# LossMatrix: Loss Matrix In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

A loss matrix is useful in Bayesian decision theory for selecting the Bayes action, the optimal Bayesian decision, when there are a discrete set of possible choices (actions) and a discrete set of possible outcomes (states of the world). The Bayes action is the action that minimizes expected loss, which is equivalent to maximizing expected utility.

## Usage

 `1` ```LossMatrix(L, p.theta) ```

## Arguments

 `L` This required argument accepts a S x A matrix or S x A x N array of losses, where S is the number of states of the world, A is the number of actions, and N is the number of samples. These losses have already been estimated, given a personal loss function. One or more personal losses has already been estimated for each combination of possible actions a=1,...,A and possible states s=1,...,S. `p.theta` This required argument accepts a S x A matrix or S x A x N array of state prior probabilities, where S is the number of states of the world, A is the number of actions, and N is the number of samples. The sum of each column must equal one.

## Details

Bayesian inference is often tied to decision theory (Bernardo and Smith, 2000), and decision theory has long been considered the foundations of statistics (Savage, 1954).

Before using the `LossMatrix` function, the user should have already considered all possible actions (choices), states of the world (outcomes unknown at the time of decision-making), chosen a loss function L(theta, alpha), estimated loss, and elicited prior probabilities p(theta | x).

Although possible actions (choices) for the decision-maker and possible states (outcomes) may be continuous or discrete, the loss matrix is used for discrete actions and states. An example of a continuous action may be that a decision-maker has already decided to invest, and the remaining, current decision is how much to invest. Likewise, an example of continuous states of the world (outcomes) may be how much profit or loss may occur after a given continuous unit of time.

The coded example provided below is taken from Berger (1985, p. 6-7) and described here. The set of possible actions for a decision-maker is to invest in bond ZZZ or alternatively in bond XXX, as it is called here. A real-world decision should include a mutually exhaustive list of actions, such as investing in neither, but perhaps the decision-maker has already decided to invest and narrowed the options down to these two bonds.

The possible states of the world (outcomes unknown at the time of decision-making) are considered to be two states: either the chosen bond will not default or it will default. Here, the loss function is a negative linear identity of money, and hence a loss in element `L[1,1]` of -500 is a profit of 500, while a loss in `L[2,1]` of 1,000 is a loss of 1,000.

The decision-maker's dilemma is that bond ZZZ may return a higher profit than bond XXX, however there is an estimated 10% chance, the prior probability, that bond ZZZ will default and return a substantial loss. In contrast, bond XXX is considered to be a sure-thing and return a steady but smaller profit. The Bayes action is to choose the first action and invest in bond ZZZ, because it minimizes expected loss, even though there is a chance of default.

A more realistic application of a loss matrix may be to replace the point-estimates of loss with samples given uncertainty around the estimated loss, and replace the point-estimates of the prior probability of each state with samples given the uncertainty of the probability of each state. The loss function used in the example is intuitive, but a more popular monetary loss function may be -log(E(W | R)), the negative log of the expectation of wealth, given the return. There are many alternative loss functions.

Although isolated decision-theoretic problems exist such as the provided example, decision theory may also be applied to the results of a probability model (such as from `IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, `PMC`), or `VariationalBayes`, contingent on how a decision-maker is considering to use the information from the model. The statistician may pass the results of a model to a client, who then considers choosing possible actions, given this information. The statistician should further assist the client with considering actions, states of the world, then loss functions, and finally eliciting the client's prior probabilities (such as with the `elicit` function).

When the outcome is finally observed, the information from this outcome may be used to refine the priors of the next such decision. In this way, Bayesian learning occurs.

## Value

The `LossMatrix` function returns a list with two components:

 `BayesAction` This is a numeric scalar that indicates the action that minimizes expected loss. `E.Loss` This is a vector of expected losses, one for each action.

## Author(s)

Statisticat, LLC. software@bayesian-inference.com

## References

Berger, J.O. (1985). "Statistical Decision Theory and Bayesian Analysis", Second Edition. Springer: New York, NY.

Bernardo, J.M. and Smith, A.F.M. (2000). "Bayesian Theory". John Wiley \& Sons: West Sussex, England.

Savage, L.J. (1954). "The Foundations of Statistics". John Wiley \& Sons: West Sussex, England.

`elicit`, `IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, `PMC`, and `VariationalBayes`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23``` ```library(LaplacesDemon) ### Point-estimated loss and state probabilities L <- matrix(c(-500,1000,-300,-300), 2, 2) rownames(L) <- c("s[1]: !Defaults","s[2]: Defaults") colnames(L) <- c("a[1]: Buy ZZZ", "a[2]: Buy XXX") L p.theta <- matrix(c(0.9, 0.1, 1, 0), 2, 2) Fit <- LossMatrix(L, p.theta) ### Point-estimated loss and samples of state probabilities L <- matrix(c(-500,1000,-300,-300), 2, 2) rownames(L) <- c("s[1]: Defaults","s[2]: !Defaults") colnames(L) <- c("a[1]: Buy ZZZ", "a[2]: Buy XXX") L p.theta <- array(runif(4000), dim=c(2,2,1000)) #Random probabilities, #just for a quick example. And, since they must sum to one: for (i in 1:1000) { p.theta[,,i] <- p.theta[,,i] / matrix(colSums(p.theta[,,i]), dim(p.theta)[1], dim(p.theta)[2], byrow=TRUE)} Fit <- LossMatrix(L, p.theta) Fit ### Point-estimates of loss may be replaced with samples as well. ```