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# 数学代写|统计计算作业代写Statistical Computing代考|Approximate Bayesian Computation

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## 数学代写|统计计算作业代写Statistical Computing代考|Basic Approximate Bayesian Computation

In this section we describe a basic version of the $\mathrm{ABC}$ method. We start the presentation by describing the method as an algorithm, and then give the required explanations to understand why this algorithm gives the desired result.
Algorithm 5.1 (basic Approximate Bayesian Computation)
input:
data $x^{} \in \mathbb{R}^{n}$ the prior density $\pi$ for the unknown parameter $\theta \in \mathbb{R}^{p}$ a summary statistic $S: \mathbb{R}^{n} \rightarrow \mathbb{R}^{q}$ an approximation parameter $\delta>0$ randomness used: samples $\theta_{j} \sim p_{\theta}$ and $X_{j} \sim p_{X \mid \theta}\left(\cdot \mid \theta_{j}\right)$ for $j \in \mathbb{N}$ output: $\theta_{j_{1}}, \theta_{j_{2}}, \ldots$ approximately distributed with density $p_{\theta \mid X}\left(\theta \mid x^{}\right)$
1: $s^{} \leftarrow S\left(x^{}\right)$
2: for $j=1,2,3, \ldots$ do
3: $\quad$ sample $\theta_{j} \sim p_{\theta}(\cdot)$
4: $\quad$ sample $X_{j} \sim p_{X \mid \theta}\left(\cdot \mid \theta_{j}\right)$
5: $\quad S_{j} \leftarrow S\left(X_{j}\right)$
6: if $\left|S_{j}-s^{*}\right| \leq \delta$ then
7: output $\theta_{j}$
8: end if
9: end for

In the algorithm, the summary statistic $S$ is assumed to take values in $\mathbb{R}^{q}$. The dimension $q$ is typically much smaller than the dimension $n$ of the data, and often $q$ equals the number $p$ of parameters. The distance $\left|S_{j}-s^{}\right|$ in line 6 of the algorithm is the Euclidean norm in $\mathbb{R}^{q}$. Since the algorithm considers the summary statistic $s^{}=S\left(x^{*}\right)$ instead of the full data, the method can only be expected to work if $S(\theta)$ contains ‘enough’ information about $\theta$. The optimal case for this is if $S$ is a sufficient statistic, as described in the following definition.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Approximate Bayesian Computation with regression

The basic ABC method as described in the previous section can be computationally very expensive. Many variants of $\mathrm{ABC}$, aiming to reduce the computational cost, are used in application areas. In this section we describe one approach to constructing such improved variants of ABC. This approach is based on the idea of accepting a larger proportion of the samples and then to numerically compensate for the systematic error introduced by the discrepancy between the sampled values $s_{j}=S\left(X_{j}\right)$ and the observed value $s^{}=S\left(x^{}\right)$.

The method discussed here is based on the assumption that the samples $\theta_{j}$ can be written as
$$\theta_{j} \approx f\left(S_{j}\right)+\varepsilon_{j}$$
for all accepted $j$, where $f(s)=\mathbb{E}(\theta \mid S=s)$ and the $\varepsilon_{j}$ are independent of each other and of the $S_{j}$. If this relation holds at least approximately, we can use the modified samples
\begin{aligned} \tilde{\theta}{j} &=f\left(s^{}\right)+\varepsilon{j} \ &=f\left(S_{j}\right)+\varepsilon_{j}+f\left(s^{ }\right)-f\left(S_{j}\right) \ &=\theta_{j}+f\left(s^{}\right)-f\left(S_{j}\right) \end{aligned} instead of $\theta_{j}$ in order to transform samples corresponding to $S=S_{j}$ into samples corresponding to the required value $S=s^{}$. This idea is made more rigorous by the following result.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|BASIC APPROXIMATE BAYESIAN COMPUTATION

data $x^{} \in \mathbb{R}^{n}$ the prior density $\pi$ for the unknown parameter $\theta \in \mathbb{R}^{p}$ a summary statistic $S: \mathbb{R}^{n} \rightarrow \mathbb{R}^{q}$ an approximation parameter $\delta>0$ randomness used: samples $\theta_{j} \sim p_{\theta}$ and $X_{j} \sim p_{X \mid \theta}\left(\cdot \mid \theta_{j}\right)$ for $j \in \mathbb{N}$ output: $\theta_{j_{1}}, \theta_{j_{2}}, \ldots$ approximately distributed with density $p_{\theta \mid X}\left(\theta \mid x^{}\right)$
1: $s^{} \leftarrow S\left(x^{}\right)$
2: for $j=1,2,3, \ldots$ do
3: $\quad$ sample $\theta_{j} \sim p_{\theta}(\cdot)$
4: $\quad$ sample $X_{j} \sim p_{X \mid \theta}\left(\cdot \mid \theta_{j}\right)$
5: $\quad S_{j} \leftarrow S\left(X_{j}\right)$
6: if $\left|S_{j}-s^{*}\right| \leq \delta$ then
output $\theta_{i}$\text {

7: } \quad \text { output } \theta_{j}

8: 结束 if
9: 结束

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|APPROXIMATE BAYESIAN COMPUTATION WITH REGRESSION

θj≈F(小号j)+ej

\begin{aligned} \tilde{\theta} {j} &=f\lefts^{}\右s^{}\右+\varepsilon{j} \ &=f\leftS_{j}\右S_{j}\右+\varepsilon_{j}+f\left(s^{ }\right)-f\leftS_{j}\右S_{j}\右\ &=\theta_{j}+f\left(s^{ }\right)-f\leftS_{j}\右S_{j}\右\end{aligned} 而不是θj为了变换对应的样本小号=小号j成对应于所需值 $S=s^{ }$ 的样本。下面的结果使这个想法更加严格。