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Revising the Estimate
Quantitative Analysis · Chapter 6: Bayesian Revision and UtilitySection 6.1 · lesson 26 of 5511 min · free with a free account
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In this lesson
How new evidence and updated probabilities sharpen business decisions.
- Define prior, likelihood, joint, marginal and posterior probability
- Compute the posterior probability of a state given new evidence
- Organize the calculation as a probability tree and as a table
- Interpret the posteriors in a business decision context
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Everything said in the video, section by section (1,637 words). Read along, or search it.
Bayesian Revision and Utility · Section 6.1
Revising the Estimate
Every forecast a business makes is provisional. Then the world answers back. A focus group reports. A pilot launch returns its first numbers. The real question is what you do with that answer. Ignore it, and you throw away information you paid for. Over-react to it, and you mistake a noisy signal for the truth. Bayesian updating is the disciplined path between those two failures. It takes your starting belief and the reliability of the evidence, and returns a revised belief that has moved by the right amount. This section builds that machinery piece by piece. By the end you will revise a market forecast two different ways. And you will get the same answer both times.
Why Estimates Move
Real Decisions Unfold Over Time
Real decisions unfold over time. A firm forms an initial view of the market. That starting view has a technical name. It is the prior probability. Then the evidence starts arriving. A focus group, a pilot launch, an early sales report, a competitor's announcement. Each of those should shift what the firm believes. But shift it in a principled way, not by mood or by gut feel. The mechanism that does the shifting is Bayesian updating. Its output is the posterior probability, the revised estimate that already contains the new evidence. Notice the sequence. Prior first, evidence second, posterior last. The order matters, because the prior is what the evidence acts on.
The Mechanism
From Prior to Posterior
Think of Bayesian updating as a short pipeline: five quantities, computed in a fixed order. You start with the prior, your belief before any evidence arrives. Next comes the likelihood, which describes the evidence source itself. Multiply those two together, and you get a joint probability. One joint for every state of nature under consideration. Add the joints up, and you have the marginal probability of the evidence. That total is the denominator you are about to divide by. Finally, joint divided by marginal gives the posterior. Prior in, posterior out, with two arithmetic steps in between. Learn the order once, and every Bayes problem starts to look the same.
Blocks One And Two
Prior and Likelihood
Start with the prior. It is the initial estimate of the probability of each state, formed before any new evidence. Where does it come from? Often from history: forty per cent of similar products had high demand. It can also come from expert judgment, or from the two combined. The prior captures everything currently known about the state. The likelihood is a different animal entirely. It asks how probable the observed evidence is, given each possible state. Suppose demand is truly high. A good survey returns a favorable signal eighty per cent of the time. That number is the likelihood. Here is the part people most often get wrong. Likelihoods are not the decision-maker's beliefs. They are properties of the process that generates the evidence. The reliability of a diagnostic test, the accuracy of a market survey: they come from the design of the source.
Blocks Three And Four
Joint and Marginal Probability
The third building block is the joint probability. It is the probability that the state and the evidence both occur, and it comes from the multiplication rule. Take the likelihood of the evidence given the state, and multiply by the prior. Why bother with an intermediate quantity at all? Because the joints can be summed across states, and that sum is exactly what you need next. That makes the joint the natural unit of computation here. Which brings us to the fourth block, the marginal probability of the evidence. It is the total probability of seeing that evidence at all. Across every possible state of nature, not just the one you care about. You obtain it by summing the joint probabilities. This is sometimes called the law of total probability. It is the denominator in Bayes' rule, and it is what forces the posteriors to sum to one.
Block Five
The Posterior: Bayes' Rule
Now the block we actually wanted all along. The posterior is the updated probability of each state, given the evidence. Bayes' rule assembles it from the four components before it. Take the likelihood of the evidence given the state, and multiply by the prior for that state. Then divide by the marginal probability of the evidence. In plain words, that is the joint divided by the total. It helps to read the two halves of the fraction separately. The numerator says how often the state and the evidence appear together. The denominator says how often the evidence could have appeared at all. Their ratio is the share of evidence-occurrences traceable to this one state. That is precisely what probability of the state, given the evidence, means.
