added the likelihood exp

statinf.tex

@@ -14,7 +14,11 @@ is the evidence of the model $\mathcal{M}$. Bayesian model selection involves co
 
 
 \subsubsection{Problems}
 \subsubsection{Problems}
 \begin{enumerate}
 \begin{enumerate}
-\item Compute the posterior distribution $p(H_0 | d, \mathcal{M}_1)$ of the Hubble constant $H_0$ using the Supernova Cosmology project data $z \leq 0.1$. Assume the Hubble's law [Eq.~\eqref{eq:Hubble_law}] as the model $\mathcal{M}_1$. Assume uniform prior for $H_0$ in the interval $(10, 100)$ km/s/Mpc. 
+\item Compute the posterior distribution $p(H_0 | d, \mathcal{M}_1)$ of the Hubble constant $H_0$ using the Supernova Cosmology project data $z \leq 0.1$. Assume the Hubble's law [Eq.~\eqref{eq:Hubble_law}] as the model $\mathcal{M}_1$. Assume uniform prior for $H_0$ in the interval $(10, 100)$ km/s/Mpc. You can use a Gaussian likelihood with $\sigma = 1$.  That is, 
+\begin{equation}
+p(d | H_0, \mathcal{M}_1) = \exp \left( - \sum_i \big |d_i - d_L(z_i, \mathcal{M}_1) \big |^2 \right), 
+\end{equation}
+where $d_i$ and $z_i$ are the samples of the luminosity distance and redshift from the data and $d_L(z_i, \mathcal{M}_1)$ is the relation between luminosity distance and redshift predicted by model $\mathcal{M}_1$ evaluated at redshift $z_i$. 
 \item Repeat the analysis using the full data set. What are the differences that you see in the posterior?
 \item Repeat the analysis using the full data set. What are the differences that you see in the posterior?
 \item Compute the posterior distribution  $p(H_0, \Omega_M | d, \mathcal{M}_2)$ of the Hubble constant $H_0$ and matter density $\Omega_M$ using the $\Lambda$CDM model $\mathcal{M}_2$ [Eq.~\eqref{eq:lcdm}]. You can compute the posterior on a 2-dimensional grid. Assume uniform priors for $H_0$ in the interval $(10, 100)$ km/s/Mpc and for $\Omega_M$ in the interval (0, 1). 
 \item Compute the posterior distribution  $p(H_0, \Omega_M | d, \mathcal{M}_2)$ of the Hubble constant $H_0$ and matter density $\Omega_M$ using the $\Lambda$CDM model $\mathcal{M}_2$ [Eq.~\eqref{eq:lcdm}]. You can compute the posterior on a 2-dimensional grid. Assume uniform priors for $H_0$ in the interval $(10, 100)$ km/s/Mpc and for $\Omega_M$ in the interval (0, 1). 
 \item Compute the likelihood ratio (Bayes factor) between the Hubble's law and $\Lambda$CDM model by computing the evidences [Eq.~\eqref{eq:evidence}] of the two models using a  numerical integration method that we learned in Sec.~\ref{sec:integr}. 
 \item Compute the likelihood ratio (Bayes factor) between the Hubble's law and $\Lambda$CDM model by computing the evidences [Eq.~\eqref{eq:evidence}] of the two models using a  numerical integration method that we learned in Sec.~\ref{sec:integr}.