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Over a period of 100 days, weather forecasters issue forecasts for rain on a daily basis. Based on the forecasts and actual observations, you are supposed to help them finding out if the forecasts were any good. A forecast is simply one of the following three possible statements: “no rain”, “maybe rain”, or “certainly rain”. The forecasters record the number of days they issued each of these three forecasts, and also the corresponding number of days it rained. The results are collected below:

Forecast: "No rain" , Days with rain: 4, Days without rain: 18

Forecast: "Maybe rain" , Days with rain: 25, Days without rain: 22

Forecast: "Rain" , Days with rain: 25, Days without rain: 6

Show that $n_{ij}$ has a multinomial distribution with unknown probabilities $p_{ij}$, which are the unknowns parameters in this problem. Write down the likelihood of the data nij as a function of $p_{ij}$ and N.

This is a past exam paper and I have no clue how to answer this in a way that could get me marks! I've written the multinomial distribution:

$\frac{n!}{X_1!...X_n!}p_1^{x_1}*p_2^{x_2}...*p_n^{x_n}$

But I can't work out how to do this question for the life of me! Can anyone help?

user145712
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1 Answers1

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For every day, there 6 exclusive events that may occur. The events are encoded as tuples $\{\mathrm{forecast}, \mathrm{observation}\}$. The six events can be organized in a table

$$ \begin{array}{r|ll} \text{forecast}\backslash \text{obs.} & \text{no rain} & \text{rain} \\ \hline \\ \text{no rain} & m_{n,n} & m_{n,r} \\ \text{maybe rain} & m_{m,n} & m_{m,r} \\ \text{rain} & m_{r,n} & m_{r,r} \end{array} $$ Where $m_{fo}$ denote the number of days with given forecast and actual observation. $$ \sum_{f,o} m_{f,o} = N $$ Because of exclusivity, the total number of observations of each type equals to the number of days the experiment was conducted. Given the probabilities of each of the exclusive events $p_{f,o}$, which must total to 1, the probability of any instance of this table is given by the multinomial distribution: $$ \Pr\left(M_{n,n}=m_{n,n}, \ldots, M_{r,r}=m_{r,r}\right) = \frac{N!}{m_{n,n}! \cdots m_{r,r}!} p_{n,n}^{m_{r,r}} \cdots p_{r,r}^{m_{r,r}} $$

Sasha
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