Introduction to Bayes theorem : Introductory video
Before watching the video read the questions below. As you watch the video try to answer them
Questions
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For the first problem in the video - the one about the female engineers - is the quantity you are being asked to calculate a conditional or an absolute probability?
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For the female engineers problem you are given two conditional probabilities and one absolute probability. What are these the probabilities of?
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For the question about the medical test there are two random variables: there is a Bernoulli random variable that tells you whether you have
the disease and a Bernoulli random variable that tells you whether or not the test result was positive. Are these random variables independent of each other? Explain your reasoning.
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Give a statement of Bayes theorem.
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Suppose you now administered two tests to the patient to determine whether or not they have the disease. Test A comes up positive in $P(T_1=1|D=1)$ percent of diseased
patients and positive in $P(T_1=1|D=0)$ percent of healthy patients. Test B by contrast comes up positive in $P(T_2=1|D=1)$ percent of diseased patients and positive
in $P(T_2=1|D=0)$ percent of healthy patients. Draw a Venn diagram showing the various possible categories each individual could be in. If we can calculate the probability
that a person has the disease given they had positive results for the two tests using
$P(D=1|T_1=1 \wedge T_2=1) = \frac{P(T_1=1|D=1)P(T_2=1|D=1)P(D=1)}{ P(T_1=1|D=1)P(T_2=1|D=1)P(D=1) + P(T_1=0|D=1)P(T_2=1|D=1)P(D=1) + P(T_1=1|D=1)P(T_2=0|D=1)P(D=1) + P(T_1=0|D=1)P(T_2=0|D=1)P(D=1) }$
what further assumption have we made? How would we use Bayes theorem to calculate this conditional probability with this assumption relaxed?
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Consider a pair of discrete random variables, $X$ and $Y$, that can both take values between $0$ and $n$. Use Bayes theorem to derive an expression for the conditional probability
$P(X=a|Y=y)$ in terms of the set of absolute probabilities, $P(X=x_j)$, for getting each possible value for $X$ and the set of conditional probabilities $P(Y=y|X=x_j)$ for getting $Y=y$
given that $X=x_j$. In doing this you will need to use summation notation.
- For the first problem in the video - the one about the female engineers - is the quantity you are being asked to calculate a conditional or an absolute probability?
- For the female engineers problem you are given two conditional probabilities and one absolute probability. What are these the probabilities of?
- For the question about the medical test there are two random variables: there is a Bernoulli random variable that tells you whether you have the disease and a Bernoulli random variable that tells you whether or not the test result was positive. Are these random variables independent of each other? Explain your reasoning.
- Give a statement of Bayes theorem.
- Suppose you now administered two tests to the patient to determine whether or not they have the disease. Test A comes up positive in $P(T_1=1|D=1)$ percent of diseased patients and positive in $P(T_1=1|D=0)$ percent of healthy patients. Test B by contrast comes up positive in $P(T_2=1|D=1)$ percent of diseased patients and positive in $P(T_2=1|D=0)$ percent of healthy patients. Draw a Venn diagram showing the various possible categories each individual could be in. If we can calculate the probability that a person has the disease given they had positive results for the two tests using $P(D=1|T_1=1 \wedge T_2=1) = \frac{P(T_1=1|D=1)P(T_2=1|D=1)P(D=1)}{ P(T_1=1|D=1)P(T_2=1|D=1)P(D=1) + P(T_1=0|D=1)P(T_2=1|D=1)P(D=1) + P(T_1=1|D=1)P(T_2=0|D=1)P(D=1) + P(T_1=0|D=1)P(T_2=0|D=1)P(D=1) }$ what further assumption have we made? How would we use Bayes theorem to calculate this conditional probability with this assumption relaxed?
- Consider a pair of discrete random variables, $X$ and $Y$, that can both take values between $0$ and $n$. Use Bayes theorem to derive an expression for the conditional probability $P(X=a|Y=y)$ in terms of the set of absolute probabilities, $P(X=x_j)$, for getting each possible value for $X$ and the set of conditional probabilities $P(Y=y|X=x_j)$ for getting $Y=y$ given that $X=x_j$. In doing this you will need to use summation notation.