SUPPLEMENT TO POSTERIOR CONTRACTION AND CREDIBLE SETS FOR FILAMENTS OF REGRESSION FUNCTIONS. North Carolina State University

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1 Submitted to the Annals of Statistics SUPPLEMENT TO POSTERIOR CONTRACTION AND CREDIBLE SETS FOR FILAMENTS OF REGRESSION FUNCTIONS By Wei Li and Subhashis Ghosal North Carolina State University The supplementary file contains proofs for Lemma 5.7, 8., and 8.4. Proof of Lemma 5.7. Let be an arbitrary point on ˆL. Thus Υ = ˆL and Υ t L for some t >. Note that inf y L y Υ Υ t t as V =. Let D Υ,V ft := d dt fυ t = fυ t T V Υ t and D 2 Υ,V ft := fυ t, V Υ t, V Υ t = fυ t T V Υ tv Υ t + V Υ t T HfΥ tv Υ t, where the second line is due to the chain rule. A Taylor epansion yields that D Υ,V ft D Υ,V ft = t t D 2 Υ,V f t for some t between and t. In particular, since D Υ,V ft =, letting t =, we obtain D Υ,V f = t D 2 Υ,V f t. Furthermore, D Υ,V f = D Υ,V f D Υ, ˆV ˆf = fυ T V Υ ˆfΥ T ˆV Υ sup f T V ˆf T ˆV sup Csup f T V ˆV + sup f ˆf T ˆV V ˆV + sup f ˆf C sup V ˆV. By the uniform continuity of f, V, V and Hf and the continuity of Υ t in t, without loss of generality, we can make t small enough imsart-aos ver. 24//6 file: Filaments_AOS_supp_9Mar28.te date: March 9, 28

2 2 LI, W. AND GHOSAL, S. see comments after the proof. Thus we have DΥ 2,V f t DΥ 2,V ft < η/2. By Assumption A3, DΥ 2,V ft > η, and hence DΥ 2,V f t DΥ 2,V ft DΥ 2,V f t DΥ 2,V ft > η/2. Therefore, inf y L y t C η sup V ˆV. Thus d ˆL L C η sup V ˆV. Similarly, dl ˆL C η sup V ˆV. Therefore, HausL, ˆL C η sup [,] 2 V ˆV. Recall that V = Gdf 2. Now since G is a fied Lipschitz continuous function, it is easy to get the upper bound for sup [,] 2 V ˆV in terms of the supremum distance of the derivatives of f ˆf. In the proof, t can be made arbitrarily small in the limit. To see this, if Assumption A5 holds for the f or more precisely, Υ, then Υ t ˆΥ ˆt = Υ t = Υ t Υ > C G t. Since ˆΥ ˆt Υ t can be made arbitrarily small due to the closedness in supremum norm see previous theorems, t can be made arbitrarily small. Proof of Lemma 8.. These results can be directly adapted from [] by noting the highest degree of derivatives in each epression. We shall show the rates for sup Hf Hf F, sup d 2 f d 2 f F and sup V V F. Notice that Hf Hf F = f 2, f 2, 2 + 2f, f, 2 + f,2 f,2 2 /2, 4 /2 f 2, f 2, + 2 f, f, + f,2 f,2. The rate for sup Hf Hf F then follows easily. Also, the contraction rates for d 2 f d 2 f F follows from the inequality d 2 f d 2 f F 6 /2 r: r =3 Dr f D r f. imsart-aos ver. 24//6 file: Filaments_AOS_supp_9Mar28.te date: March 9, 28

