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\fancyhead[C]{COMP 136 - 2021s - HW2 Submission}
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{\Large{\bf Student Name: TODO}}

{\Large{\bf Collaboration Statement:}}

Turning in this assignment indicates you have abided by the course Collaboration Policy:

\url{www.cs.tufts.edu/comp/136/2021s/index.html#collaboration-policy}

Total hours spent: TODO

I consulted the following resources:
\begin{itemize}
\item TODO
\item TODO
\item $\ldots$	
\end{itemize}

FYI Official instructions for all problems can be found at:

\url{www.cs.tufts.edu/comp/136/2021s/hw2.html}


\tableofcontents

\newpage
\officialdirections{
\subsection*{1a: Problem Statement}

Consider the estimator $\hat{\sigma}^2$:
\begin{align}
\hat{\sigma}^2(x_1, \ldots x_N) = \frac{1}{N} \sum_{n=1}^N (x_n - \mu_{\text{true}})^2
\end{align}
Compute the expected value of this estimator under the sampling distribution of the observations $x$. You should assume each $x_n$ are drawn i.i.d from $x_n \sim \mathcal{N}( \mu_{\text{true}}, \sigma^2_{\text{true}})$.  
}

\subsection{1a: Solution}

TODO


\newpage
\officialdirections{
\subsection*{1b: Problem Statement}
Given your result above, describe if the estimator is biased or unbiased.
}

\subsection{1b: Solution}

TODO



\officialdirections{
\subsection*{1c: Problem Statement}
How does your answer about bias of this estimator in 1b compare to what we learned about the bias of the maximum likelihood estimator for the variance? Explain why the answers might be the same or different.
}

\subsection{1c: Solution}
TODO




\newpage
\officialdirections{
\subsection*{2a: Problem Statement}

Show that the SGD estimator is an *unbiased* estimator of the whole dataset gradient. That is, show:
\begin{align}
\mathbb{E}_{i \sim U}[ \hat{G}(\theta, x_i) ] = G(\theta, x)
\end{align}
}

\subsection{2a: Solution}
TODO




\newpage
\officialdirections{
\subsection*{2b: Problem Statement}
How will the updated value $\theta_1$ produced by one step of whole dataset gradient descent (which has no randomness) compare to the one-step update $\theta_1$ from SGD? Why is it important that the SGD estimator is unbiased, if we want to find a solution to our original objective?
}

\subsection{2b: Solution}
TODO






\newpage

\newpage
\officialdirections{
\subsection*{3a: Problem Statement}
Given the log PDF below, show that vector $x$ has a multivariate Gaussian distribution.
\begin{align}
\log p(x) = \text{const} - \frac{1}{2} x^T A x + b^T x
\end{align}
where $A$ is an $M \times M$ symmetric and positive definite matrix, and $b$ any vector of size $M$.
}


\subsection{3a: Solution}
TODO




\newpage
\officialdirections{
\subsection*{4a: Problem Statement}
Show that we can write $S_{N+1}^{-1} = S_N^{-1} + vv^T$ for some vector $v$.
}

\subsection{4a: Solution}
TODO

\newpage
\officialdirections{
\subsection*{4b: Problem Statement}
\begin{align}
(A + vv^T)^{-1} &= A^{-1} - \frac{ (A^{-1}v)(v^T A^{-1})}{ 1 + v^T A^{-1} v}
\end{align}
Substitute $A = S_N^{-1}$ and $v$ from 4a, then simplify to write an expression for $S_{N+1}$ in terms of $S_{N}$.
}

\subsection{4b: Solution}
TODO

\newpage
\officialdirections{
\subsection*{4c: Problem Statement}
Show that $\sigma^2_{N+1}(x_*) - \sigma^2_{N}(x_*) = \phi(x_*)^T \left[ S_{N+1} - S_{N} \right] \phi(x_*)$.
}

\subsection{4c: Solution}
TODO


\newpage
\officialdirections{
\subsection*{4d: Problem Statement}
Finally, plug your result from 4b into 4c, plus the fact that $S_N$ must be positive definite, to show that:
\begin{align}
\sigma_{N+1}^2(x_*) \leq \sigma_N^2(x_*), \quad \text{or equivalently} \quad \sigma_{N+1}^2(x_*) - \sigma_N^2(x_*) \leq 0
\end{align}
}

\subsection{4d: Solution}
TODO

\end{document}
