The Variational Approach in Filtering and Correlated Noise
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918426594770944 |
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| author | Srinivasan, Sharan Gupta, Vijay Honnappa, Harsha |
| author_facet | Srinivasan, Sharan Gupta, Vijay Honnappa, Harsha |
| contents | The variational formulation of nonlinear filtering due to Mitter and Newton characterizes the filtering distribution as the unique minimizer of a free energy functional involving the relative entropy with respect to the prior and an expected energy. This formulation rests on an absolute continuity condition between the joint path measure and a product reference measure. We prove that this condition necessarily fails whenever the signal and observation diffusions share a common noise source. Specifically we show that the joint and product measures are mutually singular, so no choice of reference measure can salvage the formulation. We then introduce a conditional variational principle that replaces the prior with a reference measure that preserves the noise correlation structure. This generalization recovers the Mitter--Newton formulation as a special case when the noises are independent, and yields an explicit free energy characterization of the filter in the linear correlated-noise setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_03001 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Variational Approach in Filtering and Correlated Noise Srinivasan, Sharan Gupta, Vijay Honnappa, Harsha Probability Systems and Control Optimization and Control The variational formulation of nonlinear filtering due to Mitter and Newton characterizes the filtering distribution as the unique minimizer of a free energy functional involving the relative entropy with respect to the prior and an expected energy. This formulation rests on an absolute continuity condition between the joint path measure and a product reference measure. We prove that this condition necessarily fails whenever the signal and observation diffusions share a common noise source. Specifically we show that the joint and product measures are mutually singular, so no choice of reference measure can salvage the formulation. We then introduce a conditional variational principle that replaces the prior with a reference measure that preserves the noise correlation structure. This generalization recovers the Mitter--Newton formulation as a special case when the noises are independent, and yields an explicit free energy characterization of the filter in the linear correlated-noise setting. |
| title | The Variational Approach in Filtering and Correlated Noise |
| topic | Probability Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2604.03001 |