class: center, middle, inverse, title-slide # STA 360/602L: Module 2.2 ## Operationalizing data analysis; selecting priors ### Dr. Olanrewaju Michael Akande --- ## Outline - Operationalizing data analysis - Example: rare events - Selecting priors and potential problems --- ## Operationalizing data analysis How should we approach data analysis in general? -- - .hlight[Step 1]. State the question. -- - .hlight[Step 2]. Collect the data. -- - .hlight[Step 3]. Explore the data. -- - .hlight[Step 4]. Formulate and state a modeling framework. -- - .hlight[Step 5]. Check your models. -- - .hlight[Step 6]. Answer the question. --- ## Example: rare events - .hlight[Step 1]. State the question: + What is the prevalence of an infectious disease in a small city? + Why? High prevalence means more public health precautions are recommended. -- - .hlight[Step 2]. Collect the data: + Suppose you collect a small random sample of 20 individuals. -- - .hlight[Step 3]. Explore the data: + Let `\(Y\)` denote the unknown number of infected individuals in the sample. --- ## Example: rare events - .hlight[Step 4]. Formulate and state a modeling framework: + Parameter of interest: `\(\theta\)` is the fraction of infected individuals in the city. + Sampling model: a reasonable model for `\(Y\)` can be `\(\textrm{Bin}(20,\theta)\)` <img src="img/binomial_histograms.png" width="500px" height="370px" style="display: block; margin: auto;" /> --- ## Example: rare events - .hlight[Step 4]. Formulate and state a modeling framework: + Prior specification: information from previous studies — infection rate in “comparable cities” ranges from 0.05 to 0.20 with an average of 0.10. So maybe a standard deviation of roughly 0.05? + What is a good prior? The **expected value** of `\(\theta\)` close to 0.10 and the **standard deviation** close to 0.05. + Possible option: `\(\mbox{Beta}(3.5,31.5)\)` or maybe even `\(\mbox{Beta}(3,32)\)`? <img src="2-2-selecting-priors_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" /> --- ## Quick beta-binomial recap - Binomial likelihood: .block[ .small[ `$$p(y | \theta) = {n \choose y} \theta^y(1-\theta)^{n-y}$$` ] ] -- - `\(+\)` Beta Prior: .block[ .small[ `$$\pi(\theta) = \frac{1}{B(a,b)} \theta^{a-1} (1-\theta)^{b-1} = \textrm{Beta}(a,b)$$` ] ] -- - `\(\Rightarrow\)` Beta posterior: .block[ .small[ `$$\pi(\theta | y) = \frac{1}{B(a+y,b+n-y)} \theta^{a+y-1} (1-\theta)^{b+n-y-1} = \textrm{Beta}(a+y,b+n-y).$$` ] ] -- - Recall: If `\(\theta \sim \textrm{Beta}(a,b)\)`, then + `\(\mathbb{E}[\theta] = \frac{a}{a+b}\)` + `\(\mathbb{V}[\theta] = \frac{ab}{(a+b)^2(a+b+1)}\)` --- ## Example: rare events - .hlight[Step 4]. Formulate and state a modeling framework: + Under `\(\mbox{Beta}(3,32)\)`, `\(\Pr(\theta < 0.1) \approx 0.67\)`. + Posterior distribution for the model: `\(\pi(\theta | Y=y) = \textrm{Beta}(a+y,b+n-y)\)` + Suppose `\(Y=0\)`. Then, `\(\pi(\theta | Y=y) = \textrm{Beta}(3,32+20)\)` <img src="2-2-selecting-priors_files/figure-html/unnamed-chunk-4-1.png" style="display: block; margin: auto;" /> --- ## Example: rare events - .hlight[Step 5]. Check your models: + Compare performance of posterior mean and posterior probability that `\(\theta < 0.1\)`. + Under `\(\mbox{Beta}(3,52)\)`, - `\(\Pr(\theta < 0.1 | Y=y) \approx 0.92\)`. More confidence in low values of `\(\theta\)`. - For `\(\mathbb{E}(\theta | Y=y)\)`, we have .block[ .small[ $$ `\begin{split} \mathbb{E}(\theta | y) & = \dfrac{a+y}{a+b+n} = \dfrac{3}{52} = 0.058.\\ \end{split}` $$ ] ] - Recall that the prior mean is `\(a/(a+b)=0.09\)`. Thus, we can see how that contributes to the prior mean. .block[ .small[ $$ `\begin{split} \mathbb{E}(\theta | y) & = \dfrac{a+b}{a+b+n} \times \textrm{prior mean} + \dfrac{n}{a+b+n} \times \textrm{sample mean}\\ & = \dfrac{a+b}{a+b+n} \times \dfrac{a}{a+b} + \dfrac{n}{a+b+n} \times \dfrac{y}{n}\\ & = \dfrac{35}{52} \times \dfrac{3}{35} + \dfrac{20}{52} \times \dfrac{0}{n} = \dfrac{3}{52} = 0.058.