Prediction is essentially the same except that we don't have the evidence here, which

leads to essentially the same algorithm.

But since our evidence kind of peters out in the past, we are reaching stationary points

where we've essentially destroyed all the information we have by the observations.

So making predictions into the future is possible, but making long, long, long, long term prediction

still possible, but we're actually not using any information there.

We're basically predicting the kind of fixed points of the system.

Which is fine, and there's nothing we can do about it.

So that's something to keep in mind.

The next algorithm we looked at, the next fundamentally different algorithm we looked

at was the smoothing algorithm where we have two phases.

Essentially we start at the beginning and we filter until some kind, some time xk for

which we want to have a smooth estimation, a better estimation given the information

that comes after k.

So what we do is essentially we filter all the way to now, now being the last time we

have information, and then kind of play the transition model backwards to use the information

we have in this latter part.

And it's not very surprising that we get a recursive algorithm again, but what we're

getting here is kind of a backwards message.

Which we again can extract from this huge system of equations.

And if you look at this you can see the smoothing part of the algorithm on top and then you

go backwards across the transition model and get better values here.

Or in other words in this example, knowing that there was a second umbrella actually

tells us something about rain on the first day.

Why?

Because we've built in the 70% staying tendency into our model.

That actually says if there's rain on day two, which we've estimated to be 90% or 88%,

then that tells us something about day one, which we can kind of back propagate into this

here.

That's how the algorithm works.

Here's a straightforward implementation of that, which is an iterative way of doing it.

Which is what you would all do if given enough time and the task to implement this idea.