Use of uninformative priors to initialize state estimation for dynamical systems

Worthy, Johnny L.; Holzinger, Marcus J.

doi:10.1016/j.asr.2017.06.040

Cited by 5 publications

(3 citation statements)

References 22 publications

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“…Given the minimal information defined above, prior probabilities must be invariant to the choice of parameters, exchange of variables, and changes to location and scale [1][2][3][8][9][10][11]13,14,16,[26][27][28][29].…”

Section: Invariance Of Prior Probabilitiesmentioning

confidence: 99%

“…If there is such a case, equations Equations ( 31) and ( 32) remain valid, but there will be no unique median. The prior over scales (Equation ( 22)) can be derived as the unique distribution exhibiting invariance following scale transformation (Equation ( 18)), exemplifying the method of "transformation groups" [1,[8][9][10][11][27][28][29]. Here we transform the scale of all variables simultaneously p(x 1 , .…”

Section: Proof 1 From the Joint Distribution P(x 1 X 2 )mentioning

confidence: 99%

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The Objective Bayesian Probability that an Unknown Positive Real Variable Is Greater Than a Known Is 1/2

Fiorillo

Kim

2021

Philosophies

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If there are two dependent positive real variables x1 and x2, and only x1 is known, what is the probability that x2 is larger versus smaller than x1? There is no uniquely correct answer according to “frequentist” and “subjective Bayesian” definitions of probability. Here we derive the answer given the “objective Bayesian” definition developed by Jeffreys, Cox, and Jaynes. We declare the standard distance metric in one dimension, d(A,B)≡|A−B|, and the uniform prior distribution, as axioms. If neither variable is known, P(x2<x1)=P(x2>x1). This appears obvious, since the state spaces x2<x1 and x2>x1 have equal size. However, if x1 is known and x2 unknown, there are infinitely more numbers in the space x2>x1 than x2<x1. Despite this asymmetry, we prove P(x2<x1∣x1)=P(x2>x1∣x1), so that x1 is the median of p(x2|x1), and x1 is statistically independent of ratio x2/x1. We present three proofs that apply to all members of a set of distributions. Each member is distinguished by the form of dependence between variables implicit within a statistical model (gamma, Gaussian, etc.), but all exhibit two symmetries in the joint distribution p(x1,x2) that are required in the absence of prior information: exchangeability of variables, and non-informative priors over the marginal distributions p(x1) and p(x2). We relate our conclusion to physical models of prediction and intelligence, where the known ’sample’ could be the present internal energy within a sensor, and the unknown the energy in its external sensory cause or future motor effect.

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Section: Invariance Of Prior Probabilitiesmentioning

confidence: 99%

Section: Proof 1 From the Joint Distribution P(x 1 X 2 )mentioning

confidence: 99%

The Objective Bayesian Probability that an Unknown Positive Real Variable Is Greater Than a Known Is 1/2

Fiorillo

Kim

2021

Philosophies

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“…However, a probabilistic interpretation of the RSO's initial state can lead to a uniform probability distribution on the admissible region, thus producing a description of the RSO's state that is not inferred from the information available to the analyst. It thus appears that probability density functions (pdfs) are inappropriate tools to describe admissible regions, as argued in a recent study (Worthy III and Holzinger, 2017b). Another example is the exploitation of human-based or semantic data sources such as TLEs or natural languages statements: they could provide a wealth of information regarding RSOs, but the lack of statistical information on their accuracy/truthfulness makes their probabilistic representation, and thus their integration to a Bayesian tracking filter, difficult and largely unexplored to this day.…”

Section: Introductionmentioning

confidence: 99%

Physics and human-based information fusion for improved resident space object tracking

Delande

Houssineau

Jah

2018

Advances in Space Research

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Maintaining a catalog of Resident Space Objects (RSOs) can be cast in a typical Bayesian multi-object estimation problem, where the various sources of uncertainty in the problem -the orbital mechanics, the kinematic states of the identified objects, the data sources, etc. -are modeled as random variables with associated probability distributions. In the context of Space Situational Awareness, however, the information available to a space analyst on many uncertain components is scarce, preventing their appropriate modeling with a random variable and thus their exploitation in a RSO tracking algorithm. A typical example are human-based data sources such as Two-Line Elements (TLEs), which are publicly available but lack any statistical description of their accuracy. In this paper, we propose the first exploitation of uncertain variables in a RSO tracking problem, allowing for a representation of the uncertain components reflecting the information available to the space analyst, however scarce, and nothing more. In particular, we show that a human-based data source and a physics-based data source can be embedded in a unified and rigorous Bayesian estimator in order to track a RSO. We illustrate this concept on a scenario where real TLEs queried from the U.S. Strategic Command are fused with realistically simulated radar observations in order to track a Low-Earth Orbit satellite.

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Multi-Fidelity Orbit Determination with Systematic Errors

et al. 2021

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Use of uninformative priors to initialize state estimation for dynamical systems

Cited by 5 publications

References 22 publications

The Objective Bayesian Probability that an Unknown Positive Real Variable Is Greater Than a Known Is 1/2

The Objective Bayesian Probability that an Unknown Positive Real Variable Is Greater Than a Known Is 1/2

Physics and human-based information fusion for improved resident space object tracking

Multi-Fidelity Orbit Determination with Systematic Errors

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