and $m$ denotes measurements.

Step 2: Multiply today's prior by likelihood

Use the latest measurement to form a new estimate somewhere between this measurement and what we predicted in Step 1. The next posterior is the result of multiplying the Gaussian computed in Step 1 (a.k.a. today's prior) and the likelihood, which is also modeled as a Gaussian $N (m_{t}, σ_{m}^{2})$ :

2a: add information from prior and likelihood

To find the posterior variance, we first compute the posterior information (which is the inverse of the variance) by adding the information provided by the prior and the likelihood:

\frac{1}{σ_{s_{t}}^{2}} = \frac{1}{D^{2} σ_{s_{t - 1}}^{2} + σ_{p}^{2}} + \frac{1}{σ_{m}^{2}}

Now we can take the inverse of the posterior information to get back the posterior variance.

2b: add means from prior and likelihood

To find the posterior mean, we calculate a weighted average of means from prior and likelihood, where each weight, $g$ , is just the fraction of information that each Gaussian provides!

\begin{aligned} g_{p r i o r} & = \frac{{i n f o r m a t i o n}_{p r i o r}}{{i n f o r m a t i o n}_{p o s t e r i o r}} \\ g_{l i k e l i h o o d} & = \frac{{i n f o r m a t i o n}_{l i k e l i h o o d}}{{i n f o r m a t i o n}_{p o s t e r i o r}} \\ {\bar{μ}}_{t} & = g_{p r i o r} D μ_{s_{t - 1}} + g_{l i k e l i h o o d} m_{t} \end{aligned}