Bayesian Networks.

Here is a probabilistic model that describes what it might mean when a person sneezes, e$.$g$.$ depending on whether they have a cold, or whether a cat is present and they are allergic. Scratches on the furniture would be evidence that a cat had been present.

$\backslash$table$[[$Cold $-$ Fake,Cold $-$ True$],[95,\mathrm{.}05]]$

$\backslash$table$[[\backslash$table$[[$Cat Presence $-],[$False$]],\backslash$table$[[$Cat Presence $-],[$True$]]],[98,\mathrm{.}02]]$

$\backslash$table$[[$Cat Presence,$\backslash$table$[[$Allergic Reaction$],[-$ False$]],\backslash$table$[[$Allergic Reaction$],[-$ True$]]],[$False$,-95,\mathrm{.}05],[$True$,25,\mathrm{.}75]]$

$\backslash$table$[[$Cold$,$Allergic Reaction,Sneere $-$ False,Sreere $-$ True$],[$False$,$False,$90,-01],[$False$,$True,$3,7],[$True$,$False,$2,8],[$True$,$True,$\mathrm{.}1,\mathrm{.}9]]$

$\backslash$table$[[$Cat Presence,Scratches $-$ False,Scratches $-$ True$],[$False$,-95,\mathrm{.}05],[$True$,5,5]]$

a$)$ Using Equation $13\mathrm{.}2$ in the textbook $($p$.415),$ write out the expression for the joint probability for any state $($i$.$e$.$ combination of truth values for the $5$ variables in this problem$).[$Note: Use capital letters for names of variables, and lower$-$case to indicate truth value, e$.$g$.$ 'cold' means Cold$=$T$,$ and $\text{'}-$cold' means Cold$=$F$.]$

$P($ Cold,Sneeze,Allergic,Scratches, Cat $)=?$

b$)$ Use the equation above to calculate the joint probability that the person sneezes, but does not have a cold, has a cat, is allergic, and there are scratches on the furniture:

$P(-$cold,sneeze,allergic,scratches,cat $)=?$

c$)$ Use normalization to calculate the conditional probability that a person has cat, given that they sneeze and are allergic to cats, but do not have a cold, and there are scratches on the furniture.

$P(cat|-$ cold,sneeze,allergic,scratches $)=?$

d$)$ Use Bayes' Rule to re$-$write the expression for P$($cat$|$scratches$).$ Look up the values for the numerator in the table above.

$P(cat|$ scratches $)=$

e$)$ The denominator in the answer for $($d$)$ would require marginalization over how many joint probabilities? Write out the expressions for these $($i$.$e$.$ expand the denominator, but you don't have to calculate the actual values$).$