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Chapter 3

# Chapter 3 - 3 3.1 Likelihood and Suciency Introduction...

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§ 3 Likelihood and Sufficiency § 3.1 Introduction 3.1.1 Sample data: X = ( X 1 , . . . , X n ) Postulated parametric family of probability functions (i.e. either mass functions or pdf’s) : { p ( · | θ ) : θ Θ } e.g. — X 1 , . . . , X n are iid Poisson( θ ): p ( x 1 , . . . , x n | θ ) = n Y i =1 P ( X i = x i | θ ) = n Y i =1 e - θ θ x i x i ! , θ (0 , ) . X 1 , . . . , X n are iid N ( μ, σ 2 ): p ( x 1 , . . . , x n | θ ) = n Y i =1 1 2 πσ 2 exp - ( x i - μ ) 2 2 σ 2 θ = ( μ, σ ) ( -∞ , ) × (0 , ) . If we observe X = x = ( x 1 , . . . , x n ), what can we learn from x about the true value of θ ? We wish to answer this question by means of statistical inference . 3.1.2 Raw sample data, i.e. x , contain information relevant to our inference about θ . We want to extract such information completely yet “economically”. This amounts to finding an efficient way to “summarize” data. Answer: Likelihood function — a mathematical device summarizing all information available in x which is relevant to θ . It measures the plausibility of each θ Θ being the true θ that gives rise to x . 3.1.3 Suppose the random vector X has a probability function belonging to the parametric family { p ( · | θ ) : θ Θ } . Definition. Given that X is observed (realised) to be x , the likelihood function of θ is defined to be x ( θ ) = p ( x | θ ) , i.e. the probability function of X , evaluated at X = x , but considered as a function of θ . 17

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The loglikelihood function is S x ( θ ) = ln x ( θ ) , which gives equivalent information but is more convenient to work with than x ( θ ). 3.1.4 Common special case — X iid: X is a random sample, i.e. X = ( X 1 , . . . , X n ) are iid with each X i probability function f ( x | θ ). Then the (joint) probability function of X is p ( x | θ ) = n Y i =1 f ( x i | θ ) and so x ( θ ) = p ( x | θ ) = n Y i =1 f ( x i | θ ) , where x = ( x 1 , . . . , x n ) is the realisation of X . 3.1.5 When examining a likelihood function x ( θ ), we are mainly interested in the relative likelihoods of different values of θ . The actual magnitude of the likelihood function itself is unimportant. Without loss of information we may ignore multiplicative factors in the likelihood function that do not depend on θ . For example, if ( X 1 , . . . , X n ) iid Poisson( θ ), we may take the likelihood function to be x ( θ ) = e - θ i x i , omitting the factor ( Q i x i !) - 1 . 3.1.6 Example § 3.1.1 There are two decks of playing cards: deck A — conventional, has 52 cards; deck B — has 13 cards of the same suit ( ). Your friend draws two cards randomly without replacement from the same deck and gets an ace and a king of hearts. Without asking your friend, which deck do you think your friend is more likely to draw the cards from? Denote each card by ( N, S ) = ( rank, suit ) . In the experiment, the sample X = { ( N 1 , S 1 ) , ( N 2 , S 2 ) } is observed to be x = { ( A, ) , ( K, ) } . If we label deck A by θ = 0 and deck B by θ = 1 , then 18
the parametric family consists of two members, Deck A ( θ = 0) : p (( N 1 , S 1 ) , ( N 2 , S 2 ) | 0) = 52 2 - 1 , Deck B ( θ = 1) : p (( N 1 , S 1 ) , ( N 2 , S 2 ) | 1) = 13 2 - 1 , if S 1 = S 2 = , 0 , otherwise.

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Chapter 3 - 3 3.1 Likelihood and Suciency Introduction...

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