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Prof.~David Draper \\
Department of Statistics \\
University of California, Santa Cruz

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\textbf{\large STAT 131: Quiz 3} \textit{[15 total points]}

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Name: \underline{\hspace*{5.85in}}

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Bayes's Theorem is the only known approach to learning from data that satisfies two important properties: (a) it's logically internally consistent (meaning that it cannot produce contradictory conclusions such as \{An unknown quantity $\theta$ of interest to me cannot be negative, but Bayes's Theorem says that my best estimate of $\theta$ is $-2.3$\}, and (b) it combines information external and internal to your dataset in such a way that no extraneous information is inadvertently smuggled into your answer. However, it's possible to use Bayes's Theorem in a way that defeats its ability to help you learn from the world around you. The result we'll explore below in this quiz that illustrates this was called \textit{Cromwell's Rule} by the great British Bayesian statistician Dennis Lindley (1923--2013).

Let $U$ be a true-false proposition whose truth status is Unknown to you, and let $D$ be another true-false proposition (representing Data) whose truth status is known to you; an example would be $U =$ (person $P$ really is infected with COVID--19) and $D =$ (this PCR screening test \textit{says} that person $P$ is infected with COVID--19) (note that $U$ and $D$ are not the same; the PCR test could be wrong). Recall that Bayes's Theorem in this situation says, assuming that $P ( D ) > 0$, that
\begin{equation} \label{bayes-1}
P ( U \given D ) = \frac{ P ( U ) \, P ( D \given U ) }{ P ( D ) } \, ,
\end{equation}
in which $P ( U )$ is your \textit{prior} information about the truth of $U$ (in the example above, this would be the prevalence of COVID--19 among people similar to person $P$ in all relevant ways).

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\item[(a)]

Show that if you assume that $P ( U ) = 0$, then you would have to conclude that $P ( U \given D ) = 0$, no matter how the data information $D$ came out \textit{[5 points]}.

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\item[(b)]

Show that if you assume that $P ( U ) = 1$, then you would have to conclude that $P ( U \given D ) = 1$, no matter how the data information $D$ came out \textit{[5 points]}.

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\item[(c)]

Briefly explain in what sense your results in (a) and (b) imply that 

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\textit{Putting prior probability 0 or 1 on anything renders Bayes's Theorem incapable of helping you learn from data.}

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What practical conclusion should we draw about the assignment of prior probabilities? Explain briefly \textit{[5 points]}.

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