class: center, middle, inverse, title-slide # STA 360/602L: Module 2.6 ## Loss functions and Bayes risk ### Dr. Olanrewaju Michael Akande --- ## Bayes estimate - As we've seen by now, having posterior distributions instead of one-number summaries is great for capturing uncertainty. -- - That said, it is still very appealing to have simple summaries, especially when dealing with clients or collaborators from other fields, who desire one. -- - Can we obtain a single estimate of `\(\theta\)` based on the posterior? Sure! -- - .hlight[Bayes estimate] is the value `\(\hat{\theta}\)`, that minimizes the Bayes risk. --- ## Bayes estimate - .hlight[Bayes risk] is defined as the expected loss averaged over the posterior distribution. -- - Put differently, a .hlight[Bayes estimate] `\(\hat{\theta}\)` has the lowest posterior expected loss. -- - That's fine, but what does expected loss mean? -- - .hlight[Frequentist risk] also exists but we won't go into that here. --- ## Loss functions - A .hlight[loss function] `\(L(\theta,\delta(y))\)` is a function of a parameter `\(\theta\)`, where `\(\delta(y)\)` is some .hlight[decision] about `\(\theta\)`, based on just the data `\(y\)`. -- - For example, `\(\delta(y) = \bar{y}\)` can be the decision to use the sample mean to estimate `\(\theta\)`, the true population mean. -- - `\(L(\theta,\delta(y))\)` determines the penalty for making the decision `\(\delta(y)\)`, if `\(\theta\)` is the true parameter; `\(L(\theta,\delta(y))\)` characterizes the price paid for errors. --- ## Loss functions - A common choice for example, when dealing with point estimation, is the .hlight[squared error loss], which has .block[ `$$L(\theta,\delta(y)) = (\theta - \delta(y))^2.$$` ] -- - Bayes risk is thus .block[ `$$\rho(\theta,\delta) = \mathbb{E}\left[\ L(\theta,\delta(y)) \ | y\right] = \int L(\theta,\delta(y)) \cdot \pi(\theta|y) \ d\theta,$$` ] and we proceed to find the value `\(\hat{\theta}\)`, that is, the decision `\(\delta(y)\)`, that minimizes the Bayes risk. --- ## Bayes estimator under squared error loss - Turns out that, under squared error loss, the decision `\(\delta(y)\)` that minimizes the posterior risk is the posterior mean. -- - Proof: Let `\(L(\theta,\delta(y)) = (\theta - \delta(y))^2\)`. Then, .block[ $$ `\begin{aligned} \rho(\theta,\delta) & = \int L(\theta,\delta(y)) \cdot \pi(\theta|y) \ d\theta.\\ & = \int (\theta - \delta(y))^2 \cdot \pi(\theta|y) \ d\theta.\\ \end{aligned}` $$ ] -- - Expand, then take the partial derivative of `\(\rho(\theta,\delta)\)` with respect to `\(\delta(y)\)`. -- - <div class="question"> To be continued on the board! </div> --- ## Bayes estimator under squared error loss - .block[ $$ `\begin{aligned} \rho(\theta,\delta) & \int (\theta - \delta(y))^2 \cdot \pi(\theta|y) \ d\theta.\\ \end{aligned}` $$ ] - Easy to see then that `\(\delta(y) = \mathbb{E}[\theta | x]\)` is the minimizer. -- - Well that's great! The posterior mean is often very easy to calculate in most cases. -- - In the beta-binomial case for example, the Bayes estimate under squared error loss is just .block[ `$$\hat{\theta} = \frac{a+y}{a+b+n},$$` ] the posterior mean. --- ## What about other loss functions? - Clearly, squared error is only one possible loss function. An alternative is .hlight[absolute loss], which has .block[ `$$L(\theta,\delta(y)) = |\theta - \delta(y)|.$$` ] -- - Absolute loss places less of a penalty on large deviations & the resulting Bayes estimate is **posterior median**. -- - Median is actually relatively easy to estimate. --- ## What about other loss functions? - Recall that for a continuous random variable `\(Y\)` with cdf `\(F\)`, the median of the distribution is the value `\(z\)`, which satisfies .block[ `$$F(z) = \Pr(Y\leq z) = \dfrac{1}{2}= \Pr(Y\geq z) = 1-F(z).$$` ] -- - As long as we know how to evaluate the CDF of the distribution we have, we can solve for `\(z\)`. -- - Think R! --- ## What about other loss functions? - For the beta-binomial model, the CDF of the beta posterior can be written as .block[ `$$F(z) = \Pr(\theta\leq z | y) = \int^z_0 \textrm{Beta}(\theta| a+y, b+n-y) d\theta.