# Role of future measurements on Bayes Filter

This is a hard to explain question. If what I wrote makes no sense to you please let me know so that I can fix it.

Suppose a zone where the weather changes like a Markov chain between sunny, cloudy and rainy. You can't observe it directly, but you have a sensor that gives some information. However this sensor is faulty, so that (say) if the weather is sunny it has a 0.8 probability of reporting sunny and a 0.2 probability of reporting cloudy, and so on (you have a complete matrix describing the errors). You use Bayes Filter to find a probability distribution for the weather, and you keep updating this distribution as you keep receiving measurements from the sensor.

But would our distributions change if we were "in the future"? That is, imagine we are looking at the measurements from last week, and we see a day on which the sensor measured rainy. We look at the sensor error info and we see that on a rainy day the sensor measures rainy, without error. If we calculate now the distribution probability for the day previous to the rainy one we should assign a probability of 1 to rain and 0 to the other weather (because if the sensor says it had rained it means that it had rained). In the first scenario, though, you wouldn't have obtained this distribution.

So my question is: given a complete series of measurements on a Hidden Markov chain how can we calculate the probability distributions for the states of the system at different times?

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That is, imagine we are looking at the measurements from last week, and we see a day on which the sensor measured rainy. We look at the sensor error info and we see that on a rainy day the sensor measures rainy, without error. If we calculate now the distribution probability for the day previous to the rainy one we should assign a probability of 1 to rain and 0 to the other weather (because if the sensor says it had rained it means that it had rained).
No. We have "it is rainy => sensor shows rainy", but the sensor could show rainy at sunny or cloudy days, too (you did not exclude this here).

I don't understand what you mean with "future", as all observations are in the past.

For reasonable matrices for transitions and measurements (there are degenerate cases where everything is screwed up), and for a measurement series length that goes to infinity, I would expect that the current weather does not depend (in a relevant way) on the weather in the distant past. You can assume anything and it won't influence your result in any relevant way.

What I mean is that our calculations on the weather probability distribution for day n could be more accurate if we had the measured weather from the days n - 1, n and n + 1 than it would if we only had n - 1 and n. The case of rainy was just an extreme one that I used to show what I meant. If we could be completely sure that the day n + 1 rained then that would affect our belief of the weather at day n.

I am curious about how we would add the information on day n+1 to the other ones, but I'm having a hard time putting my question into words. Would it be better if I talked about a specific example?

Stephen Tashi