Pre algebra questions and answers

Recent questions in Pre-Algebra
hetriamhageh6k20 2022-05-12 Answered

How do you solve 2.8 h 8.4 ?

Alisa Durham 2022-05-12 Answered

I have been reading a paper: Wang X, Golbandi N, Bendersky M, Metzler D, Najork M. Position bias estimation for unbiased learning to rank in personal search.
I am wondering how they derive the M-step in an EM algorithm that has been applied to the problem in their paper.
This is the data log-likelihood function:
log P ( L ) = ( c , q , d , k ) L c log θ k γ q , d + ( 1 c ) log ( 1 θ k γ q , d ) .
where observed data are (c:click, q:query, d:document, k:result position) rows, and θ k is the probability a search result at rank k is examined by the user (assumed to be independent of query-document pair), and γ q , d is the probability a search result, i.e., a query-document pair is actually relevant (assumed to be independent of result position).
Here are some hidden variable probability expressions (E means examination and R means relevant, they assume that a click event implies a search result is both examined and relevant):
P ( E = 1 , R = 1 C = 1 , q , d , k ) = 1 P ( E = 1 , R = 0 C = 0 , q , d , k ) = θ k ( t ) ( 1 γ q , d ( t ) ) 1 θ k ( t ) γ q , d ( t ) P ( E = 0 , R = 1 C = 0 , q , d , k ) = ( 1 θ k ( t ) ) γ q , d ( t ) 1 θ k ( t ) γ q , d ( t ) P ( E = 0 , R = 0 C = 0 , q , d , k ) = ( 1 θ k ( t ) ) ( 1 γ q , d ( t ) ) 1 θ k ( t ) γ q , d ( t )
The paper has derived the M-step update formulas:
θ k ( t + 1 ) = c , q , d , k I k = k ( c + ( 1 c ) P ( E = 1 c , q , d , k ) ) c , q , d , k I k = k γ q , d ( t + 1 ) = c , q , d , k I q = q , d = d ( c + ( 1 c ) P ( R = 1 c , q , d , k ) ) c , q , d , k , I q = q , d = d
however, how to get these two formulas?
Note: M-step is usually derived from this formula:
Q ( θ θ ( t ) ) = E Z X , θ ( t ) [ log L ( θ ; X , Z ) ]
θ ( t + 1 ) = a r g m a x θ   Q ( θ θ ( t ) )
where θ are parameters (i.e., θ k and γ q , d in this case), X are data (c,q,d,k in this case), and Z are hidden variables (E and R?).

lnwlf1728112xo85f 2022-05-11 Answered

How do you solve x 2 < 5 8 ?

Alaina Holt 2022-05-11 Answered

How do you solve 7 x + 8 3 x - 16 ?

Micah Haynes 2022-05-09 Answered

Reduce 3/9*13/12+12*8.

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