$$` ] -- - Then, if `\(\hat{\theta}\)` is the median, we have that `\(F(\hat{\theta}) = 0.5\)`. -- - To solve for `\(\hat{\theta}\)`, apply the inverse CDF `\(\hat{\theta} = F^{-1}(0.5)\)`. -- - In R, that's simply ```r qbeta(0.5,a+y,b+n-y) ``` -- - For other popular distributions, switch out the beta. --- ## Loss functions and decisions - Loss functions are not specific to estimation problems but are a critical part of decision making. -- - For example, suppose you are deciding how much money to bet ($A) on Duke in the next UNC-Duke men's basketball game. -- - Suppose, if Duke + loses (y = 0), you lose the amount you bet ($A) + wins (y = 1), you gain B per $1 bet -- - <div class="question"> What is a good sampling distribution for y here? </div> -- - Then, the loss function can be characterized as .block[ .small[ `$$L(A,y) = A(1-y) - y(BA),$$` ] ] with your action being the amount bet A. -- - <div class="question"> When will your bet be "rational"? </div> --- ## How much to bet on Duke? - `\(y\)` is an unknown state, but we can think of it as a new prediction `\(y_{n+1}\)` given that we have data from win-loss records `\((y_{1:n})\)` that can be converted into a Bayesian posterior, .block[ .small[ `$$\theta \sim \textrm{beta}(a_n,b_n),$$` ] ] -- with this posterior concentrated slightly to the left of 0.5, if we only use data on UNC-Duke games (UNC men lead Duke 139-112 all time). -- - Actually, it might make more sense to focus on more recent head-to-head data and not the all time record. -- - In fact, we might want to build a model that predicts the outcome of the game using historical data & predictors (current team rankings, injuries, etc). -- - However, to keep it simple for this illustration, go with the posterior above. --- ## How much to bet on Duke? - The Bayes risk for action A is then the expectation of the loss function, .block[ .small[ `$$\rho(A) = \mathbb{E}\left[\ L(A,y) | \ y_{1:n}\right].$$` ] ] -- - To calculate this as a function of `\(A\)` and find the optimal `\(A\)`, we need to marginalize over the **posterior predictive distribution** for `\(y\)`. -- - <div class="question"> Why are we using the posterior predictive distribution here instead of the posterior distribution? </div> -- - As an aside, recall from Module 2.3 that .block[ .small[ `$$p(y_{n+1}|y_{1:n}) = \dfrac{a_n^{y_{n+1}} b_n^{1-y_{n+1}}}{a_n + b_n}; \ \ \ y_{n+1}=0,1.$$` ] ] -- - Specifically, that the posterior predictive distribution here is `\(\textrm{Bernoulli}(\hat{\theta})\)`, with .block[ .small[ `$$\hat{\theta} = \dfrac{a_n}{a_n + b_n}$$` ] ] -- - By the way, what do `\(a_n\)` and `\(b_n\)` represent? --- ## How much to bet on Duke? - With the loss function `\(L(A,y) = A(1-y) - y(BA)\)`, and using the notation `\(y_{n+1}\)` instead of `\(y\)` (to make it obvious the game has not been played), the Bayes risk (expected loss) for bet `\(A\)` is .block[ $$ `\begin{split} \rho(A) & = \mathbb{E}\left[\ L(A,y_{n+1}) \ | y_{1:n}\right]\\ \\ & = \mathbb{E}\left[A(1-y_{n+1}) - y_{n+1}(BA) \ | y_{1:n}\right]\\ \\ & = A \ \mathbb{E}\left[1-y_{n+1} | \ y_{1:n}\right] - (BA) \ \mathbb{E}\left[y_{n+1} | y_{1:n}\right]\\ \\ & = A \ \left(1 - \mathbb{E}\left[y_{n+1} | y_{1:n}\right] \right) - (BA) \ \mathbb{E}\left[y_{n+1} | y_{1:n}\right]\\ \\ & = A \ \left(1 -\mathbb{E}\left[y_{n+1} | y_{1:n}\right] \ (1+B) \right). \end{split}` $$ ] --- ## How much to bet on Duke? - Hence, your bet is rational as long as .block[ $$ `\begin{split} \mathbb{E}\left[y_{n+1} | \ y_{1:n}\right](1+B) > 1\\ \\ \dfrac{a_n (1+B)}{a_n + b_n} > 1. \end{split}` $$ ] -- - Clearly, there is no limit to the amount you should bet if this condition is satisfied (the loss function is clearly too simple). -- - Loss function needs to be carefully chosen to lead to a good decision - finite resources, diminishing returns, limits on donations, etc. -- - Want more on loss functions, expected loss/utility, or decision problems in general? Consider taking a course on decision theory (STA623?). --- class: center, middle # What's next? ### Move on to the readings for the next module!