3 Since V V = Gd 2 f d 2 f Gd 2 f d 2 f, which is 2 by 2 matri, straightforward calculation gives its, element G,, d 2 ff 3, G,, d 2 f f 3,, + G,, d 2 ff 2, G,, d 2 f f 2,, + G,, d 2 ff,2 G,, d 2 f f,2. The absolute value of the first summand is bounded by the sum of G,, d 2 ff 3, G,, d 2 ff 3,, 3 and G,, d 2 ff 3, G,, d 2 f f 3,. Noting that d 2 f contracts to d 2 f uniformly in, hence {d 2 f : [, ] 2 } Q δ with posterior probability tending to, the first term is bounded by a constant multiple of f 3, f 3,, in view of the result 4 of Remark 8.. By the same remark and assumption f α, <, the second term is bounded by d 2 f d 2 f f 3,. Using similar arguments for the second and third summand, one can see that the absolute value of the, element of V V is bounded by f 3, f 3, + f 2, f 2, + f,2 f,2. Dealing the rest elements of V V similarly, we can see that V V F r: r =3 Dr f D r f. Proof of Lemma 8.4. We shall consider only two separate cases i r = 2, and ii r =,. i. For r = 2, similarly for r =, 2, a, t = t t J t j G sb j Υ, sb j2 Υ 2, sds j Υ, s B j2 Υ 2, sds j Υ, s ds, imsart-aos ver. 24//6 file: Filaments_AOS_supp_9Mar28.te date: March 9, 28

4 4 LI, W. AND GHOSAL, S. the last line follows by J 2 j 2 = B j 2 2 = and noticing that B j,q = q q 2B j,q 2 t j t j q +2t j t j q + + q q 2B j,q 2 t j t j q +t j t j q + q q 2B j,q 2 t j t j q +2t j t j q + q q 2B j 2,q 2 t j 2 t j q +t j t j q, which implies J j = B j Υ, s 4J 2. Therefore, a, t J 2. Net, let S j = [t,j q 2, t,j ], S j2 = [t 2,j2 q 2, t 2,j2 ] and j,j 2 s := {s : Υ s S j S j2 }. Turn to a, t 2, which is = = t G s G s 2 G sb j Υ, sb j2 Υ 2, sds B B j Υ, s B j2 Υ 2, s dsds j Υ, sb j2 Υ 2, s j Υ, s B j2 Υ 2, s B j Υ, s B j2 Υ 2, s dsds j,j 2 s j,j 2 s B j Υ, s B j2 Υ 2, s j Υ, s B j2 Υ 2, s dsds. The last equality is obtained as follows. Since for any fi j, j 2, B j is supported on S j and B j2 is supported on S j2. So j,j 2 s j,j 2 s = {s, s : Υ s S j S j2 and Υ s S j S j2 }. Notice that for n large, S j S j2 J and Υ s S j S j2 and Υ s S j S j2 implies that Υ s Υ s J. By Assumption A5, C G s s s s Υ s Υ s /s s J, and hence s s J. Therefore, for n large enough, above quantity can be further bounded by a constant multiple of { s s < CJ } B j Υ, s B j2 Υ 2, s j Υ, s B j2 Υ 2, s dsds. imsart-aos ver. 24//6 file: Filaments_AOS_supp_9Mar28.te date: March 9, 28

5 Noting B j2 2 and J 2 j 2 = B j 2 =, it can be further bounded by { s s < CJ } B j Υ, s B j2 Υ 2, s B j Υ, s dsds J J 2 J t { s s < CJ } j j = j Υ, s ds. j Υ, sb j Υ, s dsds From argument used in bounding a, t, we have J j = B j Υ, s J 2. This completes the proof for a, t 2 J 3. For the third result, we write where a, t, t := and a 2 t,, := a, t a, t 2 a, t, t 2 + a 2 t,, 2, t First, note that a, t, t 2 = GΥ sb 2, J,J 2 Υ sds GΥ sb 2, J,J 2 Υ sds t J 2 J 4J 2 ds t = J 3 t t, t 5 GΥ sb 2, J,J 2 Υ sds, GΥ sb 2, J,J 2 Υ sds. 2 GΥ sb j Υ, sb j2 Υ 2, sds where the second line follows by a similar argument used to bound a, t 2. Net, a 2 t,, 2 is given by [ GΥ sb j Υ, sb j2 Υ 2, s GΥ sb j Υ, sb j2 Υ 2, sds] 2, imsart-aos ver. 24//6 file: Filaments_AOS_supp_9Mar28.te date: March 9, 28

