Bayesian Decision Theory is a wonderfully useful tool that provides a formalism for decision making under uncertainty. It is used in a diverse range of applications including but definitely not limited to finance for guiding investment strategies or in engineering for designing control systems. In what follows I hope to distill a few of the key ideas in Bayesian decision theory. In particular I will give examples that rely on simulation rather than analytical closed form solutions to global optimization problems. My hope is that such a simulation based approach will provide a gentler introduction while allowing readers to solve more difficult problems right from the start.

I will break the basics of decision theory into 5 parts. The first part is to give a formal definition to the possible decisions we are trying to choose between. Next we have to quantify the information we are using to make the decision. Third we have to decide how to quantify how good/bad a decision is given our information. Fourth, I will discuss how to tie all these things together to make an optimal decision when there is uncertainty in our information. Finally, I will give some greater context by discussing closed form solutions to global optimization problems and the connection to engineering control theory.

Framing the decision space I first need to introduce some formal way in which we discuss “decisions”. For a given problem, imagine that there is a space in which all of my possible decisions live. This space can be discrete, continuous, multivariate, or any number of crazy things based on the problem at hand. In what follows I will denote our decision space (regardless of what exactly it is) by \(\mathcal{A}\) and a decision \(a\in \mathcal{A}\). Examples: Part 1 Lets say I am trying to decide a price at which to list a used phone I want to sell. In this case I may denote my decision space as the entire positive real line such that \(a \in [0, +\infty)\) . Lets say I am trying to choose between two different brands of breakfast cereal. In this case I may denote my decision space as simply the set \({a_1, a_2}\) , corresponding to a decision to pick either the first or second cereal. Suppose I am trying to choose a dosage of a drug for a clinical trial. In this case my decision may be 1 dimensional, continuous and taking on any positive real value ( \(a \in [0, +\infty)\) ; just like in the first example). As a more complicated example, suppose I am trying to decide on a path I should drive a toy helicopter. In this case my decision space consists of 3 spatial dimensions as well as a temporal dimension. As a much more complicated example, suppose I am trying to cluster 20 individuals into 4 groups of equal size (each with 5 individuals in them). In this case my decision space is a combinatorial space of partitions. In all these cases the first thing to do is to identify some mathematical representation of our decisions.

The other information that helps us make a decision In Bayesian decision theory we are concerned with choosing between these different decision based on some information. Like our decision space, there is tremendous flexibility in what our information is (univariate, multivariate, continuous, discrete, etc…). Whatever that information is I will denote the space that our information lives in by \(\Theta\) and a piece of information (one element of this space) by \(\theta\) such that \(\theta \in \Theta\). Examples: Part 2 Following the phone listing example above: I may want to use a model fit to previous closed online listings to predict the probability that my phone will sell at a given listing price. In this case my information may be the probability that my phone will sell at the specified price such that \(\Theta \in [0, 1]\) . Breakfast Cereal: I may want to use the grams of sugar per serving of each of the two cereals as the information I use to make my decision. In this case I may have \(\Theta \in \mathcal{R}^{2+}\) (e.g., the positive quadrant of 2 dimensional real space). Drug Dosage: I may want to use knowledge of a smooth function relating the adverse event rate of patients to the drug dosage to make a decision. In this case I will have \(\Theta \in \mathcal{C}^\infty(\mathcal{R})\) (don’t worry if you are not familiar with notation of function spaces). Basically our information in this case is a function \(f\) that takes in a drug dosage and outputs an event rate. Toy helicopter: I may want to use knowledge of the location of obstacles in 3D space to pick this path. There are many ways of describing this type of information and it can get complicated quickly. Once we have this information we can start asking about which decision is “best”.

