Assumed-density filter

In Bayesian inference, assumed-density filtering (ADF), sometimes also called canonical-form filtering, is an approximation of recursive Bayesian estimation (RBE) that can be used when the posterior distribution that would otherwise be obtained with RBE would end up on an increasingly complex functional form over time, which makes RBE intractable. In ADF, the true posterior is simplified after each measurement update by being projected onto a selected parametric family of distributions (the "assumed density" or "canonical form"), often by moment matching. This keeps the functional form of the posterior distribution constant which avoids the problem of increasing complexity seen with RBE.

Examples

The following table gives examples of some common assumed-density filters:

Assumed density	Filters
$p(x;\theta )={𝒩}(x;\mu ,\mathrm{\Sigma })$	Kalman filters, extended Kalman filters, unscented Kalman filters
$p(x;\theta )=\sum _{i=1}^N{w}^{(i)}\delta (x-x^{(i)})$	Particle filters
$p(x;\theta )=\sum _{i=1}^N{w}^{(i)}{𝒩}(x_{{𝓁}};{\mu }^{(i)},{\mathrm{\Sigma }}^{(i)})\delta (x_{\text{n}}-x_{\text{n}}^{(i)})$	Rao–Blackwellized particle filters

where ${𝒩}$ is the multivariate normal distribution, $\delta$ is the multivariate Dirac distribution, $\{w^{(i)}\}$ is a set of convex weights, and $x_{{𝓁}}$ and $x_{\text{n}}$ are the parts of the vector $x$ in the linear dimensions and in the non-linear dimensions, respectively.

External sources

• Assumed Density Filtering on YouTube
• 6.3 Assumed density filters on YouTube

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