(*maisthemarginalwrtBickel'sprior--withpointmass."th"istheta*)ma[x_,m_,ep_]:=ep*1/(Sqrt[2Pi])*Exp[-1/2x^2]+(1-ep)*Integrate[1/(mSqrt[2Pi])*Cos[Pith/(2m)]^2*Exp[-1/2(x-th)^2],{th,-m,m}];
(*maderisthederivativeofma,-numericalderivative-andneededforBayesrule;recall:delta(x)=x+marginal'(x)/marginal(x)istheBayesruleifthelikelihoodisnormal(asitis)*)mader[x_,m_,ep_]:=(ma[x+0.001,m,ep]-ma[x-0.001,m,ep])/0.002;
(*Thesamenumericalworkfortheuniform[-m,m]prior,eventhoughtherulewehaveisexact*)mao[x_,m_,ep_]:=ep*1/(Sqrt[2Pi])*Exp[-1/2x^2]+(1-ep)*Integrate[1/(2mSqrt[2Pi])*Exp[-1/2(x-th)^2],{th,-m,m}];(*derivative*)madero[x_,m_,ep_]:=(mao[x+0.001,m,ep]-mao[x-0.001,m,ep])/0.002;
(*Now,Itookm=20,ep=0.8;youmaytrydifferentvalues*)m=20;ep=0.8;Plot[{x,x+madero[x,m,ep]/mao[x,m,ep],x+mader[x,m,ep]/ma[x,m,ep]},{x,-m-5,m+5},PlotStyle{{GrayLevel[0]},{Hue[5/8]},{Hue[0]}},AspectRatioAutomatic](*Black=Diagonal;Blue=uniform;Red=Bickel*)
⁃Graphics⁃