OptimalDecisionMaking.qmd

---
title: "Optimal Decision Making"
author: "Adrien Osakwe"
format: html
editor: visual
---

## Decision-Making

Many statistical procedures involve some form of *decision-making* where actions are taken given the observed data. Examples include

-   parameter estimation

-   hypothesis testing

-   prediction/classification

-   model selection

We can denote a function $T$ which is used as an estimator of a statistic of interest from the random variable $Y$. Consider, the mean, order statistics, or the empirical cdf. We can also consider the model space $\mathbb{F}$. We can then examine a loss function $L(..)$ which will determine the accuracy with which we report $T$ when the ground truth comes from $F \in \mathbb{F}$.

For example, with the cdf $F_y$ we can have that

$$
\mu = \int yF_y(dy)
$$

And define the loss function as $L(T,F_y) = (T - \mu)^2$ to represent the loss in reporting the estimator $T$ when the value of interest is $\mu$.

The *optimal decision* is the decision which minimizes the expected loss with respect to the distribution $F_y$. In a parametric analysis defined by $\theta$. We have that

1.  $\theta$ is considered a fixed constant and the data as random in a **frequentist** setting
2.  The data are fixed and $\theta$ **is random** in a Bayesian setting

### Frequentist calculation

Back to our previous example, in a frequentist setting the optimal decision for $T$ would be

$$
argmin_{\text{T}}\mathbb{E}_{F_y}[(T- \mu)^2] = argmin_{\text{T}}\{\mathbb{E}_{F_y}[(T -\mathbb{E}_{F_y}[T])^2] + (\mathbb{E_{F_y}[T] - \mu)^2}) \}
$$

$$
= argmin_{\text{T}}\{Var_{F_y}[T] + (\mathbb{E_{F_y}[T] - \mu)^2})\}
$$

This tells us that we need to take into account the variance of T and the squared bias to define the optimal T.

### Kullback-Leibler Divergence/Loss

In Bayesian settings, we usually look at the Kullback-Leibler Loss which is used to measure the difference between two distributions $F_0$ and $F_1$.

$$
KL(F_0,F_1) = \int \log\{\frac{dF_0(y)}{dF_1(y)}\}dF_0(y)
$$

This is defined when $F_1$ is absolutely continuous with respect to $F_0$. That is, the probability mass at a given point for $F_1$ is zero whenever it is zero for $F_0$. In essence, we can express the KL as an expectation:

$$
KL(f_0,f_1) = \mathbb{E}_{f_0}[log\{\frac{f_0(Y)}{f_1(Y)}\}]
$$

It is important to note that:

1.  KL is always non-negative
2.  $KL(F_0,F_1) \neq KL(F_1,F_0)$
3.  KL can only be zero if both distributions are identical.

This is applicable to both discrete and continuous distributions. In a parametric setting, we can expect to have pdf $f(y;\theta)$ where $\theta_0$ represents the optimal, data-generating model. We can then write KL as

$$
KL(\theta_0,\theta) = \int\{\frac{f(y;\theta_0)}{f(y;\theta)}\}f(y;\theta_0)dy
$$

and aim to report an estimator $\hat{\theta} = T(Y)$ for the true value $\theta_0$

### Decision Theory Concepts

The key parts of a decision problem are:

-   a decision d, selected from a set of $D$ decisions must be made.

-   there is a true state of nature, $v(\theta)$, which lies in the set $\gamma$ , that is defined bby the data generating model, $F_Y(y;\theta)$ .

-   there is a loss function $L(d,v)$ for decision $d$ and state $v$ which records the loss when for a given state $v$ and decision $d$.

With these components, we aim to **minimize the expected loss**.

In the case of estimation, the decision required is the estimation of the parameter $\theta$ and the true state of nature is the parameter's value, that is, $v(\theta) \equiv \theta$. If data is available, the optimal decision is usually a function of the observed data. If the decision takes the form of a statistic, we have $d(y) \equiv T(y) = t_n$ with associated loss $L(t_n,\theta)$. The corresponding random variable will be $T_n \equiv T(Y)$.

**Frequentist Risk (loss):** The expected loss associated with $d(Y)$ over the distribution of $Y|\theta$.

$$
R_n(d,\theta) = \mathbb{E}_{F_Y}[L(T_n,\theta)] = \int_yL(T(y),\theta)f_Y(y;\theta)dy
$$

**Bayes Risk (average):** Is the expected risk over the **prior** distribution:

$$
R_n(d) = \mathbb{E}_{\pi_0}[R_n(d,\theta)] = \mathbb{E}_{\pi_0}[\mathbb{E}_{F_Y}[L(T_n,\theta)]]
$$

$$
= \int_\Theta\{ \int_yL(t_n,\theta)f_Y(y;\theta)dy\}\pi_0(\theta)d\theta  
$$

where $t_n \equiv t_n(y)$.

$$
= \int_\Theta\int_yL(t_n,\theta)f_Y(y)\pi_n(\theta)dyd\theta
$$

$$
= \int_y\{\int_\Theta L(t_n,\theta)\pi_n(\theta)d\theta\}f_Y(y)dy
$$

Following Bayes Theorem.

With the prior and fixed observable data, the optimal Bayesian decision, the **Bayes rule** is

$$
\hat{d}_B = argmin_{d \in D} R_n(d(y))
$$

Interpretation:

-   Bayes risk is minimized when the inner integral is minimized for fixed observations, regardless of their value

-   The double integral is minimized

-   or that the optimal decision should be chosen conditionally on the observed data and not averaged over all possible values.

To estimate, we need to select a statistic of interest. The minimization can be reduced to the following:

$$
argmin_{t \in \Theta}\int_\Theta L(t_n,\theta)\pi_n(\theta)d\theta
$$

Given the interpretation mentioned above. In other words, the optimal decision minimizes the Bayes risk and the **posterior expected loss**.

#### Squared-error loss

-   Returns the posterior expectation (see proof)

#### Absolute error loss

-   Returns the posterior median

#### Zero-one loss

-   Returns the posterior mode