The Loss Function So how do we determine the “best” decision? This requires that we first define some notion of what we want (what are we trying to do?). The formal object that we use to do this goes by many names depending on the field: I will refer to it as a Loss function (\(\mathcal{L}\)) but the same general concept may be alternatively called a cost function, a utility function, an acquisition function, or any number of different things. The crucial idea is that this is a function that allows us to quantify how bad/good a given decision (\(a\)) is given some information (\(\theta\)). What does it mean to quantify? By convention I mean a real number (between \(-\infty\) and \(+\infty\)). Here I will be discussing “Loss functions” and this real value will reflect the loss we feel if we choose decision \(a\) given our information \(\theta\). We may denote this loss function as \[\mathcal{L}: \Theta\times\mathcal{A} \rightarrow \mathcal{R}\] The crucial idea is that the loss function ties together our decision space (\(\mathcal{A}\)) and our information space (\(\Theta\)). I find that the hard part of decision theory is often the choice of the loss function; this is really a subjective choice that should capture what matters to you. In the following examples I will focus on just a few very simple examples. Examples: Part 3 Phone Listing: Lets say I want to maximize the amount of money I expect to make (my expected return). In this case I may choose a loss function of the following form \(\mathcal{L}(\theta, a) = -\theta a\) . Recall that in this example \(\theta\) is a probability that the phone will sell and \(a\) is the price I list it at. Sorry for any confusion caused by then negative sign; its there because in this case we want to maximize our return so I am denoting this as negative loss (e.g., minimizing negative loss is like maximizing the return). Breakfast Cereal: Lets say I want to choose the breakfast cereal with the least sugar. In this very simple situation I may choose a very simple loss function such that

\[\mathcal{L}(\theta_1, \theta_2, a) = \begin{cases} \theta_1 & \text{if } a = a_1 \\ \theta_2 & \text{if } a = a_2 \\ \end{cases}\] Drug Dosage: Lets say that we (almost certainty) want to decrease the adverse event rate. In this case I may feel more loss with a higher adverse event rate. Therefore I may choose the following simple loss function \[\mathcal{L}(f, a) = f(a).\] Notice how weird this looks, actually our information is itself a function where the input is the action! Decision theory is very flexible and can be applied in many different situations. What makes matters more complicated (and the reason why this is related to Bayesian statistics) is that we rarely know any information exactly, instead we often only have some beliefs about the information we want to use to make a decision.

Uncertainty I have written before on how crucial it can be to quantify our uncertainty in analyses or estimates. In short, it is one thing to estimate a quantity, but it can be far more powerful to not only estimate a quantity but actually quantify our uncertainty about an estimate as a distribution over possible values. Rather than just dealing with a single known value for \(\theta\in \Theta\) we now work with a probability distribution representing our belief in the “true” value of \(\theta\) which I will denote by \(p(\theta)\). To connect this to Bayesian statistics, we may have a posterior distribution over \(\theta\) conditioned on some observed data \(x\). It is because Bayesian statistics is often about calculating these posterior distributions that most Bayesian texts on decision theory discuss this uncertainty of information as \(p(\theta|x)\). However, any probability distribution over our information space can be used here. For example, we could obtain a distribution over \(\theta\) through error propagation of another measured or guessed quantity as I discussed in a previous post. So how do we figure out the loss associated with individual decisions when we don’t even know the information we want to use to make a decision? The answer is that we turn to probability theory and instead calculate the “Expected Loss” we would feel if we choose a given action given our beliefs (our probability distribution) about \(\theta\). We calculate this “Expected” loss just like we would calculate the expectation of any other function of a random variable. \[\text{Expected Loss}(a) = \int_\Theta\mathcal{L}(\theta, a)p(\theta)d\theta\] Essentially the expected loss sums up the loss we would feel for a given decision \(a\) over all possible values of our information \(\theta\) but weighted by the probability of \(\theta\). So why should we care about this? In this form the Expected Loss does some pretty remarkable things in terms of decision making. In particular, the Expected Loss balances between how probable each value of \(\theta\) is and the loss associated with each value of \(\theta\) for a given \(a\). In what follows, we will not calculate this integral directly but instead use \(N\) samples \((\theta^{(1)}, \dots, \theta^{(N)})\) from the distribution \(p(\theta)\) to approximate this integral by \[\text{Expected Loss}(a) \approx \frac{1}{N}\sum_{n=1}^N\mathcal{L}(\theta^{(n)}, a).\] This is what I meant at the start of this post when I said we would rely on simulation rather than analytical closed form solutions (more on this at the end of the post). Finally we come to the hard part. We need to pick a given value of \(a\) that is “best”. By best we will mean the decision that has the lowest expected loss. Will call the decision that minimizes the Expected Loss the “Bayes Action” and denote it \(\hat{a}\) such that \[\hat{a} \approx \underset{a\in \mathcal{A}}{\text{argmin}} \frac{1}{N}\sum_{n=1}^N\mathcal{L}(\theta^{(n)}, a). \] Why did I say this was the hard part? Because this part can get computationally intense quickly. Even if we were not using simulations and instead solving this optimization explicitly, analytical closed form solutions are only possible for a select set of loss functions and distributions \(p(\theta)\) (more on this at the end of the post). Here, rather than continuing to go through the examples in the same manner I have done so far, I will instead focus just on the first example and go start-to-finish through a full worked Bayesian decision theory problem.

