Consider the Bayesian model for (tau=left{x_{1}, ldots, x_{n} ight}) with likelihood (g(tau mid mu)) such that (left(X_{1},

Consider the Bayesian model for \(\tau=\left\{x_{1}, \ldots, x_{n}\right\}\) with likelihood \(g(\tau \mid \mu)\) such that \(\left(X_{1}, \ldots, X_{n} \mid \mu\right) \sim_{\text {idd }} \mathscr{N}(\mu, 1)\) and prior pdf \(g(\mu)\) such that \(\mu \sim \mathscr{N}(u, 1)\) for some hyperparameter \(v\). Define a sequence of densities \(w_{t}(\mu), t \geqslant 2\) via \(w_{t}(\mu) \propto w_{t-1}(\mu) g(\tau \mid \mu)\), starting with \(w_{1}(\mu)=g(\mu)\). Let \(a_{t}\) and \(b_{t}\) denote the mean and precision \({ }^{4}\) of \(\mu\) under the posterior \(g_{t}(\mu \mid \tau) \propto g(\tau \mid \mu) w_{t}(\mu)\). Show that \(g_{t}(\mu \mid \tau)\) is a normal density with precision \(b_{t}=\) \(b_{t-1}+n, b_{0}=1\) and mean \(a_{t}=\left(1-\gamma_{t}\right) a_{t-1}+\gamma_{t} \bar{x}_{n}, a_{0}=v\), where \(\gamma_{t}:=n /\left(b_{t-1}+n\right)\). Hence, deduce that \(g_{t}(\mu \mid \tau)\) converges to a degenerate density with a point-mass at \(\bar{x}_{n}\).