Notes for "Posterior consistency

作者: jjx323 | 来源:发表于2020-02-10 15:30 被阅读0次

Notes for "Posterior consistency
Posterior 前与后
likelihood and posterior
Posterior distribution given pri
贝叶斯决策理论
Dorsal 后背
Memory Consistency
Persistency & Consistency
consistency and repliccation
DDIA Ch9

H. Kekkonen, M. Lassas, S. Siltanen, Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators, Inverse Problems, 32, 2016, 085005

Page 10, line 6, Section 3 Generalised random variables

The connection between $B_V$ and $C_V$ is $B_V = C_V (I-\Delta)^s : H^{s}\rightarrow H^{s}$

Notes:
Consider $\phi, \psi\in H^s(N)$ , then we have
$\langle C_V\phi, \psi \rangle_{H^{s}\times H^{-s}} = \langle \psi, C_V^* \psi\rangle_{H^{s}\times H^{-s}},$
where $C_V^* : H^{s}\rightarrow H^{-s}$ . Since $(B_{V}\phi, \psi)_{H^s} = (\phi, B_V \psi)_{H^s}$ and $(B_{V}\phi, \psi)_{H^s} = \langle C_V\phi, \psi \rangle_{H^{s}\times H^{-s}}$ , we obtain
$\int_{N}(I-\Delta)^{s/2}\phi(x) (I-\Delta)^{s/2}B_V\psi(x) ds = \int_{N}\phi(x)C_V^{*} \psi(x)dx.$
Hence, we find that
$(I-\Delta)^{s}B_V = C_V^{*} : H^{s}\rightarrow H^{-s},$
which implies
$C_V = B_V (I-\Delta)^{s}.$

Page 11, line (from end) 7, Section 3.2

Condition $B_U \in \mathfrak{C}^{1}(H^{\tau})$ guarantees that $\mathbb{E}(U, U)_{H^{\tau}} < \infty$ .

Notes:
Taking $(\lambda_j, \{\varphi_{j}\})_{j=1}^{\infty}$ be eigensystem of $B_U$ on $H^{\tau}$ , we have
$\begin{align} \mathbb{E}(U,U)_{H^{\tau}} & = \mathbb{E}\sum_{j=1}^{\infty}(U,\varphi_{j})(U,\varphi_{j})=\sum_{j=1}^{\infty}\mathbb{E}{((U,\varphi_{j})_{H^{\tau}}(U,\varphi_{j})_{H^{\tau}})} \\ & = \sum_{j=1}^{\infty}(B_U \varphi_{j}, \varphi_{j})_{H^{\tau}} = \sum_{j=1}^{\infty}\lambda_{j} < \infty, \end{align}$
which is just the required estimation.
Here, in this part, we may see $C_U : H^{-\tau} \rightarrow H^{\tau}$ ( $\tau\leq r$ ) and $B_U : H^{\tau} \rightarrow H^{\tau}$ , which coincides with formula (3.3) and (3.4) on page 10. Taking $\nu=\nu_k$ in the first formula on page 12, we have
$k = N(\nu_k) \sim cv_k^{\frac{d}{2(r-\tau)}}(1+O(\nu_k^{-\frac{1}{2(r-\tau)}})).$
Question: The operator $B_U^{-1}$ is a self-adjoint elliptic operator with smooth coefficients (defined on closed manifold), the eigenvalues are irrelevant to the definition function space of the operator $B_U^{-1}$ ?

Proof of Lemma 3, Page 15

$\ldots$ and we can write
$B_0 = B_0 ((A^* A)B_1 - K_2) = B_1 - B_0 K_2.$

Notes:
Here, we assume that $t, t_0 > 0$ . By my understanding, the equality means that for $f\in H^{r+2t_0}\subset H^{r}$ we have
$B_0(f) = B_0((A^* A)B_1)(f) - B_0 K_2(f) = B_0 A^* A B_1(f) - B_0 K_2 (f) .$
Since $B_1(f) \in H^{r}$ , we obtain $B_0 A^* A B_1(f) = B_1 (f)$ and
$B_0(f) = B_1(f) + B_0K_2(f)$
holds for every $f\in H^{r+2t_0}$ . Because for any $r\in \mathbb{R}$ , we can deduce that the corresponding $B_0 : C^{\infty} \rightarrow C^{\infty}$ is continuous. Hence, we find that
$B = B_1 \, \text{mod} \, \Psi^{-\infty}$
holds on $H^{r}(N)$ for any $r\in \mathbb{R}$ . That is to say,
$B = B_1 \, \text{mod} \, \Psi^{-\infty}$
holds on space $\mathcal{D}'$ , which implies $B\in H\Psi^{2t_0, 2t}$ .

2020/02

Notes for "Posterior consistency
H. Kekkonen, M. Lassas, S. Siltanen, Posterior consistenc...
Posterior 前与后
Posterior. This describes something that is closer to the...
likelihood and posterior
https://mml-book.github.io/[https://mml-book.github.io/] ...
Posterior distribution given pri
The posterior distribution of parameter \theta given obse...
贝叶斯决策理论
Bayesian Bayes's Theorem prior: likelihood: posterior: Op...
Dorsal 后背
Dorsal. This term can be used instead of posterior and me...
Memory Consistency
Memory Consistency is talking about how to deal with prob...
Persistency & Consistency
The key words for today are these two, which seem are sim...
consistency and repliccation
分布式系统中最重要问题是数据的复制，其问题之一是保持副本的一致性。好处：通过应用复制和缓存来缓解可伸缩性问题坏处：...
DDIA Ch9
Consistency Guarantees 主要还是要了解更强的 consistency models， onc...