Fully Worked Example: What price should I sell my used phone for? Imagine I have an unscratched used phone that was made in 2014 that I want to sell. I might go online and look at some closed eBay listings and see what sold and what didn’t. Lets pretend I believe that there are 3 key variables that dictate whether a phone sold or not: whether it was scratched or not, the year it was made, and the price it was listed at. Given this information I want to figure out what price I should list my phone at. In particular I want to maximize my expected return (the expected amount of money I will make), this defines my loss function as I will discuss below. Here is some pretend data (from 6 closed eBay listings) from such an exercise where I have denoted whether a phone sold with a 1 and if it didn’t with a 0 (same for whether or not it was scratched). library(tidyverse) library(brms) d <- data.frame(sold = c(1, 1, 0, 0, 1, 1), scratched = c(0, 0, 1, 0, 0, 0), year = c(2014, 2015, 2010, 2014, 2015, 2016), price = c(50, 70, 40, 100, 90, 100)) d ## sold scratched year price ## 1 1 0 2014 50 ## 2 1 0 2015 70 ## 3 0 1 2010 40 ## 4 0 0 2014 100 ## 5 1 0 2015 90 ## 6 1 0 2016 100 Lets fit a very simple Bayesian Logistic Regression model to this data. Here is the model: \[\begin{align} y_{\text{sold}, i} &\sim \text{Bernoulli}(\pi_i) \\ \pi_i &= \text{Logit}^{-1}(\eta_i) \\ \eta_i &= \beta_0 + \beta_\text{1}x_{\text{scratched},i} + \beta_\text{2}x_{\text{year},i} + \beta_\text{3}x_{\text{price},i} \end{align}\] We will use a default uniform prior on \(\beta_0\) and we will place informative priors on the other coefficients. For example, since I believe that having a scratch would likely decrease the probability that a phone sold I will use the following prior \(\beta_1 \sim \text{Normal}(-1, 1)\). fit <- brm(sold ~ scratched + year + price, data = d, family = bernoulli(link = "logit"), prior = c(set_prior("normal(-1,1)", class="b", coef="scratched"), set_prior("normal(1, 1)", class="b", coef="year"), set_prior("normal(-2, 1)", class="b", coef="price")), silent=TRUE, refresh = -1) ## ## Gradient evaluation took 3.9e-05 seconds ## 1000 transitions using 10 leapfrog steps per transition would take 0.39 seconds. ## Adjust your expectations accordingly! ## ## ## ## Elapsed Time: 0.056959 seconds (Warm-up) ## 0.035268 seconds (Sampling) ## 0.092227 seconds (Total) ## ## ## Gradient evaluation took 6e-06 seconds ## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds. ## Adjust your expectations accordingly! ## ## ## ## Elapsed Time: 0.059337 seconds (Warm-up) ## 0.02873 seconds (Sampling) ## 0.088067 seconds (Total) ## ## ## Gradient evaluation took 6e-06 seconds ## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds. ## Adjust your expectations accordingly! ## ## ## ## Elapsed Time: 0.060765 seconds (Warm-up) ## 0.033488 seconds (Sampling) ## 0.094253 seconds (Total) ## ## ## Gradient evaluation took 6e-06 seconds ## 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds. ## Adjust your expectations accordingly! ## ## ## ## Elapsed Time: 0.056948 seconds (Warm-up) ## 0.03405 seconds (Sampling) ## 0.090998 seconds (Total) summary(fit) ## Family: bernoulli(logit) ## Formula: sold ~ scratched + year + price ## Data: d (Number of observations: 6) ## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1; ## total post-warmup samples = 4000 ## ICs: LOO = NA; WAIC = NA; R2 = NA ## ## Population-Level Effects: ## Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat ## Intercept -4310.14 1620.15 -7609.58 -1334.14 2505 1.00 ## scratched -1.16 1.00 -3.13 0.81 3185 1.00 ## year 2.15 0.81 0.67 3.79 2500 1.00 ## price -0.12 0.08 -0.30 0.03 2196 1.00 ## ## Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample ## is a crude measure of effective sample size, and Rhat is the potential ## scale reduction factor on split chains (at convergence, Rhat = 1). Now that we have our fitted model based on our data (that is, now that we have \(p(\theta|x)\)) we can build our loss function and use our fitted model to figure out the optimal price to list my phone at. As I said, the “loss” function I will choose is based on maximizing my expected return. Note that in this case I say “Loss function” but in reality its a function that measures the expected return and we want to maximize it. Its really just a convention as I have been calling the thing the loss function throughout this post and so I will continue to do so. However, now we want to maximize this function. This defines probably one of the simplest loss functions I could choose in this setting: \[\mathcal{L}(\pi^*(\text{price}), \text{price}) = \pi^*(\text{price})\times\text{price}.\] Notice that I made this loss function in terms of \(\pi^*(\text{price})\) which I use to denote the probability that my phone will sell if I list it as that particular price. Just like one of our previous examples, in this case our information is actually a function of our action! It is a function that takes in the price and outputs a value between 0 and 1 (a probability). While this may seem weird and bewildering, notice that we can simply use our fitted model to predict \(\pi^*(\text{price})\). I will implement this expected loss calculation as an R function next. loss <- function(price){ # First Create the input data to predict with out model - we want to predict whether or not our phone will sell our.phone <- data.frame(scratched=0, year=2014, price=price) # Next, for each posterior sample from out model, predict whether or not our phone would sell at the given price. This will give a vector of 0's and 1's, did the phone sell in each posterior sample. Think of each posterior sample as a simulation. pp <- posterior_predict(fit, newdata=our.phone) # Next calculate the expected return for each of these posterior simulations mean(pp*price) } Now that we have a function implementing the calculation of expected loss for each price we can optimize this to find the listing price that maximizes our expected return. Again because our “loss” function is something that in this case we want to maximize I will instead pass it to R’s optim function as a negated version of itself (add a negative sign) as the optim function defaults to minimizing a passed function. (op <- optim(50, function(x) -loss(x))) ## $par ## [1] 71.71875 ## ## $value ## [1] -58.41492 ## ## $counts ## function gradient ## 177 NA ## ## $convergence ## [1] 10 ## ## $message ## NULL Therefore we find that the optimal listing price should be 72 and at this value we expect to make 58 dollars (or whatever your favorite currency is). Finally lets look at what this expected return/loss looks like evaluated at a number of different listing prices. x <- 10:110 # Listing prices to evaluate l <- sapply(x, loss) plot(x, l, xlab = "Listing Price", ylab = "Expected Return") So whats the intuition behind this result? Basically if we list our phone for a low-enough value (e.g., for 10) we are almost certain that the phone will sell but we won’t make much money for it and on average we can expect to make about 10. At the other extreme if we list our phone for too much (e.g., for 110) then we are unlikely to sell it but if we do sell it, it will make much more money. Where is the best balance in terms of maximizing the expected return? Right around 71 where we can expect to make about 58 on average, just as we saw from our black-box optimization using the optim function.