class: center, middle, inverse, title-slide # STA 360/602L: Module 3.4 ## The normal model: conditional inference for the mean ### Dr. Olanrewaju Michael Akande --- ## Normal model - Suppose we have independent observations `\(Y = (y_1,y_2,\ldots,y_n)\)`, where each `\(y_i \sim \mathcal{N}(\mu, \sigma^2)\)` or `\(y_i \sim \mathcal{N}(\mu, \tau^{-1})\)`, with unknown parameters `\(\mu\)` and `\(\sigma^2\)` (or `\(\tau\)`). -- - Then, the likelihood is .block[ .small[ $$ `\begin{split} P(Y| \mu,\sigma^2) & = \prod_{i=1}^n \dfrac{1}{\sqrt{2\pi}} \tau^{\frac{1}{2}} \ \textrm{exp}\left\{-\frac{1}{2} \tau (y_i-\mu)^2\right\}\\ & \propto \tau^{\frac{n}{2}} \ \textrm{exp}\left\{-\frac{1}{2} \tau \sum_{i=1}^n (y_i-\mu)^2\right\}\\ & \propto \tau^{\frac{n}{2}} \ \textrm{exp}\left\{-\frac{1}{2} \tau \sum_{i=1}^n \left[ (y_i-\bar{y}) - (\mu - \bar{y}) \right]^2 \right\}\\ \\ & \propto \tau^{\frac{n}{2}} \ \textrm{exp}\left\{-\frac{1}{2} \tau \left[ \sum_{i=1}^n (y_i-\bar{y})^2 + \sum_{i=1}^n(\mu - \bar{y})^2 \right] \right\}\\ & \propto \tau^{\frac{n}{2}} \ \textrm{exp}\left\{-\frac{1}{2} \tau \left[ \sum_{i=1}^n (y_i-\bar{y})^2 + n(\mu - \bar{y})^2 \right] \right\}\\ & \propto \tau^{\frac{n}{2}} \ \textrm{exp}\left\{-\frac{1}{2} \tau s^2(n-1) \right\} \ \textrm{exp}\left\{-\frac{1}{2} \tau n(\mu - \bar{y})^2 \right\}.\\ \end{split}` $$ ] ] --- ## Likelihood for normal model - Likelihood: .block[ .large[ `$$P(Y| \mu,\sigma^2) \propto \tau^{\frac{n}{2}} \ \textrm{exp}\left\{-\frac{1}{2} \tau s^2(n-1) \right\} \ \textrm{exp}\left\{-\frac{1}{2} \tau n(\mu - \bar{y})^2 \right\},$$` ] ] where + `\(\bar{y} =\sum_{i=1}^n y_i\)` is the sample mean; and + `\(s^2 = \sum_{i=1}^n (y_i-\bar{y})^2/(n-1)\)` is the sample variance. - Sufficient statistics: + Sample mean `\(\bar{y}\)`; and + Sample sum of squares `\(SS = s^2(n-1) = \sum_{i=1}^n (y_i-\bar{y})^2\)`. -- - MLEs: + `\(\hat{\mu} = \bar{y}\)`. + `\(\hat{\tau} = n/SS\)`, and `\(\hat{\sigma}^2 = SS/n\)`. --- ## Inference for mean, conditional on variance - We can break down inference problem for this two-parameter model into two one-parameter problems. -- - First start by developing inference on `\(\mu\)` when `\(\sigma^2\)` is known. Turns out we can use a conjugate prior for `\(\pi(\mu|\sigma^2)\)`. We will get to unknown `\(\sigma^2\)` in the next module. -- - For `\(\sigma^2\)` known, the normal likelihood further simplifies to .block[ .small[ `$$\propto \ \textrm{exp}\left\{-\frac{1}{2} \tau n(\mu - \bar{y})^2 \right\},$$` ] ] leaving out everything else that does not depend on `\(\mu\)`. -- - For `\(\pi(\mu|\sigma^2)\)`, we consider `\(\mathcal{N}(\mu_0, \sigma_0^2)\)`, i.e., `\(\mathcal{N}(\mu_0, \tau_0^{-1})\)`, where `\(\tau_0^{-1} = \sigma_0^2\)`. -- - Let's derive the posterior `\(\pi(\mu|Y,\sigma^2)\)`. --- ## Inference for mean, conditional on variance - First, the prior `\(\pi(\mu|\sigma^2) = \mathcal{N}(\mu_0, \tau_0^{-1})\)` can be written as .block[ .small[ $$ `\begin{split} \Rightarrow \pi(\mu|\sigma^2) \ & = \ \dfrac{1}{\sqrt{2\pi}} \tau_0^{\frac{1}{2}} \cdot \textrm{exp}\left\{-\frac{1}{2} \tau_0 (\mu-\mu_0)^2) \right\} \\ \\ & \propto \ \textrm{exp}\left\{-\frac{1}{2} \tau_0 (\mu^2 - 2\mu\mu_0 + \mu_0^2) \right\} \\ \\ & \propto \ \textrm{exp}\left\{-\frac{1}{2} \tau_0 (\mu^2 - 2\mu\mu_0) \right\}.\\ \end{split}` $$ ] ] -- - **When the normal density is written in this form, note the following details in the exponent.** + First, we must have `\(\mu^2 - 2\mu\)`, and whatever term we see multiplying `\(2\mu\)` must be the mean, in this case, `\(\mu_0\)`. + Second, the precision `\(\tau_0\)` is outside the parenthensis. --- ## Inference for mean, conditional on variance - Now to the posterior: .block[ .small[ `$$\pi(\mu|Y,\sigma^2) \ \propto \ \pi(\mu|\sigma^2) P(Y| \mu,\sigma^2) \ \propto \ \textrm{exp}\left\{-\frac{1}{2} \tau_0 (\mu - \mu_0)^2 \right\}\ \textrm{exp}\left\{-\frac{1}{2} \tau n(\mu - \bar{y})^2 \right\}$$` ] ] -- - Expanding out squared terms .block[ .small[ `$$\Rightarrow \pi(\mu|Y,\sigma^2) \ \propto \ \textrm{exp}\left\{-\frac{1}{2} \tau_0 (\mu^2 - 2\mu\mu_0 + \mu_0^2) \right\}\ \textrm{exp}\left\{-\frac{1}{2} \tau n(\mu^2 - 2\mu\bar{y} + \bar{y}^2) \right\}$$` ] ] -- - Ignoring terms not containing `\(\mu\)` .block[ .small[ $$ `\begin{split} \Rightarrow \pi(\mu|Y,\sigma^2) \ & \propto \ \textrm{exp}\left\{-\frac{1}{2} \tau_0 (\mu^2 - 2\mu\mu_0) \right\}\ \textrm{exp}\left\{-\frac{1}{2} \tau n(\mu^2 - 2\mu\bar{y}) \right\}\\ \\ & = \ \textrm{exp}\left\{-\frac{1}{2} \left[\tau_0 (\mu^2 - 2\mu\mu_0) + \tau n(\mu^2 - 2\mu\bar{y}) \right] \right\}\\ \\ & = \ \textrm{exp}\left\{-\frac{1}{2} \left[ \mu^2(\tau n + \tau_0) - 2\mu(\tau n\bar{y} + \tau_0\mu_0) \right] \right\}.\\ \end{split}` $$ ] ] --- ## Inference for mean, conditional on variance - This sort of looks like a normal kernel but we need to do a bit more work to get there. -- - Particularly, we need to have it be of the form `\(b(\mu^2 - 2\mu a)\)`, so that we have `\(a\)` as the mean and `\(b\)` as the precision. -- - We have .block[ .small[ $$ `\begin{split} \pi(\mu|Y,\sigma^2) \ & \propto \ \textrm{exp}\left\{-\frac{1}{2} \left[ \mu^2(\tau n + \tau_0) - 2\mu(\tau n\bar{y} + \tau_0\mu_0) \right] \right\}\\ \\ & = \ \textrm{exp}\left\{-\frac{1}{2} \cdot (\tau n + \tau_0) \left[ \mu^2 - 2\mu \left( \frac{\tau n\bar{y} + \tau_0\mu_0}{\tau n + \tau_0} \right) \right] \right\}.