YRM Talk

#Plot1: marginal likelihood - here the maximum is attained # over the parabola. Outside of the maximum set the # function is decreasing with n exponentially fast var('u,v') plot3d(exp(-15*(u-v^2)^2),(u,-1,1),(v,-1,1)) 

#Plot2A: regular case in 3d - if the maximum is unique # we can use the standard Laplace's approximation. # In a small neighborhood of $\hat{theta}$ use # quadratic approximation. var('u,v') p1=plot3d(exp(-2*(.5*(u-1/2)^2+2*(v-1/2)^2)),(u,0,1),(v,0,1),color=(0.7,0,0),opacity=0.3) p2=plot3d(exp(-10*(.5*(u-1/2)^2+2*(v-1/2)^2)),(u,0,1),(v,0,1),color=(.7,.7,0),opacity=0.5) p3=plot3d(exp(-50*(.5*(u-1/2)^2+2*(v-1/2)^2)),(u,0,1),(v,0,1),color=(0.1,.8,0.1),opacity=0.7) p4=plot3d(exp(-200*(.5*(u-1/2)^2+2*(v-1/2)^2)),(u,0,1),(v,0,1),color=(.1,0.1,0.8),opacity=1) show(p1+p2+p3+p4) 

#Plot2B: regular case in 3d - the same as (2A) but we # show a section of the plot in u=0.5 which contains # the maximum var('u,v') p1=plot3d(exp(-2*(.5*(u-1/2)^2+2*(v-1/2)^2)),(u,0,.5),(v,0,1),color=(0.7,0,0)) p2=plot3d(exp(-10*(.5*(u-1/2)^2+2*(v-1/2)^2)),(u,0,.5),(v,0,1),color=(.7,.7,0)) p3=plot3d(exp(-50*(.5*(u-1/2)^2+2*(v-1/2)^2)),(u,0,.5),(v,0,1),color=(0.1,.8,0.1)) p4=plot3d(exp(-200*(.5*(u-1/2)^2+2*(v-1/2)^2)),(u,0,.5),(v,0,1),color=(.1,0.1,0.8)) show(p1+p2+p3+p4) 

#Plot3: a concrete singular example - the normalized likelihood # in the case when the maximum set is singular var('u,v') p1=plot3d(exp(-10*(u^2-v^3)^2),(u,-1,1),(v,-1,1),color=(0.9,0.9,0.9),opacity=0.3) p2=plot3d(exp(-50*(u^2-v^3)^2),(u,-1,1),(v,-1,1),color=(0.9,0.9,0.5),opacity=0.6) p3=plot3d(exp(-150*(u^2-v^3)^2),(u,-1,1),(v,-1,1),color=(0.1,0.5,0.1)) show(p1+p2+p3) 

#Plot4: a fiber of $\hat{p} if covariance is nonzero - # the model is unidentifiable but the maximum # parameter set is non-singular u, v = var('u,v') f1 = (u, 1/(10*u), v) f2 = (u, 1/(10*u), v) p1 = parametric_plot3d(f1, (u,-1,-0.1), (v,-1,1), texture="orange") p2 = parametric_plot3d(f2, (u,0.1,1), (v,-1,1), texture="violet") show(p1+p2) 

#Plot5: a fiber of $\hat{p} if covariance is zero - # the maximum parameter set is singular u, v = var('u,v') f1 = (u, 0, v) f2 = (0, u, v) p1 = parametric_plot3d(f1, (u,-1,1), (v,-1,1), texture="orange") p2 = parametric_plot3d(f2, (u,-1,1), (v,-1,1), texture="violet") show(p1+p2) 

#Plot6: the marginal likelihood if the covariance is nonzero - # for large n the behavior is nice var('u,v') p1=plot3d(exp(-20*(u*v-1/10)^2),(u,-1,1),(v,-1,1),color=(.8,.8,.8),opacity=0.3) p2=plot3d(exp(-200*(u*v-1/10)^2),(u,-1,1),(v,-1,1),color=(.9,.9,.9),opacity=0.5) p3=plot3d(exp(-500*(u*v-1/10)^2),(u,-1,1),(v,-1,1),color=(0.1,0.3,0.5)) show(p1+p2+p3) 

#Plot7: the marginal likelihood if the covariance is zero - # the largest contribution to the integral comes from # a neighborhood of the singularity var('u,v') plot3d(exp(-50*u^2*v^2),(u,-1,1),(v,-1,1),color=(0.1,0.3,0.5))