Difference between prediction and update on Bayesian Filters

[carllacan]

Hi.

I have a couple ofsimple question about Bayesian Filters, just to check that I'm correctly grasping everything.

I've been thinking on the difference between the prediction and update steps. I understand the "Physical" difference: in the first one we calculate the probability of the world being in a certain state given that the system has performed certain action u, that is $p(x_t) = p(x_t|u_t, x_{t-1})p(x_{t-1})$, and in the second step we calculate the probability that the world is in a certain state given that the system has measured some quantity z, that is $p(x_t) = p(x_t|z_t, x_{t-1})p(x_{t-1})$.

Now, what troubles me is that even when both are the same calculation (finding the probability of some event throgh conditional probabilities on other event) they are performed on different ways in the Bayesian Filter algorithm:

Algorithm Bayes filter(bel(x[SUB]t−1[/SUB] ), u[SUB]t[/SUB] , z[SUB]t[/SUB] ):
    for all x[SUB]t[/SUB] do
        [u]bel[/u](x[SUB]t[/SUB] ) = ∫ p(x[SUB]t[/SUB] | u[SUB]t[/SUB] , x[SUB]t−1[/SUB] ) bel(x[SUB]t−1[/SUB] ) dx
        bel(xt ) = η p(z[SUB]t[/SUB] | x[SUB]t[/SUB] ) [u]bel[/u](x[SUB]t[/SUB] )
    endfor
    return bel(x[SUB]t[/SUB] )

I've come to the conclusion that this is due that while we know $p(x_t|u_t, x_{t-1})$, that is, the results of our actions, we don't usually have direct information about $p(x_t|z_t, x_{t-1})$, but rather just $p(z_t | x_t)$,so we use Bayes Theorem to "transform" from one conditional probability to the other one. Does this make any sense?

Also, minor side question: on the third line of the algorithm there is an integral. Over which variable are we integrating there, x_t or x_t-1? Or both?

Thank you for your time. I would be grateful to receive any kind of correction about terminology or formality, be as nitpicking as you can :-)

 

[D H]

The predict step doesn't use Bayes' theorem. It uses physics / math to advance the state. It also changes the covariance matrix. The covariance matrix should grow due to the prediction. It's the update step where Bayes' theorem comes into play. This should once again touch on both the state estimate and it's uncertainty. The update should decrease the uncertainty.

 

[carllacan]

The predict step doesn't use Bayes' theorem. It uses physics / math to advance the state. It also changes the covariance matrix. The covariance matrix should grow due to the prediction. It's the update step where Bayes' theorem comes into play. This should once again touch on both the state estimate and it's uncertainty. The update should decrease the uncertainty.

Yes, that's what I mean, the prediciton doesn't use Bayes' Theorem because we "know" the conditional probability, but in the update step we use Bayes because we need to express the conditional probability in terms of what we know. Is that so?

 

[FactChecker]

I think the description in http://en.wikipedia.org/wiki/Recursive_Bayesian_estimation may help. It clearly distinguishes between the prediction and update calculations. IMO it's notation is better, and shows that the integral is over x_t-1

 

[blue_raver22]

Hello,

Thank you for your questions about Bayesian filters. You are correct in understanding the difference between the prediction and update steps. The prediction step uses the previous state and the action taken to calculate the probability of the current state, while the update step uses the current state and the measurement to update the probability of the current state.

The reason for the difference in calculation methods is due to the nature of the information available. As you mentioned, in the prediction step, we have direct information about p(x_t|u_t, x_{t-1}), but in the update step, we only have information about p(z_t | x_t). This is why we use Bayes Theorem to transform the conditional probabilities in order to update the probability of the current state.

As for your side question, the integral is taken over the variable xt, as indicated by the limits of integration in the algorithm. This is because we are calculating the probability of the current state, xt, based on the previous state and the action taken, xt-1 and ut.

I hope this clarifies any confusion you had about the difference between prediction and update steps in Bayesian filters. Your understanding and terminology are correct, and I appreciate your attention to detail. If you have any further questions, please don't hesitate to ask.

Best,