At A Glance
Five Components, Two Layouts
So there are five components, and each has a job. Prior: what you believed before. Likelihood: how good the evidence is. Joint: prior times likelihood. Marginal: the sum of those joints. Posterior: joint divided by marginal, and you are done. How you lay the calculation out is up to you, and two organizations dominate in practice. The probability tree draws each state as a branch. Each evidence outcome hangs off it as a sub-branch, with joints at the leaves. It is visually intuitive, and it is the better tool for explaining the logic. The tabular method puts everything in a single grid instead. It scales far better when there are many states to track. The mathematics is identical either way. Only the bookkeeping changes.
Worked Example
Updating With a Market Survey
Let us put all of that to work on a real decision. A firm is considering launching a new product. Historical data on similar launches gives the prior. Demand turns out high in forty per cent of cases, and low in sixty. Before committing, the firm commissions a market survey. Past performance tells us the survey is reliable, but not perfect. When demand is truly high, it returns a favorable signal eighty per cent of the time. When demand is truly low, it still returns favorable twenty per cent of the time. That twenty per cent is the false-positive rate. The result arrives, and it is favorable. The question is how far that should move the firm's belief. We want the posterior probability of high demand, given a favorable survey.
Method 1
The Probability Tree
The tree comes first, because it shows the structure. Draw two levels. The first level branches on the state of nature, high demand or low. Those branches carry the priors, zero point four zero and zero point six zero. The second level branches on the survey result, favorable or unfavorable, and carries the likelihoods. Multiply along a path, and the leaf holds a joint probability. Two paths lead to a favorable signal. High and favorable: zero point four times zero point eight, which is zero point three two. Low and favorable: zero point six times zero point two, which is zero point one two. Add them, and the marginal probability of a favorable signal is zero point four four. Now think in counts, because it makes the division obvious. Out of a hundred launches, forty-four produce a favorable survey. Thirty-two of those come from genuinely high demand. Twelve are false positives. So the posterior for high demand is thirty-two over forty-four. That is about zero point seven two seven. Low demand takes the remaining zero point two seven three.
Method 2
The Tabular Method
Now the same calculation, laid out as a table. One row per state of nature, so two rows here. The columns are prior, likelihood, joint and posterior. Fill the first two columns straight from the problem statement. High demand takes a prior of zero point four zero, likelihood zero point eight zero. Low demand takes a prior of zero point six zero, likelihood zero point two zero. Multiply across each row to fill the joint column: zero point three two, and zero point one two. Total the joint column. Zero point four four, the marginal probability of the evidence. Then divide each joint by that total to get the posteriors. Zero point seven two seven for high demand, zero point two seven three for low. The table carries two checks you should always run. The priors must sum to one, and the posteriors must sum to one. The likelihood column, though, has no reason to sum to anything in particular.
Interpretation
What the Posterior Tells the Firm
Step back and read what the numbers actually say. Before the survey, the firm gave high demand a forty per cent chance. After a favorable result, that belief jumps to seventy-two point seven per cent. A shift of roughly thirty-three percentage points, from one piece of evidence. The probability of low demand falls by exactly the same amount. But look at where the belief stopped. It did not go all the way to certainty. The survey is informative, and it is imperfect. So a twenty-seven point three per cent chance remains that demand is genuinely low. In that case, the favorable signal was simply a false positive. Apply Bayes' rule directly and you land in the same place: zero point eight times zero point four, over zero point four four. That is zero point three two over zero point four four, about zero point seven two seven. Formula, tree and table are three ways of writing one computation.
Key Takeaways
What to Carry Forward
Pull the threads together. Bayesian updating turns prior beliefs into posterior beliefs. It does that by combining them with the likelihood of the evidence you saw. Five blocks, and the order never changes: prior, likelihood, joint, marginal, posterior. Remember that likelihoods describe the evidence source, not your beliefs. A survey's accuracy, a medical test's sensitivity, a model's hit rate. When many states are in play, reach for the table rather than the tree. And always check that your posteriors sum to one. If they do not, you mis-typed a likelihood, or forgot to divide by the marginal. Last, and most easily forgotten: decide on the posteriors. Reverting to the prior because the survey was imperfect throws information away. Next section, those posteriors go straight into a decision tree.
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