Appendx Proof of heorem he multvarate Gauan probablty denty functon for random vector X (X,,X ) px exp / / x x mean and varance equal to the th dagonal term of, denoted he margnal dtrbuton of X Gauan wth he decon maker rk-neutral and therefore /he maxmze the total expected proft A pror, the decon maker chooe the project wth potve expected value If we denote the et of a pror proftable project a, then the pror value PV max 0, Suppoe there an nformaton ource about the project; X E, ~, E 0 Let denote a ubet of the project If the decon maker able to oberve nformaton about the ubet (for free), then the value wth th nformaton y max E X,0 y py dy max E X y,0 py dy y From the properte of the multvarate Gauan dtrbuton, ee eg Anderon (003), E X y y Let u denote X E y a W W a lnear combnaton of Gauan random varable and therefore ha margnal dtrbuton, where p (w ) (, )
h lnear combnaton the un-varate varable of nteret n the mult-dmenonal ntegral requred for the value of nformaton computaton Margnalzng out the remanng dmenon of nformaton, the value wth nformaton can be mplfed to max w,0 p(w ) dw E max W, 0 w he followng reult from Schlaffer (959) and Bckel (008) For a Gauan varable W wth mean m and varance, Emax W,0 m ( m ) m Ung th reult and ubtractng the pror value from the value wth nformaton VOI( ) Snce, we get the requred reult Proof of Corollary For VOI, there nformaton regardng all project, therefore he reult follow by ubttutng and, and recognzng that e e Proof of heorem he proof of heorem how that the VOI( ) can be wrtten a VOI( ) Conder agan a Gauan varable W wth mean m and varance We wll fnd partal dervatve of the expreon m( m ) m wth repect to any parameter
d m m m m m dm d dm / d d dm / m m m d d dm / d d dm / d Snce d x dx x and d x d x x exp x exp x x, dx dx the dervatve mplfe to m m d m dm m / / m dm d m dm m m m d d d d d dm d m m d d Sentvty to the mean If m, Sentvty to the tandard devaton If, m m d m m dm m m d m m d () Sentvty to From the reult above (entvty to the mean), the dervatve wth repect to the mean of any project d,, 0 d d 0 VOI( ) maxmum at 0 and decreae a the mean get further from 0 () Sentvty to he meaurement noe affect the equvalent tandard devaton but doe not affect any mean parameter For a parameter that only affect the equvalent tandard devaton, 3
d d d d d For the lat tep n the computaton above, we have ued the followng reult, whch we proved before part () (entvty to the tandard devaton) d d For the remander of the proof, we wll compute the dervatve of the equvalent varance wth repect to q, e d d, and then replace t back n the expreon d d he dervatve of the nvere of an nvertble matrx M dm d dm d M M, where the matrx dervatve work for every entry of the matrx Here, parameter n,, therefore, for ome d d d d d d If q = t for, then d a matrx of 0 except the th dagonal element, whch equal d Matrce and are both ymmetrc alo ymmetrc nce matrx nveron b e, retan the ymmetrc property hu, by replacng d d d b b b 0 d In ummary, we get negatve effect and the dervatve dvoi( ) b d 4
() Sentvty to he pror tandard devaton affect the equvalent tandard devaton but doe not affect any mean parameter Smlar to (), for a parameter that only affect the equvalent tandard devaton, d d d d d Conder frt the pecal cae of VOI he dervatve of agan the crucal part he followng matrx dentte hold by the Sherman-Woodbury-Morron formula, ee eg Henderon and Searle (98) and herefore, S For the dervatve of d e e d d e e e e d d d If q =, d a matrx whch nonzero only for the element n row and column Defnng d d B d de d e, whch non-negatve defnte, the above expreon can be wrtten a e B e e Be Snce both and are potve defnte, th dervatve wll be potve, and VOI ncreae a a functon of the pror uncertanty he cae wth PVOI can be vewed a a pecal cae of total nformaton where ome t go to nfnty he ame proof therefore hold 5