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\usetheme{Madrid}
\author{Aleksandr Petrosyan}
\date{\today}
\title{Cosmological Parameter estimation using Bayesian accelerated Machine learning.}
\hypersetup{
 pdfauthor={Aleksandr Petrosyan},
 pdftitle={Cosmological Parameter estimation using Bayesian accelerated Machine learning.},
 pdfkeywords={},
 pdfsubject={},
 pdfcreator={Emacs 26.3 (Org mode 9.1.9)}, 
 pdflang={English}}
\begin{document}

\maketitle

\begin{frame}[label={sec:orge6b6c72}]{What is \(\Lambda\)CDM.}
It's the accepted standard model of Cosmology. 
Has the following main parameters:
\begin{itemize}
\item \(\Omega_{b}h^{2}\) --- Physical baryon density.
\item \(\Omega_{c}h^{2}\) --- Physical dark matter density.
\item \(\tau_{0}\) --- Age of the universe.
\item \(n_{s}\) --- Scalar spectral index.
\item \(\tau_\text{reio}\) --- Re-ionization optical depth.
\item \(\Delta_{R}^{2}\) --- Curvature fluctuation amplitude.
\end{itemize}
\end{frame}

\begin{frame}[label={sec:orga7c365a}]{How do we measure it?}
Have Planck, WMap and many other sources of data.

Do Bayesian inference. Long discussion what this actually means. 

The gist: 
\begin{itemize}
\item Data using \(\chi^{2}\)defines \({\cal L}\) --- likelihood.
\item Prior information (other experiments) give us \(\pi\) --- the prior.
\item From Bayes' theorem, compute \emph{evidence} \({\cal Z}\) and \emph{posterior} \({\cal P}\).
\item From posterior, \emph{marginalise} the \emph{calibration parameters}.
\end{itemize}
\end{frame}

\begin{frame}[label={sec:orgf5b3dfe}]{Example of posterior.}
  Started with a \alert{Gaussian}. Then Gave an \alert{offset}.
  \(\ln {\cal Z} = -62 \pm 1\) exactly what we expect. Except for
  \alert{one} case.
  \includegraphics[width=.75\linewidth]{./illustrations/convergence.pdf}
  
\end{frame}
\begin{frame}[label={sec:org083fab8}]{How do we do it?}
We are spoiled for choice. But can only use Monte-Carlo in many
dimensions (lots of parameters).

The fastest methods given below in order of stability/speed. 
\begin{block}{Monte-Carlo methods}
\begin{itemize}
\item Nested sampling.
\item Hamiltonian Monte-Carlo.
\item Metropolis-Hastings.
\end{itemize}
\end{block}
\end{frame}


\begin{frame}[label={sec:orgcf084f1}]{Can we make it go fast?}
Yes and no. 

If we give the prior as the posterior, we will converge the
fastest. But the evidence is all wrong. The posterior is also wrong. 

We can mitigate this, by using an \emph{intuitive Gaussian}. But it gets
the wrong answer if we were wrong about either the place or the size
of the Posterior.
\begin{block}{My Intuitive Gaussian.}
Take \(\tilde{\pi}(\theta) = {\cal P}(\theta)\), change \({\cal L}\), such that. 
\begin{equation*}
\tilde{\cal L}(\theta)\tilde{\pi}(\theta) = {\cal L}(\theta)\pi(\theta)
\end{equation*}
\end{block}
\end{frame}



\begin{frame}[label={sec:org958d1e9}]{Can we safely make it go fast? Posterior re-partitioning.}

\begin{block}{Chen-Ferroz-Hobson PPR.}
\begin{equation*}
\tilde{\pi}(\bm{\bm{\theta}};\beta) = \cfrac{\pi(\bm{\theta})^{\beta}}{Z(\beta)\{\pi\}},
\end{equation*}
\begin{equation*}
 Z(\beta)\{\pi\} = \int_{\bm{\theta} \in \Psi}		\pi(\bm{\bm{\theta}})^{\beta}d\bm{\bm{\theta}}.
 \end{equation*}
\begin{equation*}
\tilde{\cal L}(\bm{\theta}) = {\cal L}(\bm{\theta}) Z(\beta)\{\pi\} \cdot		\pi^{1-\beta}(\bm{\theta}).
\end{equation*}
\end{block}
This can mitigate the issues of size, but not place.

\end{frame}



\begin{frame}[label={sec:org73e18f6}]{Works well, but not if we have an offset.}
\begin{center}
\includegraphics[width=.9\linewidth]{./illustrations/convergence.pdf}
\end{center}
\end{frame}

\begin{frame}[label={sec:orgb21b8c0}]{My discovery.}
\begin{block}{My Stochastic superpositional isometric model mixture.}
\begin{equation*}
  \tilde{\pi}(\bm{\theta}; \beta)  \triangleq \begin{cases}
	\tilde{\pi}_{1}(\bm{\theta}) & \text{with probability } \beta_{1},\\
	& \vdots,\\
	\tilde{\pi}_{n}(\bm{\theta}) & \text{with probability } (1- \sum_{i}^{m}\beta_{i}),
	\end{cases}
\end{equation*}
\begin{equation*}
  \tilde{\cal L}(\bm{\theta}; \bm{\beta})  \triangleq
  \begin{cases}
	\tilde{\cal L}_{1}(\bm{\theta}) &  \text{with probability } \beta_{1},\\
		    &\vdots,\\
	\tilde{\cal L}_{m}(\bm{\theta}) & \text{with probability} (1- \sum_{i}^{m}\beta_{i}).
\end{cases}
\end{equation*}
\begin{equation*}
  \tilde{\pi}(\bm{\theta}; \bm{\beta}) = \tilde{\pi}(\bm{\theta})_{i} \Leftrightarrow \tilde{\cal L}(\bm{\theta}; \bm{\beta}) = \tilde{\cal L}_{m}(\bm{\theta}; \bm{\beta}), 
\end{equation*}
\end{block}
\end{frame}

\begin{frame}
  \frametitle{This is what it looks like}
  \input{./illustrations/mixture_2.tex}
\end{frame}

\begin{frame}
  \frametitle{This is what it looks like}
\input{./illustrations/mixture_3.tex}
\end{frame}


\begin{frame}
  \frametitle{This is what it looks like}
  \input{./illustrations/mixture_4.tex}
\end{frame}


\begin{frame}[label={sec:orgcde55a9}]{This is really FAST.}
\begin{figure}
\input{./illustrations/benchmark.tex}
\end{figure}
\end{frame}

\begin{frame}[label={sec:org34912c4}]{And Accurate.}
\begin{figure}
\input{./illustrations/evidence-drift.tex}
\end{figure}
\end{frame}

\begin{frame}[fragile,label={sec:org7a68446}]{What this can do?}
 Examples: 
\begin{itemize}
\item Can accelerate if you know \alert{roughly} what to expect.
\item Can make more robust if you are wrong about the where.
\item Can make more robust if you are unsure about biases.
\item This can allow you to chain models.
\item You can run a \texttt{CosmoChord} on a laptop, \texttt{Cobaya} on a desktop.
\end{itemize}
\end{frame}
\end{document}