\\ \end{split}` $$ ] ] which now looks like the kernel of a normal distribution. --- ## Posterior with precision terms - Again, the posterior is .block[ $$ `\begin{split} \pi(\mu|Y,\sigma^2) \ & \propto \ \textrm{exp}\left\{-\frac{1}{2} \cdot (\tau n + \tau_0) \left[ \mu^2 - 2\mu \left( \frac{\tau n\bar{y} + \tau_0\mu_0}{\tau n + \tau_0} \right) \right] \right\}.\\ \end{split}` $$ ] -- - So, in terms of precision, we have .block[ `$$\mu|Y,\sigma^2 \sim \mathcal{N}(\mu_n, \tau_n^{-1})$$` ] where .block[ `$$\mu_n = \dfrac{\tau n\bar{y} + \tau_0\mu_0}{\tau n + \tau_0}$$` ] and .block[ `$$\tau_n = \tau n + \tau_0.$$` ] --- ## Posterior with precision terms - As mentioned before, Bayesians often prefer to talk about precision instead of variance. -- - We have + `\(\tau\)` as the sampling precision (how close the `\(y_i\)`'s are to `\(\mu\)`). + `\(\tau_0\)` as the prior precision (our prior belief about the uncertainty about `\(\mu\)` around our prior guess `\(\mu_0\)`). + `\(\tau_n\)` as the posterior precision -- - From the posterior, we can see that, _the posterior precision equals the prior precision plus the data precision_. -- - That is, once again, the posterior information is a combination of the prior information and the information from the data. --- ## Posterior with precision terms: combining information - Posterior mean is weighted sum of prior information plus data information: .block[ $$ `\begin{split} \mu_n & = \dfrac{n\tau\bar{y} + \tau_0\mu_0}{\tau n + \tau_0}\\ & = \dfrac{\tau_0}{\tau_0 + \tau n} \mu_0 + \dfrac{n\tau}{\tau_0 + \tau n} \bar{y} \end{split}` $$ ] -- - Recall that `\(\sigma^2\)` (and thus `\(\tau\)`) is known for now. -- - If we think of the prior mean as being based on `\(\kappa_0\)` prior observations from a similar population as `\(y_1,y_2,\ldots,y_n\)`, then we might set `\(\sigma_0^2 = \frac{\sigma^2}{\kappa_0}\)`, which implies `\(\tau_0 = \kappa_0 \tau\)`, and then the posterior mean is given by .block[ $$ `\begin{split} \mu_n & = \dfrac{\kappa_0}{\kappa_0 + n} \mu_0 + \dfrac{n}{\kappa_0 + n} \bar{y}. \end{split}` $$ ] --- ## Posterior with variance terms - In terms of variances, we have .block[ `$$\mu|Y,\sigma^2 \sim \mathcal{N}(\mu_n, \sigma_n^2)$$` ] where .block[ `$$\mu_n = \dfrac{ \dfrac{n}{\sigma^2}\bar{y} + \dfrac{1}{\sigma^2_0} \mu_0}{\dfrac{n}{\sigma^2} + \dfrac{1}{\sigma^2_0}}$$` ] and .block[ `$$\sigma^2_n = \dfrac{1}{\dfrac{n}{\sigma^2} + \dfrac{1}{\sigma^2_0}}.$$` ] -- - It is still easy to see that we can re-express the posterior information as a sum of the prior information and the information from the data. --- class: center, middle # What's next? ### Move on to the readings for the next module!