Bayesian updating with conjugate prior (specific example)

This question deals with Bayesian updating with conjugate prior.Suppose we have a prior distribution of N~(5, 3) and then we observe 5 data points (8, 9, 10, 8, 7) (assumed to be taken randomly from a N~(9, 3) distribution). What would be the posterior after these observations in the form of N~(x, y)? I read the Wikipedia article on conjugate priors, but I want to have a more precise understanding of how to solve this specific problem. If there is no way to solve it without assuming some things, can you please explain what needs to be known and solve under very simple assumptions? Thank you in advance.

• Where is the likelihood? and what are the parameters? – niandra82 Sep 27 '16 at 9:34
• If the information is all that is given to work with, is there any way to approximate the posterior? – statstudent Sep 27 '16 at 10:44
• @Kelly if you know that the data comes from N(9, 3) distribution, then there is no need to estimate anything since you know the parameters in advance (9, 3). What niandra82 was saying is that you need to specify priors and likelihood, what you did is you only specified prior for mean. – Tim Sep 27 '16 at 11:01
• When you say N~(a,b) presumably that indicates a normal distribution with mean a ... but is b the variance or the standard deviation in this notation? – Glen_b Sep 27 '16 at 11:15
• @Kelly if you do not know the distribution of your data (i.e. you are not willing to assume it), then you cannot use classical Bayesian inference since for this you would need to assume a likelihood function (i.e. distribution) for your data. If you do not assume distribution for your data then what parameters you want to estimate? N(a,b) is prior for what? – Tim Sep 27 '16 at 11:25

$$\underbrace{p(\theta \mid X)}_\text{posterior} = \frac{\overbrace{p(X \mid \theta)}^\text{likelihood} \, \overbrace{p(\theta)}^\text{prior}}{\underbrace{p(X)}_\text{normalizing constant}}$$
To make it more concrete, you can estimate $\mu$ and $\sigma^2$ parameters from normal distribution (i.e. normal likelihood) using data $X$, assuming normal prior for $\mu$ with hyperparameters $\mu_0$ and $\sigma^2_0$, and uniform prior for $\sigma^2$ with hyperparameters $a$ and $b$, to obtain posterior distributions for $\mu$ and $\sigma^2$:
$$X \sim \mathrm{Normal}(\mu, \sigma^2) \\ \mu \sim \mathrm{Normal}(\mu_0, \sigma^2_0) \\ \sigma^2 \sim \mathrm{Uniform}(a,b)$$