Find your next learning step
A posterior distribution does not choose a point estimate until a loss function says which mistakes matter. Under absolute-error loss, every conditional median is a Bayes action. If the posterior places exactly half its mass on each side of an interval, the whole interval can minimise risk. That flat optimum is a mathematical feature of the decision, not a software failure.
Open the chapter contents · Probability root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Name the state, observation, action and loss
Let Θ be an unknown real quantity and Y the information observed. After seeing Y=y, choose an action a and incur absolute loss L(Θ,a)=|Θ−a|. The posterior risk is r(a|y)=E[|Θ−a||Y=y].
A Bayes action minimises this conditional expectation. It is conditional on the stated posterior and loss; it is not automatically the posterior mean, mode, most likely category or best action under a different cost structure.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Characterise the full median set
A number m is a posterior median when P(Θ≤m|y)≥1/2 and P(Θ≥m|y)≥1/2. These two inequalities allow an interval of medians when the cumulative distribution reaches one half and stays there across a gap.
Moving a slightly to the right changes absolute risk according to the balance of posterior mass on the two sides. Risk falls while more than half the mass lies to the right, rises while more than half lies to the left, and can remain flat where both sides carry exactly one half.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work a discrete non-unique example
Suppose the posterior assigns probabilities 0.50, 0.30 and 0.20 to Θ=0, 4 and 10. Every action from 0 through 4 is a median. Pair the mass at zero with all mass above four: within that interval, moving right increases one half of the loss and decreases the other half by the same amount.
| Action a | Loss at Θ=0 | Loss at Θ=4 | Loss at Θ=10 | Posterior risk |
|---|---|---|---|---|
| 0 | 0 | 4 | 10 | 3.2 |
| 2 | 2 | 2 | 8 | 3.2 |
| 4 | 4 | 0 | 6 | 3.2 |
| 6 | 6 | 2 | 4 | 4.4 |
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Prove optimality without differentiating a corner
Absolute loss has a corner at Θ=a, so ordinary differentiation can hide point masses. A safer argument compares the risk at a and a+δ. Posterior values below a add roughly δ each, values above a+δ subtract roughly δ each, and mass inside the short interval supplies the remaining one-sided term.
The subgradient condition is 0 in the interval [P(Θ
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Change one probability and watch uniqueness return
Shift 0.01 posterior mass from Θ=4 to Θ=0. The probabilities become 0.51, 0.29 and 0.20. Now Θ=0 is the unique median because more than half the mass sits at that point; actions immediately to its right increase risk.
| Action a | Risk before shift | Risk after shift | Decision |
|---|---|---|---|
| 0 | 3.20 | 3.16 | New unique minimum |
| 2 | 3.20 | 3.20 | No longer optimal |
| 4 | 3.20 | 3.24 | No longer optimal |
| 6 | 4.40 | 4.44 | Clearly suboptimal |
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Extend absolute loss to asymmetric errors
If underestimation and overestimation have different costs, use a tilted absolute or pinball loss. The optimal action becomes a posterior quantile at a level determined by the cost ratio, rather than necessarily the 0.5 quantile.
This makes the decision boundary explicit. A delivery promise, reserve threshold or safety stock may rationally target an upper quantile because late or low predictions cost more than conservative ones. The probability level should come from consequences, not convenience.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Connect the established probability routes
The posterior mean and MMSE guide shows why squared loss selects the conditional mean. The Tweedie formula guide reconstructs that mean in an additive Gaussian model.
Use How Quantiles Work for cumulative-probability definitions and the quantile-regression guide for pinball loss and conditional predictive quantiles.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Separate a conditional median from nearby ideas
The median of the prior, the median of raw observations and the posterior median after conditioning on y answer different questions. A sample median is a statistic of data; a posterior median is a decision summary of a conditional distribution for Θ.
Medians also do not generally commute with nonlinear transformations. If a parameter is transformed, derive the posterior decision on the scale where loss is incurred. Componentwise medians in several dimensions need not minimise Euclidean distance or respect constraints.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
Non-uniqueness does not mean every number is equally good. It occurs on the median set, and actions outside it have larger posterior risk. A numerical optimiser returning one endpoint has selected one Bayes action, not proved uniqueness.
Posterior optimality remains model dependent. Prior sensitivity, likelihood misspecification, finite computation and distribution shift can change the median set. Absolute loss is robust to large errors relative to squared loss, but it does not remove model error or make uncertainty vanish.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use an evidence and source routine
Record the target, conditioning information, posterior mass or density, action space, loss scale, cost asymmetry, full median set, chosen tie rule, posterior risk, sensitivity analysis and changed-context result. Report an interval of optimal actions when that is what the mathematics gives.
Oxford statistical theory notes state that the posterior median is a Bayes estimator under absolute error loss. Schwertman, Gilks and Cameron give a non-calculus proof of the median’s absolute-deviation property and make non-uniqueness visible.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does a continuous posterior have a unique median?
Often, but not automatically. A strictly increasing cumulative distribution crosses one half at one point; flat regions or mixed distributions can produce several medians.
Why not average all Bayes actions?
Averaging endpoints stays optimal for an interval-valued median set, but the averaging rule is an extra tie convention. It may matter when later constraints or nonlinear costs are introduced.
Is the posterior mode optimal under absolute loss?
Not generally. The mode concerns density height or point mass, while absolute-loss risk balances cumulative mass on the two sides.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a changed-cost transfer check
Use posterior masses 0.30 at 0, 0.45 at 4 and 0.25 at 10. Find the median and then replace symmetric absolute loss with an underestimation cost twice the overestimation cost. Determine which posterior quantile the new rule targets and how the chosen action changes.
Continue through the Probability root, the Gold Standard of Education and the How X Works hub.
Top · Contents · Previous · How X Works