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}
 \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 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}.