Widely used? average out effect Discrete Prior. Examplep. More than one observation. using MVUE (sample mean) yy 1 = 3.2, y 2 =2.2, y 3 =3.6, y 4 =4.
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1 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
2 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
3 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
4 Credible Iterval Credible iterval for μ uig the poterior kow variace preciio i the reciprocal of the variace poterior preciio = + 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
5 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
6 for Exercie # #3 (49 Lecture of 009 Fall
7 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
8 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
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