\\ \end{split}` $$ ] ] --- ## Example: rare events - .hlight[Step 6]. Answer the question: + People with low prior expectations are generally at least `\(90\%\)` certain that the infection rate is below 0.10. + `\(\pi(\theta | Y)\)` is to the left of `\(\pi(\theta)\)` because the observation `\(Y=0\)` provides evidence of a low value of `\(\theta\)`. + `\(\pi(\theta | Y)\)` is more peaked than `\(\pi(\theta)\)` because it combines information and so contains more information than `\(\pi(\theta)\)` alone. + The posterior expectation is 0.058. + The posterior mode is 0.04. - Note, for `\(\mbox{Beta}(a,b)\)`, the mode is `\((a-1)/(a+b-2)\)`. + The posterior probability that `\(\theta < 0.1\)` is 0.92. --- ## Cautionary tale: parameters at the boundary - Tuyl et al. (2008) discuss potential dangers of using priors that have `\(a < 1\)` with data that are all 0's (or `\(b < 1\)` with all 1's). They consider data on adverse reactions to a new radiological contrast agent. -- - Let `\(\theta_N\)`: probability of a bad reaction using the new agent. -- - Current standard agent causes bad reactions about 15 times in 10000, so one might think 0.0015 is a good guess for `\(\theta_N\)`. -- - How do we choose a prior distribution? --- ## Potential prior distributions - One might consider a variety of choices centered on `\(15/10000 = 0.0015\)`: + Prior 1: .hlight[Beta(1,666)] (mean 0.0015; 1 prior bad reaction in 667 administrations) + Prior 2: .hlight[Beta(0.05,33.33)] (mean 0.0015; 0.05 prior bad reactions in 33.38 administrations) + Prior 3: .hlight[Beta(1.6, 407.4)] (mode 0.0015; 409 prior administrations) + Prior 4: .hlight[Beta(1.05, 497)] (median 0.0015; 498.05 prior administrations) -- - Does it matter which one we choose? --- ## Potential prior distributions <img src="2-2-selecting-priors_files/figure-html/unnamed-chunk-5-1.png" style="display: block; margin: auto;" /> --- ## Potential prior distributions Let's zoom in: <img src="2-2-selecting-priors_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" /> --- ## Potential prior distributions - Let's take a closer look at properties of these four prior distributions, concentrating on the probability that `\(\theta_N < 0.0015\)`. -- - That is, new agent not more dangerous than old agent. <br /> | Be(1,666) | Be(0.05,33.33) | Be(1.6,407.4) | Be(1.05,497) :----------- | :---------: |:---------: |:---------: |:---------: | Prior prob | 0.632 | 0.882 | 0.222 | 0.500 Post prob (0 events) | 0.683 | 0.939 | 0.289 | 0.568 Post prob (1 event) | 0.319 | 0.162 | 0.074 | 0.213 -- - Suppose we have a single arm study of 100 subjects. -- - Consider the two most likely potential outcomes: + 0 adverse outcomes observed + 1 adverse outcome observed --- ## Problems with the priors - After just 100 trials with no bad reactions, the low weight (33.38 prior observations) prior indicates one should be 94% sure the new agent is equally safe as (or safer than) the old one. -- - The low weight prior largely assumes the conclusion we actually hope for (that the new agent is safer), thus it takes very little confirmatory data to reach that conclusion. -- - Is this the behavior we want? -- - Take home message: be very careful with priors that have `\(a < 1\)` with data that are all 0's (or `\(b < 1\)` with all 1's). -- - Given that we know the adverse event rate should be small, we might try a restricted prior e.g. Unif(0,0.1). --- class: center, middle # What's next? ### Move on to the readings for the next module!