• specify the prior distributions for $\tau$, $\tau_0$, $\sigma^2$, and $\sigma_\eta^2$
• write a short note
• a quick derivation or full conditional posterior presentation
• include your name in the author list

• present estimation of a degrees of freedom parameter of an $N$-variate Student-t distribution modelled as the IG2-scale mixture of normal.
• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

• describe what's the idea for using the Simulation smoother and precision sampler
• provide an R function for the implementation of the precision sampler (a random number generator from a multivariate normal with a band precision matrix) Copy/paste/adjust my code. And describe a bit.
• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

todos

• set up repo
• upload the website
• deploy the website
• invite collaborators
• create issues
• assign issues

Present a model in which the error terms $\epsilon_t$ follow a Student-t distribution

• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

• put a prior on the prior scale of $\sigma^2$ and present its full conditional posterior distribution with an R function executing this sampler. The prior structure should be:
  $\sigma^2|s \sim\mathcal{IG}2(s, \underline{\nu})$
  $s\sim\mathcal{G}(\underline{s},\underline{a})$
• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

Consider an extension of the model from the repo by a zero-mean autoregressive equation for $\epsilon_t$:
$$\epsilon_t = \alpha_1 \epsilon_{t-1} = \dots + \alpha_p \epsilon_{t-p} + e_t$$
where
$$e_t \sim N\left(0, \sigma_\alpha^2\right)$$
With the prior for autoregressive parameter vector $\boldsymbol\alpha = (\alpha_1,\dots,\alpha_p)'$
$$\boldsymbol\alpha\mid\sigma_{\alpha}^2 \sim N_p(0_p,\sigma_{\alpha}^2I_p)$$
and
$$\sigma_\alpha^2 \sim IG2 (s_\alpha, \nu_\alpha)$$

• present a Bayesian algorithm for the estimation of autoregressive parameters and their shrinkage level
• write a short note
• a quick derivation or full conditional posterior distributions for $\boldsymbol\alpha$ and $\sigma_\alpha^2$
• an R function with the sampler
• include your name in the author list

• put a prior on the prior scale of $\sigma^2$ and present its full conditional posterior distribution with an R function executing this sampler. The prior structure should be:
  $\sigma^2|s \sim\mathcal{IG}2(s, \underline{\nu})$
  $s\sim\mathcal{IG}2(\underline{s},\underline{\mu})$
• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

• provide an analytical solution for a joint posterior for $\tau$ and $\sigma^2$ for a simplified model in which $\tau_0 = 0$ and $\sigma^2_\eta = c\sigma^2$ where $c$ is a constant signal-to-noise ratio
• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

• implement a Student-t prior for the trend component $\tau$ that in ths baseline model is given by a multivariate normal:

$\tau|\sigma^2_{\eta}\sim\mathcal{N}_T$

$(\mathbf{0}, \sigma^2_{\eta}(\mathbf{H}'\mathbf{H})^{-1})$

• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

• implement a Laplace prior for the trend component $\tau$ that in the baseline model is given by a multivariate normal:

$\tau|\sigma^2_{\eta}\sim\mathcal{N}_T$

$(\mathbf{0}, \sigma^2_{\eta}(\mathbf{H}'\mathbf{H})^{-1})$

• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

• put a prior on the prior scale of $\tau_0$ and present its full conditional posterior distribution with an R function executing this sampler. The prior structure should be:
  $\tau|s \sim\mathcal{IG}2(\underline{\tau}, s^2)$
  $s^2\sim\mathcal{IG}2(\underline{s},\underline{\nu})$
• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

• present a model in which the state-space equation includes a drift that changes over time. You choose the form of the regime change 😄

$$\tau_t=\mu_t + \tau_{t-1}+\eta_t$$

• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list

• present a model with the autoregressive cycle component

$$\epsilon_{t}=\alpha_1\epsilon_{t-1}+\dots+\alpha_p\epsilon_{t-p}+e_t$$

$$e_t\sim\mathcal{N}(0\sigma_e^2)$$

• also estimate the initial conditions $\boldsymbol\epsilon_0 = \epsilon_0, \dots, \epsilon_{-p+1}$ following the normal prior
  $$\boldsymbol\epsilon_0 \sim\mathcal{N}_p(\mathbf{0}_p,\sigma_0^2I_0)$$
• write a short note
• a quick derivation or full conditional posteriors for the $T$-vector $\boldsymbol\epsilon$, $p$-vectors $\boldsymbol\epsilon_0$
• an R function with the sampler
• include your name in the author list

• present the extension of the full conditional posterior distributions for the parameters of the baseline model given that the shocks $\epsilon_t$ is heteroskedastic with conditional variances $\sigma_t^2$
• write a short note
• a quick derivation or full conditional posterior for $T$-vector $\tau$
• an R function with the sampler for full conditional posterior for $\tau$
• include your name in the author list

• present very concisely Bayesian forecasting with the local-level model
• write a short note
• one-period ahead predictive density presentation
• an R function with the sampler
• include your name in the author list

Cover sections (modify the titles if you will)

Predictive density

Sampling from the predictive density

• specify the full conditional posterior distributions for $\tau$, $\tau_0$, $\sigma^2$, and $\sigma^2_\eta$

Jimmy is about to join this task.

• write a short note
• a quick derivation or full conditional posterior presentation
• an R function with the sampler
• include your name in the author list