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Let $x_1,...,x_k$ distributed from $\mathcal{N}(\mu,\ 1/\tau)\ (i.i.d)$ and priors on mean and variance are in form: $$\mu|\tau \sim \mathcal{N}(\mu_0,\frac{1}{\tau}) $$ $$\tau \sim Ga(\alpha,\beta) $$ Now how calculate posterior distribution? The formal Bayesian inference says $$P(\mu|x)\propto P(\mu)f_{\mu}(x)$$ but here we have distribution $\mu|\tau$ so I guess the inference should be based on $\mu|\tau$, i.e. $$P(\mu|x,\tau)\propto P(\mu,\tau)f_{\mu,\tau}(x)$$ but I'm not sure about this and how to continue.

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You are correct in your reasoning. Your full likelihood, with some abuse of notation, is $P(x, \mu, \tau| \mu_0, \alpha, \beta) = \prod_{i=1}^{k}f(x_i|\mu,\tau)P(\mu|\mu_0, \tau)P(\tau|\alpha, \beta)$, so you can get your posterior distributions, \begin{align*} P(\mu|\mu_0, x,\tau) & \propto \prod_{i=1}^{k}f(x_i|\mu,\tau)P(\mu|\tau) \\ P(\tau|x,\alpha, \beta) & \propto \prod_{i=1}^{k}f(x_i|\mu,\tau)P(\mu|\mu_0, \tau)P(\tau|\alpha, \beta). \end{align*}

If you write out the full equations, solve for $\mu$ and $\tau$ in their respective posteriors, you will get your answer.

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