Likelihood ratios multiply odds, subject to the right conditioning assumptions.
By the end: Translate probabilities into odds and update a sequence of evidence with explicit dependence assumptions.
Probability p corresponds to odds p divided by one minus p. A probability of 0.1 gives odds 1 to 9, represented numerically as 1/9. To return from numeric odds o to probability, divide o by 1 + o. Odds of three mean probability 3/4, not probability three.
A likelihood ratio compares how likely the evidence is under the state and its alternative. If a positive occurs with probability 0.6 under H and 0.2 under not-H, its likelihood ratio is three. Multiply prior odds by three. A negative has likelihood ratio 0.4/0.8 = one half; it supports not-H in this model.
Odds express the relative weight of two exhaustive states. If H has probability 10%, its alternative has probability 90%, so the weights are one to nine. Dividing the first weight by the second gives numeric odds of one ninth. This number is not a fraction of all cases. To recover the fraction, combine the two weights. One part for H plus nine parts for its alternative gives ten total parts, so H occupies one tenth. Odds of three to one similarly mean three parts among four.
The evidence model says that a positive occurs in 60% of H cases and 20% of alternative cases. For a reference population of 1,000, the prior gives 100 expected H cases and 900 alternative cases. Restricting to positives leaves 60 from H and 180 from the alternative. Their relative weight is one to three. The posterior is therefore one quarter, or 25%. The likelihood ratio of three changed the old relative weight, one ninth, into the new relative weight, one third.
Why does the total positive count not appear in the odds multiplication? Both posterior state probabilities divide by that same total. Comparing them cancels the common denominator. What remains is the old state weight ratio multiplied by the ratio of the two positive likelihoods. This cancellation explains why odds are convenient for repeated evidence. You still need to normalize when reporting a probability. A relative weight of one third means one part for H and three for the alternative, so there are four parts altogether.
Negative evidence has its own comparison. Within H, the negative chance is 40%; within the alternative it is 80%. Their ratio is one half. Starting from the original 10% prior, a negative leaves 40 expected H cases and 720 alternative cases among 1,000 reference cases. The odds are one to eighteen and the probability is one nineteenth, about 5.26%. A negative is twice as likely under the alternative as under H, so it halves the relative support for H.
An evidence outcome with equal likelihood under both states has ratio one. For example, if an indicator appears in 30% of H cases and 30% of alternative cases, observing it preserves their prior ratio. It may be a rare observation or a common one; equal likelihoods make it uninformative for this particular comparison. Conversely, a ratio above one favors H and a ratio below one favors the alternative. The direction depends on which state occupies the numerator, so name that state whenever reporting a ratio.
Absolute likelihood and evidential strength are different quantities. A report occurring with probabilities 6% and 2% has the same ratio of three as the positive report with probabilities 60% and 20%. With the same prior, either observation gives the same posterior. The rarer report would occur in fewer cases overall, but its relative support is equal. These are probabilities of defined report events, not density heights. For continuous measurements, likelihood comparisons require compatible density models and measurement units rather than interpreting point densities as event probabilities.
| Defined report | P(report | H) | P(report | alternative) | Ratio | Posterior at prior 10% |
|---|---|---|---|---|
| Common positive | 0.6 | 0.2 | 3 | 25% |
| Rarer report | 0.06 | 0.02 | 3 | 25% |
| Uninformative indicator | 0.3 | 0.3 | 1 | 10% |
| Negative | 0.4 | 0.8 | 1/2 | 1/19 ≈ 5.26% |
Starting at 10%, a positive raises probability to 25%. A following negative lowers it to 1/7, about 14.29%. Another positive raises it to 1/3. The negative does not cancel a positive because their likelihood ratios, three and one half, are not reciprocals.
These fixed ratios can be reused only if the outcomes are conditionally independent given H and also given not-H. More generally, the next likelihood ratio must condition on all prior evidence. Repeated displays, common lab errors, and sources copying one original report are not independent confirmations.
A sequential update asks what the next observation adds after previous observations are known. The multiplication rule always allows a sequence probability to be built from a first-result probability and a second-result probability conditional on the first. Using the same fixed test rate at every step is stronger. It assumes that knowing earlier results does not change the next result's distribution once the state is fixed. That is conditional independence given H, and the model needs the corresponding assumption given the alternative as well.
Construct a clean example with 1,000 reference cases and the same prior and positive likelihoods. After one positive, the expected surviving weights are 60 for H and 180 for the alternative. With conditional independence, a second positive retains 60% of those 60 H cases, or 36, and 20% of the 180 alternative cases, also 36. Their equal weights give a 50% posterior. Equivalently, prior odds one ninth multiplied by three twice become one. The second result adds the same likelihood ratio, not the same number of percentage points.
