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carllacan

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## Homework Statement

An imperfect sensor measures the weather (which can only be either sunny, cloudy or rainy) in a fixed place. Suppose we know the weather on the first day was

*sunny*. In the following days we obtain the measurements

*cloudy, cloudy, rainy, sunny*.

What is the probability that the weather at day 5 is indeed sunny as our sensor says?

## Homework Equations

The noise of the sensor is given by a known stochastic matrix:

[itex]

S_{ij} = p(z_n = j | x_n = i)

[/itex]

where [itex]z_n[/itex] refers to a measurement and [itex]x_n[/itex] to the real state at day n, and [itex]i[/itex] and [itex]j[/itex] refer to the possible weathers (sunny, cloudy, rainy).

## The Attempt at a Solution

I just tried to find a general formulation, as I saw that I would probably need it later.

By straightforward application of Bayes Rule I've found the expression:

[itex]

p(x_n = i | z_n = j) = \frac{p(z_n = j | x_n = i)p(x_n = i)}{p(z_n = j)}

[/itex]

Now, here is where I have trouble. How can I find out the real probability that one day had a certain weather?

In a previous exercise I was presented with a similar situation and was given a stochastic matrix that described the probability of weather on a certain day based on the weather of the previous day. I am not sure that it applies also to this exercise. Is there any way to solve it with the given data?

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