Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun University of Alberta Supported by NSERC, MITACS, PIMS Lockheed Martin Naval Electronics and Surveillance System Lockheed Martin Canada, APR. Inc

Outline Introduction Simulation Studies Filtering Equations Markov Chain Approximations Model Selection Future Work

1. Introduction Motivation: Submarine tracking and fish farming Model: - Signal : (1)

- Observation : (2) Goal: to find the best estimation for the number of targets and the location of each target.

2. Simulation Studies

3. filtering equations Notations : the space of bounded continuous functions on ; : the set of all cadlag functions from into ; : the spaces of probability measures; : the spaces of positive finite measures on ; : state space of.

Let,, and. Define

The generator of Let where.

For any, we define where and

Conditions: C1. and satisfy the Lipschitz conditions. C2. C3. C4.

Theorem 1. The equation (1) has a unique solution a.s., which is an -valued Markov process.

Bayes formula and filtering equations Theorem 2. Suppose that C1-C3 hold. Then (i) (ii) where is the innovation process. (iii)

Uniqueness Theorem 3. Suppose that C1-C4 hold. Let be an - adapted cadlag process which is a solution of the Kushner-FKK equation where Then, for all a.s.

Theorem 4 Suppose that C1-C4 hold. If is an - adapted -valued cadlag process satisfying and Then, for all a.s.

4. Markov chain approximations Step 1 : Constructing smooth approximation of the observation process Step 2 : Dividing D and Let, For, let

Note that if is a rearrangement of. Let then. For, let. For, with 1 in the i-th coordinate.

Step 3 : Constructing the Markov chain approximations Method 1: — Method 1: Let. Set. One can find that and

Define as and for, define as let

─Method 2 : Let and, Then and Define as for.

Let as, take denote the integer part, set and let satisfy

Then, the Markov chain approximation is given by Theorem 5. in probability on for almost every sample path of.

5. Model selection Assume that the possible number of targets is,. Model k:,. Which model is better? Bayesian FactorsBayesian Factors Define the filter ratio processes as

The Evolution of Bayesian Factors Let and be independent and Y be Brownian motion on some probability space. Theorem 3. Let be the generator of,. Suppose that is continuous. Then is the unique measure- valued pair solution of the following system of SDEs, (3)

for, and (4) for, where is the optimal filter for model k, and

Markov chain approximations Applying the method in Section 3, one can construct Markov chain approximations to equations (3) and (4).

6. Future work Number of targets is a random variable Number of Targets is a random process