Compare a different collection mechanism. Suppose the first result is stored and a second report always reproduces it. After seeing the first positive, the second positive has conditional probability one under H and one under the alternative. Its conditional likelihood ratio is one, so the posterior stays at 25%. Two distinct reports can contain one underlying observation. Here the copying mechanism establishes dependence. Counting objects is not a way to count independent evidence.
Dependence need not be perfect copying. Imagine a second instrument that, after a first positive, is positive with probability 80% under H and 50% under the alternative. These are stipulated conditional rates after the first result, not the original marginal rates. Its added ratio is 0.8 divided by 0.5, or 1.6. The first positive left odds one third; multiplying by 1.6 gives odds eight fifteenths. Converting gives probability eight twenty-thirds, about 34.78%. Squaring the original ratio would instead produce 50%, an overstatement in this example.
| Collection mechanism | P(second + | H, first +) | P(second + | alternative, first +) | Added ratio | Posterior after both |
|---|---|---|---|---|
| Conditionally independent tests | 0.6 | 0.2 | 3 | 50% |
| Guaranteed copy | 1 | 1 | 1 | 25% |
| Dependent measurement example | 0.8 | 0.5 | 1.6 | 8/23 ≈ 34.78% |
Shared contamination can create this kind of problem even with two genuinely separate measurements. Once one instrument signals a contaminated sound sample, the second may also be unusually likely to signal. Conditioning on soundness alone has not removed the common error source. To justify the fixed-ratio update, investigate the whole collection process, including sample preparation and shared reference data. Independence after conditioning on one named variable does not follow from separate timestamps, separate staff members, or separate hardware if another common cause remains uncontrolled.
Conditional independence also differs from independence across the mixed population. In the clean example, the chance of a second positive after a first positive is 72 divided by 240, or 30%. Before any result, a positive has probability 24%. The first positive made H more prevalent among remaining cases, and H produces positives more often. Thus results are associated across all cases even though they are independent within each fixed state. The mixture explains the association; it does not contradict the conditional assumption used in the update.
A positive has probability 0.6 under H and 0.2 under not-H, so its likelihood ratio is 3.
For numeric odds o, probability is o/(1 + o).
Prior before observations. Current probability: 10%.
| Evidence processed | Odds for H | Probability |
|---|---|---|
| None | 1/9 | 10% |
| Positive | 1/3 | 25% |
| Then negative | 1/6 | 14.29% |
| Then positive | 1/2 | 33.33% |
Use the Positive, negative, positive scenario in the evidence player. The prior row is 10%. After the first positive, odds change from one ninth to one third, giving 25%. The next negative multiplies those odds by one half. Odds become one sixth, which means one H part among seven total parts. The probability is therefore one seventh, about 14.29%. A final positive triples odds to one half, giving one H part among three total parts, or about 33.33%.
A positive followed by a negative does not return to the starting point because their ratios are three and one half. Their product is 1.5, not one. To undo a ratio of three exactly would require a later conditional ratio of one third. The probabilities of the two result types are complements within each state, but ratios of complements need not be reciprocal. Under H the likelihoods are 60% and 40%; under the alternative they are 20% and 80%. Those four rates determine how much each result contributes.
The same conclusion follows by comparing whole-sequence probabilities. For positive, negative, positive under H, multiply 0.6 by 0.4 by 0.6 to get 0.144. Under the alternative, multiply 0.2 by 0.8 by 0.2 to get 0.032. The sequence likelihood ratio is 4.5. Applying that to prior odds one ninth gives one half. These products assume conditional independence under each state. They concern this ordered sequence, not the probability of any sequence containing two positives and one negative.
For the event of two positives anywhere among three independent observations, there are three possible positions for the negative. Within a fixed state, all three sequences have equal probability in this model. Their combined likelihoods are therefore 0.432 under H and 0.096 under the alternative. The ratio remains 4.5 because the same factor of three multiplies both likelihoods. That cancellation would fail if order affected test quality or if the report selected sequences differently under the competing states. Specify what was observed before calculating its likelihood.
Switch to Three positives and compare its final row with the mixed sequence. The odds after successive positives are one third, one, and three, so probabilities are 25%, 50%, and 75%. Repeated positive evidence has increased the probability at every step, but the increments are not constant. The strips measure probabilities, not likelihood ratios. Next step and the static stage list connect rows to odds calculations. Playback speed changes no probability.
As a paper comparison, put the negative first while retaining two positives. Conditional independence and unchanged test rates give the same final product of ratios, hence the same final posterior. The intermediate probabilities differ because the information arrives in a different order. The player does not offer that rearranged scenario. Its Reset keeps the chosen scenario and returns to the prior, while changing scenarios replaces the sequence. Neither operation simulates collecting a fresh independent dataset, and repeatedly replaying the same results must not count as further evidence.
