QUESTION 8 The following table provides the history of how the economist has pre

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QUESTION 8 The following table provides the history of how the economist has predicted the future of the economy True Market flat down 0.15 0. 0.15 Eco Predicts up flat 0.8 0.1 0.1 0.2 From this you can see if the economist predicts the market to be up given the market will be up is 80. The the economist will predict the market up given flat is 15. Finally, the conditional probability that the economist will predict the market being up and the market goes down is 20. This is the first line of the above table. Using this data fill out the Bayes table for the "Economist Predicts Up. The table is below. (Two decimals will work for these values) conditional probability that Predicts Up SON Condition Joint Posterior 0.30.8 C E 0.5 0.211 0.2 0.04 F total The G value will e the probability that the economist will return a "up" prediction. How many additional Bayes tables will be required to solve this problem?

Explanation / Answer

See, SON here means the "State of Nature". So, that means weather the economy will actually be up, down or flat, okay?

So, there are only 3 possiblities, and they are mutually exclusive events ie. they cannot occur together.
So, in the first column of the second table the probabilties should add up to 1, ie.
0.3 + 0.5 + A = 1 that means, A = 0.2

B is the conditional probability that the economist will predict the economy to be up, even when it is flat, as given in the first table (Row 1, Coulmn 2): So, it's the same value: B= 0.15

Now, Joint probabilities can be calculated as:

P(State of Nature) X P(Economists Prediction| State of Nature)

So, for table 2: Column 3 = Column 1 X Column 2

So, C is the probability that the economy is going to be up and the economits predicts it to be up as well:
C= 0.3* 0.8 = 0.24

D= 0.5* 0.15 = 0.075

By the same logic we can verify our value of A:

A * 0.2 should be = 0.04, So, A should be = 0.04/ 0.2 = 0.2 as we initially calculated. :)

Now G value is the total of the joint probabilities ie. 0.24 + .075 + 0.04 = 0.355, this is the marginal (or independent) probability that the economist will give an 'up prediction'.

Now, the posterior probability means the probability that the economy is up, flat or down given that the economist predicts it to be up.
So, this can be calculated as:
Joint Probability/ Total probablity of the economist predicting the ecnomy to be up
Or Simply:
Each value of Column 3 divided by the value of G (we just calculated ie. 0.355).

So, E = 0.24/ 0.355 = 0.676
F = 0.04/ 0.355 = 0.113

You can also verify the middle row value as: .075/ 0.355 = 0.211 (Correct, Yaay!)

To solve the entire problem 2 more tables will be required for 'Pridicts Flat" and "Predicts Down".

Thankyou

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