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This post is discussing Bayesian reasoning in the context of guess a specified number between 1 and 20 (both inclusive).

Consider the following example: I’m thinking of a number between 1 and 20 (both inclusive), and I want you to try and guess the number.

In this case, the hypothesis space is the set of values that observations can take, H = {1, . . . , 20}. If N = 4 values were chosen from this space, the resulting data set might look like the following:

D1 = {14, 10, 2, 18}
D2 = {4, 2, 16, 8}
D3 = {5, 11, 2, 17}
D4 = {3, 7, 2, 4}

The post defines a hypothesis space as H = {1, . . . , 20}, regarding which, what is the sample space?

Are D1, D2, D3, D4 4 different sample spaces?

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The sample space of an experiment is the set of all possible outcomes of that experiment.

For example, if you toss a die two times, the sample space of this experiment would be $$ \Omega = \{ (1, 1), (1, 2), (1, 3), ..., (1, 6), (2, 1), (2, 2), ..., (6, 1), (6, 2), ... (6, 6) \} $$

In the example with guessing the numbers, the experiment is to choose 4 numbers out of the 20 numbers possible, so the sample space would consist of all the combinations of 4 numbers from 1 to 20. The size of the sample space would be:

$$ | \Omega | = {20 \choose 4} $$

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