Re: [Bayes++] Observation model for SIR scheme

[Nicola B. <nb...@es...>]

Steven,

I modified my model inheriting from Likelihood_observe_model and copying the 
likelihood function code, but I had continuosly exceptions like "Roughening X 
not PSD" or "zero cumulative weight sum".
So I went back to the simplest example, the PV_SIR.cpp, and modified just a 
little the observation model multiplying z_pred[0] and Hx(0,0) by 2 (see 
attached file). This caused again a "zero cumulative weight sum" exception.
I looked through the library's code but I could not really understand the 
reason of the problem. What am I doing wrong?

Regards,

Nicola


On Thursday 16 Nov 2006 11:57, Michael Stevens wrote:
> On Wednesday, 15. November 2006 22:01, Nicola Bellotto wrote:
> > Steven,
> >
> > I am trying to implement a particle filter using the "SIR_schee", but I
> > have some problem with the observation model. The filter needs a model
> > derived from "Likelihood_observe_model" and four classes are already
> > implemented for linear-linearized/correlated-uncorrelated models.
>
> OK. I think you are refering to the four General_XX_observe_model classes
> in models.hpp.
>
> > Unfortunately my observation model is not linear (with uncorrelated
> > noises), indeed I used a "Correlated_addative_observe_model" for a
> > previous UKF implementation. Do you have an example of non-linear
> > observation model for the SIR_scheme?
>
> Looking at the class 'General_LzCoAd_observe_mode', the implementation only
> needs a 'Correlated_addative_observe_model' to generalise. Sadly I have
> restricted the interface to a 'Linrz_correlated_observe_model' which is
> unnecessary. I think I should change this in the future.
>
> You can easily generalise you own 'Correlated_addative_observe_model' by
> inheriting from Likelihood_observe_model and copying the addition
> likelihood function code from the class 'General_LzCoAd_observe_model'. The
> function definitions, which compute the likelihood of a correlated Gaussian
> noise are, at the end of 'bayesFltAlg.hpp'.
>
> > Do I have to define a new likelihood "L"? If so, I do
> > not understand exactly what it is...
>
> For the SIR_scheme your observe model must define a likelihood function.
>
> I like the following from Wikipedia...
>   Informally, if "probability" allows us to predict unknown outcomes based
>   on known parameters, then "likelihood" allows us to determine unknown
>   parameters based on known outcomes.
>
> The likelihood function is at the core of all Bayesian inference.
> Fundementally, all the observation models in Bayes++ define a likelihood.
> Most of the definitions restrict the form the function takes however.
>
> These restrictions (such as Gaussian noise) allow for simple numeric
> solutions. So when you use a Kalman filter, the observation likelihood is
> expressed as a few matrices rather then an arbitary likelihood function.
> The SIR_scheme, and other sampled solutions to Bayesian inference, have the
> advantage that they can find an approximate solution when an arbitary
> likelihood function is defined.
>
> How to define a likelihood function:
>
> Imagine you can define the conditional probability of some observation z,
> given the system state x as p( z | x ).
>
> If you knew x (which you generally don't!) then you can use this function
> to find the probability of a observation z.
>
> Conversely if you know z (which is true in our case) then fixing z in p( z
> | x ) defines a function of x. This is the Likelihood function, and can be
> writen L( x ). It is not a probability, for example, a continues likelihood
> function will not in general integrate to 1. I has many similar properties.
>
> All this said, you only need to use the SIR_scheme when you have a complex
> likelihood function which cannot be well approximated by any of the simpler
> models. Building likelihood function of real observations can be quite
> difficult however.  "Bayesian Multiple Target Tracking" Lawrence D Stone,
> Carl A Barlow, Thomas L Corwin is a good reference.
>
> As a starter, it can be quite a lot of fun to compare the results on
> simiple problems between the SIR_scheme and other schemes.
>
> All the best,
> 	Michael

-- 
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Nicola Bellotto
University of Essex
Department of Computer Science
Wivenhoe Park
Colchester CO4 3SQ
United Kingdom

Room: 1N1.2.8
Tel. +44 (0)1206 874094
URL: http://privatewww.essex.ac.uk/~nbello
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