6 6 LI, W. AND GHOSAL, S. which is bounded up to a multiple constant by GΥ s B j Υ, s B j Υ, s B j2 Υ 2, sds 2 GΥ s B j2 Υ 2, s B j2 Υ 2, s B j Υ, sds GΥ s GΥ s B j Υ, sb j2 Υ 2, sds 2. Bound the first term in the right hand side of above epression as B j2 Υ 2, sds 2 2 GΥ s B j Υ, s B j Υ, s B j2 Υ 2, sds GΥ s j Υ, s Υ, s Υ, s 2 2 J 2 2 J 5. GΥ 2 s B j Υ, s B j2 Υ 2, sds j Υ, s ds The second term is bounded as 2 2 GΥ s B j2 Υ 2, s B j2 Υ 2, s B j Υ, sds GΥ 2 s Υ 2, s Υ 2, s B j 2 Υ 2, s B j Υ, s GΥ 2 s B j 2 Υ 2, s B j Υ, s imsart-aos ver. 24//6 file: Filaments_AOS_supp_9Mar28.te date: March 9, 28

7 2 { s s CJ } 2 B j 2 Υ 2, s j Υ, s B j 2 Υ 2, s j Υ, s dsds J 5 2, { s s CJ }J 2 j = 7 j Υ, s j Υ, s dsds where the second line follows from the mean value theorem, the third line from the Lipschitz continuity of Υ in Remark 8. and fourth line by a similar argument used to bound a, t 2. The third term GΥ s GΥ s 2 2 J 3, B j Υ, sb j2 Υ 2, sds Υ s Υ s B j Υ, sb j2 Υ 2, sds 2 B j Υ, s B j2 Υ 2, sds where the second line holds by mean value theorem and the third line holds due to Lipschitz continuity of Υ in Remark 8. and last line holds by similar argument for a, t 2. In summary, we have that a 2 t,, 2 J 5 2 and a, t, t 2 J 3 t t. ii. Now turning to the case r =,. By similar argument we have a, t = t t J t j = G sb j Υ, sb j 2 Υ 2, sds B j Υ, s B j 2 Υ 2, s ds B j Υ, s ds J imsart-aos ver. 24//6 file: Filaments_AOS_supp_9Mar28.te date: March 9, 28

8 8 LI, W. AND GHOSAL, S. Likewise, a, t 2 = J J2 t G sb j Υ, sb j 2 Υ 2, sds can be bounded by J { s s < CJ } B j Υ, sb j 2 Υ 2, s B j Υ, s B j 2 Υ 2, s dsds B j Υ, s dsds J 2 t J 3 t t t J 3 J t j = { s s < CJ } B j Υ, s B j 2 Υ 2, s { s s < CJ } B j Υ, sb j Υ, s dsds { s s < CJ } B j Υ, s dsds j = j B j Υ, s ds which is bounded by a constant multiple of J 3. The third result a, t, t 2 J 3 t t and a 2 t,, 2 J 5 2 can be derived in a similar manner, since a, t, t 2 = t J 2 J 4J 2 ds t t 2 GΥ sb j Υ, sb j 2 Υ 2, sds which equals to J 3 t t ; where the second line follows by a similar argument used in bounding a, t 2. Net, to bound a 2 t,, 2, we need to estimate the following three terms 2 2, GΥ s B j Υ, s B j Υ, s B j 2 Υ 2, sds imsart-aos ver. 24//6 file: Filaments_AOS_supp_9Mar28.te date: March 9, 28

9 2, GΥ s B j 2 Υ 2, s B j 2 Υ 2, s B j Υ, sds GΥ 2. s GΥ s B j Υ, sb j 2 Υ 2, sds This can be done by similar argument used for the case r = 2, and we omit the details. References. [] Yoo, W. W. and Ghosal, S. 26. Supremum norm posterior contraction and credible sets for nonparametric multivariate regression. The Annals of Statistics Department of Statistics North Carolina State University Raleigh, North Carolina USA wli35@ncsu.edu Department of Statistics North Carolina State University Raleigh, North Carolina USA sghosal@stat.ncsu.edu imsart-aos ver. 24//6 file: Filaments_AOS_supp_9Mar28.te date: March 9, 28