Dicrete Prior for (μ Widely ued? average out effect Dicrete Prior populatio td i kow equally likely or ubjective weight π ( μ y ~ π ( μ l( y μ π ( μ e Examplep ( μ y Set a ubjective prior ad a gueig value of. Calculate the likelihood value uig a tadard ormal table uig ormal PDF Get the Poterior Pr ior likelihood igle obervatio: y = 3. Φ( y μ exp ( y μ See the left for a ubjective prior ad the reult. uig MVUE (ample mea yy 1 = 3., y =., y 3 =3.6, y 4 =4.1 lightly differet? roud off error igle obervatio: y 3. exp exp ( μ yi ( y μ More tha oe obervatio equetially q y oe at a time all together Lecture of 009 Fall (44
Cotiuou Prior Pi Cotiuou Prior chooe a prior which i imilar to likelihood why? Poterior Pr ior likelihood π ( μ y π ( μ Normal Likelihood Flat π (μnormal μ of y o iformatio ~ Uiform(-, π ( μ y π ( μ Normal( Poterior ~ Normal( cojugate prior: π(μ ~ Normal (m, ( (, π μ y Normal m Normal( Poterior ~ 1/ / Normal( m + / / + How to chooe a Normal prior gue a mea (m firt the, chooe a td( with empirical rule ue the equivalet ample ize large eq : the prior ifluece too much. Subjective iterpolate ad umerical itegratio refer: how to calculate itegral i R Example /1 / Norma ( m+ //1 + //1 + + mea legth of oe-year old raibow trout previou tudy ~ Normal(= Aie ~ Normal (30, 4 Barb ~ flat prior Chuck ~ trapezoidal 0, 18, 4~40(weight 1, 46 oberved data: =1, ybar=3 Lecture of 009 Fall (45
Poterior Prior Normal(30,4 π ( μ = 1 Poterior Normal (31.96,0.365 Normal (3,5.774 umerically π ( p = 1/ 6( μ 18,18 < μ < 4 = 1,4 < μ < 40 = 1/ 6( μ 46,40 < μ < 46 Lecture of 009 Fall (46
Credible Iterval Credible iterval for μ uig the poterior kow variace preciio i the reciprocal of the variace poterior preciio 1 1 1 = + poterior mea 1/ / m = m + y / / m ± zα / ukow variace etimator: t ample variace t-dit. i ued m ± tα / No-omial prior ˆ poterior i ot ormal. umerically calculate with the equal ize for the both ide. example Hypothei tetig oe-ided : H0: μ= μ0 v. Ha: μ< μ0 y Gamma π ( μ ~ ( α *, β * μ 0 uder the parameter pace i ull hypothei two-ided uig the credible iterval Lecture of 009 Fall (47
Predictive Deity For the ext obervatio margializatio proce fid the joit pdf of the ext ob. ad the parameter, give the radom ample. the parameter i treated a uiace. itegral by du to get coditioal pdf f ( y+ 1 y1,..., y = f ( y+ 1 μ g ( μ y1,..., y dμ N(, N( m, μ dμ N( m m, = + = the predictive deity i f ( y + 1 y ProcedureP d radom ample Exercie #1 y,,...,, ~ (, 1 y y y+ 1 iidnormal μ kow give the parameter, o. pdf of radom ample f ( y1,..., y, y + 1 μ let g(u e the prior of u ame a the poterior PDF of the parameter μ oe of the advatage of Bayeia f ( y+ 1, μ y1,..., y = f ( y+ 1 μ, y1,..., y g( μ y1,..., y = f ( y+ 1 μ { r..} g( μ y1,..., y Lecture of 009 Fall (48
for Exercie # #3 (49 Lecture of 009 Fall http://wolfpack.hu.ac.kr
Comparig with Frequetit tit Frequetit MVUE Comparig MSE=B +V μ ~ (, f = y N μ / Bayeia etimator E(μ b 1/ / μ b = m + / / E( μ b = ( m μ + V(μ b ; clearly maller tha V(μ f V ( μ b = ( + y example etimate milk powder weight, μ (=1, o the label machie produce μ =1015, =5, =10 Arold prior ~ N(1000, 10 Bth Beth prior ~ N(1015, 7.5 Carol prior ~ flat Frequetit feaible area: 1015±3*5 5 => (1000, 1030 Lecture of 009 Fall (50
Comparig with Frequetit tit ( Cofidece ad Credible Iterval Frequetit Bayeia y ± zα / m ± zα / 1/ / m = m + / / ukow : uig ad t-ditributio Bayeia with flat prior = Frequetit Tetig Hypothei y 1 1 = + Oe ided: H0: μ=μ 0, Ha: μ>μ 0 Samplig Dit. y ~ N( μ0,, μb ~ N( m', ' Calculate p-value y μ p = ( 0 f P z coclude / μ0 pb = π ( μ y dμ Two-ided: uig credible iterval Lecture of 009 Fall (51