The likelihood of data under one hypothesis is not its posterior probability. Another hypothesis may make those data even more likely. For composite hypotheses, different parameter values can predict different data; a Bayesian comparison must specify how their likelihoods are averaged under each hypothesis’s prior.
A prior probability of exactly zero stays zero under finite likelihood ratios. Do not use zero to mean “unlikely” if later evidence should be able to change your mind. Also inspect whether the alternatives cover the relevant possibilities. Confidence among two bad models does not establish that either is adequate.
A likelihood ratio can favor H even when the observation is unlikely under H. Suppose a report has probability 2% under H and 0.1% under the alternative. Its ratio is twenty. With prior probability 1%, the prior odds are one to ninety-nine; after the report they become twenty to ninety-nine. The posterior is twenty divided by 119, about 16.81%. Strong relative evidence increased a very small prior substantially without making H the most probable state. The likelihood of 2% was never itself a posterior probability.
The reverse limitation matters too. A report can be common under H but almost equally common under the alternative. Probabilities 90% and 80% give ratio 1.125, only a modest change in relative support. Starting from 1%, the positive weights in 10,000 reference cases are 90 under H and 7,920 under the alternative. The posterior is 90 divided by 8,010, about 1.12%. Saying that H predicted the report well omits the competing explanation, which predicted it almost as well.
| Report model | Likelihood under H | Under alternative | Ratio | Posterior at prior 1% |
|---|---|---|---|---|
| Rare but contrasting | 2% | 0.1% | 20 | 20/119 ≈ 16.81% |
| Common under both | 90% | 80% | 1.125 | 90/8,010 ≈ 1.12% |
An alternative may itself contain several mechanisms. Suppose H describes a fault, while the alternative includes ordinary sound operation and contaminated sound samples. For a constructed alternative population, let 90% be ordinary sound cases with positive chance 10%, and 10% be contaminated cases with positive chance 90%. Averaging over these subgroups gives alternative positive probability 18%. If H has positive probability 60%, the positive likelihood ratio is about 3.33. Comparing only against ordinary sound cases would give six and would omit an important route to the observation.
That average requires relevant subgroup weights. If contamination rises to 30% within the alternative, its positive probability becomes 70% of 10% plus 30% of 90%, or 34%. The ratio falls to about 1.76. This is not a change in Bayes' rule. It is a change in what the alternative predicts. When a hypothesis includes uncertain parameter values, averaging their predictions requires specified weights in the Bayesian model. Choosing only whichever parameter makes the observation most likely is a different calculation, not that weighted prediction.
A complete calculation does not prove that the alternatives cover reality. An unexpected failure mode may produce the same report through a route absent from both models. Compare predicted patterns beyond the one result used for updating. If both models predict that instrument disagreements are rare but disagreements occur often, inspect shared assumptions about measurement. A large posterior within two stipulated models measures their relative support under those assumptions. It does not supply a probability for an omitted mechanism unless the model represents that mechanism.
Finally, distinguish updating from testing and from choosing actions. A small probability of data under a null model does not supply posterior odds without alternative likelihoods and prior weights. A test's size or power describes rejection frequencies conditional on specified generating models, not the chance a reported conclusion is wrong. A posterior informs a decision, but losses determine whether risk warrants action. Each calculation needs different inputs. A likelihood ratio alone supplies neither a posterior error probability nor an action threshold.
Starting at probability 0.2, apply two conditionally independent likelihood ratios of four. What is the posterior?
Convert 0.2 to odds before multiplying.
| Stage | Odds | Probability |
|---|---|---|
| Prior | 1/4 | 20% |
| First result | 1 | 50% |
| Second result | 4 | 80% |
Initial odds are 0.2/0.8 = 1/4. Multiplication by 4 × 4 gives odds 4. The posterior is 4/5 = 80%. Multiplying probability itself by sixteen would give an invalid number above one.
One original report has likelihood ratio three. Five websites copy it exactly. Should the prior odds be multiplied by three to the fifth power?
Condition on the original report before asking what the copies add.
| Evidence | Conditional ratio |
|---|---|
| Original report | 3 |
| Each guaranteed copy | 1 |
| Combined ratio | 3, not 243 |
No. If the copies are guaranteed reproductions, they add no information about H after the original is known. Their conditional likelihood ratios are one. Apply the original ratio once. Independent investigations would require a different model and evidence of independence.
The explanations and exercises are complete without these optional links. The references distinguish probability identities from inference procedures and reporting guidance. The examples are constructed teaching models, not empirical estimates or professional decision advice.
Prior, likelihood, and posterior in reliability inference. Decision losses here are stipulated teaching assumptions, not estimated consequences.
Addition, multiplication, conditional probability, and the role of independence.