Using the data below, which of the following would be the posterior probabilities, P(S_j| U) ? ?
? $\begin{array} { c c c } \begin{array} { c } \text { States of Nature } \\( s j ) \end{array} & \begin{array} { c } \text { Prior } \\\text { Probabilities } \\P ( s j ) \end{array} & \begin{array} { c } \text { Conditional } \\\text { Probabilities } \\P ( U \mid s j ) \end{array} \\\hline s 1 & 0.65 & 0.75 \\s 2 & 0.20 & 0.35 \\s 3 & 0.15 & 0.20 \\\hline \text { Total } & 1.00 & \\\hline\end{array}$ ?

Question

Using the data below, which of the following would be the posterior probabilities, P(S_j| U) ? ?
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\begin{array} { c c c } \begin{array} { c } \text { States of Nature } \\( s j ) \end{array} & \begin{array} { c } \text { Prior } \\\text { Probabilities } \\P ( s j ) \end{array} & \begin{array} { c } \text { Conditional } \\\text { Probabilities } \\P ( U \mid s j ) \end{array} \\\hline s 1 & 0.65 & 0.75 \\s 2 & 0.20 & 0.35 \\s 3 & 0.15 & 0.20 \\\hline \text { Total } & 1.00 & \\\hline\end{array}

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Quizplus · Accepted Answer

The posterior probabilities, P(Sj|U), can be calculated using Bayes' theorem. For each state of nature Sj, the posterior probability is P(Sj|U) = [P(U|Sj) * P(Sj)] / P(U), where P(U) = Σ[P(U|Si) * P(Si)] for all i. Calculating P(U) = (0.75*0.65) + (0.35*0.20) + (0.20*0.15) = 0.4875 + 0.07 + 0.03 = 0.5875. Then, for each state, P(S1|U) = (0.75*0.65) / 0.5875 = 0.83, P(S2|U) = (0.35*0.20) / 0.5875 = 0.12, and P(S3|U) = (0.20*0.15) / 0.5875 = 0.05.

Using the Data Below, Which of the Following Would Be