# 贝叶斯统计推断|STA 145/STAT 625/ECON 7960/STAT 6574/Bayesian Statistical Inference代写

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$$y_{t} \sim \operatorname{Normal}\left(\log \left(q K P_{t}\right), \sigma^{2}\right), \quad t=1, \ldots, n=23,$$
where $q, K$, and $P_{t}$ denote the “catchability parameter,” “carrying capacity” of the environment, and biomass in year $t$ expressed as a proportion of the carrying capacity, respectively. In addition, the biomass dynamics are given by
$$\log P_{t}=f\left(P_{t-1}\right)+u_{t}, \quad f\left(P_{t-1}\right)=\log \left[P_{t-1}+r P_{t-1}\left(1-P_{t-1}\right)-\frac{C_{t-1}}{K}\right],$$
where $u_{t} \sim \operatorname{Normal}\left(0, \omega^{2}\right), r$ is the “intrinsic growth rate,” and $C_{t-1}$ denotes the total catch, in kilotonnes, during year $t-1$. To avoid the issues discussed above, we will express this, instead, as
$$\log P_{t} \sim \operatorname{Normal}\left(f\left(P_{t-1}\right), \omega^{2}\right), \quad t=1, \ldots, n .$$

## STAT 770/BIOS 805/BIOSTAT695/PSY 525/625/SOCI612/STA 4504/STA 517 COURSE NOTES ：

$$\psi_{i j}=\frac{D}{V_{i}} \exp \left(-\frac{C L_{i}}{V_{i}} t_{i j}\right)$$
This is the unique solution to the following differential equation and initial condition, at time $t=t_{i j}$ :
$$\frac{d C(t)}{d t}=-\frac{C L_{i}}{V_{i}} C(t), \quad C(t=0)=\frac{D}{V_{i}}$$

# 贝叶斯统计推断 | Bayesian Statistical Inference 代写 ISYE 6420代考

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Deviance Information Criterion
$$\mathrm{DIC}=D(\bar{\theta})+2 p_{D}$$
which is based on $p(y \mid \theta)$ and is trivial to compute.
Akaike Information Criterion
$$\mathrm{AIC}=-2 \log p(y \mid \hat{\phi})+2 p_{\phi}$$

where $p_{\phi}$ is the number of hyperparameters. This relies on being able to integrate out the $\theta$ ‘s to give
$$p(y \mid \phi)=\int_{\Theta} p(y \mid \theta) p(\theta \mid \phi) d \theta .$$

## ISYE 6420COURSE NOTES ：

$$y_{t}=c+\sum_{i=1}^{q} \phi_{i} \epsilon_{t-i}+\epsilon_{t}, \quad t=q+1, \ldots, n .$$
This is denoted $\operatorname{MA}(q)$, and the early time points are handled in the same way as for the $\mathrm{AR}$ model. $\operatorname{AR}(p)$ and $\operatorname{MA}(q)$ may be combined to form an $\operatorname{ARMA}(p, q)$ model:
$$y_{t}=c+\sum_{i=1}^{p} \theta_{i} y_{t-i}+\sum_{i=1}^{q} \phi_{i} \epsilon_{t-i}+\epsilon_{t}$$