Suppose we are to make a two-class classification decision based on observations of a scalar random variable x. Suppose we collect N such observations, x₁, i=1,K,N and suppose they are independent and identically distributed (i.i.d.). Let us further suppose the following PDFs for a single observation of x under two states of nature (classes) @ and @2₂: p(x|@)= 1 -x²/2 2π p(x| @₂) = -√2e²(x-µ)²/2 a) Write the general expression given in class for the minimum-error-rate decision rule Explain briefly in words how we derived this rule. b) Beginning from the definition you stated in part (a), derive the minimum-error-rate decision rule for N i.i.d. observations of x, given the PDFs specified above. Assume that μ = 2 and P(@₂)=eP(@) (e = 2.71828....). Hints: Use the natural logarithm (ln) to simplify your answer. Your final answer should be of the form g(x) >T, where x=(x₁,K,x) and T is a threshold. c) Now write your answer to part (b) for the case when only one observation is obtained (i.e., N=1). Your decision rule should again be of the form g(x) >T.
Suppose we are to make a two-class classification decision based on observations of a scalar random variable x. Suppose we collect N such observations, x₁, i=1,K,N and suppose they are independent and identically distributed (i.i.d.). Let us further suppose the following PDFs for a single observation of x under two states of nature (classes) @ and @2₂: p(x|@)= 1 -x²/2 2π p(x| @₂) = -√2e²(x-µ)²/2 a) Write the general expression given in class for the minimum-error-rate decision rule Explain briefly in words how we derived this rule. b) Beginning from the definition you stated in part (a), derive the minimum-error-rate decision rule for N i.i.d. observations of x, given the PDFs specified above. Assume that μ = 2 and P(@₂)=eP(@) (e = 2.71828....). Hints: Use the natural logarithm (ln) to simplify your answer. Your final answer should be of the form g(x) >T, where x=(x₁,K,x) and T is a threshold. c) Now write your answer to part (b) for the case when only one observation is obtained (i.e., N=1). Your decision rule should again be of the form g(x) >T.
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Bayes Decision Theory
Transcribed Image Text:Suppose we are to make a two-class classification decision based on observations of a scalar
random variable x. Suppose we collect N such observations, x;, i=1,K,N and suppose
they are independent and identically distributed (i.i.d.). Let us further suppose the following
PDFs for a single observation of x under two states of nature (classes) @ and @₂:
=e=x²12
p(x|w) = -
1
2π
1
p(x|@₂) = - =e-(x-μ)²³/2_
√√2π
a) Write the general expression given in class for the minimum-error-rate decision rule.
Explain briefly in words how we derived this rule.
b) Beginning from the definition you stated in part (a), derive the minimum-error-rate
decision rule for N i.i.d. observations of x, given the PDFs specified above. Assume that
μ = 2 and P(@₂)=eP(w) (e = 2.71828....). Hints: Use the natural logarithm (ln) to
simplify your answer. Your final answer should be of the form g(x) >T, where
x = (x₁,K,x) and T is a threshold.
c) Now write your answer to part (b) for the case when only one observation is obtained
(i.e., N=1). Your decision rule should again be of the form g(x) >T.
d) Roughly sketch the PDFs p(x|@) and p(x|@₂), and indicate the threshold T (from part
(c)) on the graph. Explain in words why this value of T' is logical.
e) In general, when a parameter of the PDFs is unknown (such as in this problem), how
can you specify the decision rule? (Answer briefly. Don't work out the specific solution
for this problem).
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