The Central Idea
Suppose \(H\) is a hypothesis and \(E\) is observed evidence.
Bayesian updating can be written in odds form as
\[ \boxed{ \text{posterior odds} = \text{prior odds} \times \text{likelihood ratio} }. \]Symbolically,
\[ \boxed{ \frac{P(H\mid E)}{P(H^c\mid E)} = \frac{P(H)}{P(H^c)} \cdot \frac{P(E\mid H)}{P(E\mid H^c)} }. \]Probability and Odds Are Different Representations
A probability compares an event with the entire sample space.
Odds compare the event directly with its complement.
If
\[ P(A)=p, \]then
\[ P(A^c)=1-p. \]The odds in favor of \(A\) are
\[ \boxed{ \frac{p}{1-p} }. \]Odds in Favor
If
\[ P(A)=0.75, \]then
\[ P(A^c)=0.25. \]The odds in favor of \(A\) are
\[ \frac{0.75}{0.25}=3. \]This may be written
\[ \boxed{3:1}. \]For every one unit of probability mass assigned to \(A^c\), three units are assigned to \(A\).
Odds Against
The odds against \(A\) reverse the comparison:
\[ \frac{P(A^c)}{P(A)}. \]If the odds in favor of \(A\) are
\[ 3:1, \]then the odds against \(A\) are
\[ \boxed{1:3}. \]Convert Probability to Odds
If
\[ P(A)=p, \]then the odds in favor of \(A\) are
\[ \boxed{ O(A)=\frac{p}{1-p} }. \]Here \(O(A)\) denotes the odds in favor of \(A\).
Example: Probability to Odds
Suppose
\[ P(A)=0.60. \]Then
\[ P(A^c)=0.40. \]Therefore,
\[ O(A)=\frac{0.60}{0.40}=1.5. \]The odds are
\[ \boxed{1.5:1} \]or equivalently
\[ \boxed{3:2}. \]Convert Odds to Probability
Suppose the odds in favor of \(A\) are
\[ r:1. \]Then
\[ \boxed{ P(A)=\frac{r}{1+r} }. \]More generally, if the odds are
\[ a:b, \]then
\[ \boxed{ P(A)=\frac{a}{a+b} }. \]Example: Odds to Probability
Suppose the odds in favor of \(A\) are
\[ 4:1. \]Then
\[ P(A)=\frac{4}{4+1}. \]Therefore,
\[ \boxed{P(A)=0.80}. \]Another Odds-to-Probability Example
Suppose the odds in favor of \(A\) are
\[ 2:3. \]Then
\[ P(A)=\frac{2}{2+3}. \]Thus,
\[ \boxed{P(A)=0.40}. \]Prior Odds
If \(H\) is a hypothesis, its prior probability is
\[ P(H). \]The corresponding prior odds are
\[ \boxed{ O_{\text{prior}}(H) = \frac{P(H)}{P(H^c)} }. \]Since
\[ P(H^c)=1-P(H), \]we can also write
\[ O_{\text{prior}}(H) = \frac{P(H)}{1-P(H)}. \]Posterior Odds
After observing evidence \(E\), the posterior probability is
\[ P(H\mid E). \]The corresponding posterior odds are
\[ \boxed{ O_{\text{post}}(H\mid E) = \frac{P(H\mid E)}{P(H^c\mid E)} }. \]The Likelihood Ratio
The likelihood ratio compares how likely the observed evidence is under two competing hypotheses.
For \(H\) versus \(H^c\),
\[ \boxed{ LR = \frac{P(E\mid H)}{P(E\mid H^c)} }. \]It measures how much more strongly the evidence is associated with \(H\) than with its alternative.
Interpreting a Likelihood Ratio Greater Than One
If
\[ LR>1, \]then
\[ P(E\mid H)>P(E\mid H^c). \]The observed evidence is more likely under \(H\) than under \(H^c\).
Therefore, the evidence increases the odds in favor of \(H\).
Interpreting a Likelihood Ratio Less Than One
If
\[ 0The evidence therefore lowers the odds in favor of \(H\).
Interpreting a Likelihood Ratio Equal to One
If
\[ LR=1, \]then
\[ P(E\mid H)=P(E\mid H^c). \]The evidence is equally likely under both possibilities.
It does not change the odds:
\[ \boxed{ O_{\text{post}}=O_{\text{prior}} }. \]Deriving the Odds Form of Bayes' Theorem
Bayes' theorem gives
\[ P(H\mid E) = \frac{P(E\mid H)P(H)}{P(E)} \]and
\[ P(H^c\mid E) = \frac{P(E\mid H^c)P(H^c)}{P(E)}. \]Divide the first equation by the second:
\[ \frac{P(H\mid E)}{P(H^c\mid E)} = \frac{P(E\mid H)P(H)} {P(E\mid H^c)P(H^c)}. \]Separate the Two Ratios
Rewrite:
\[ \frac{P(H\mid E)}{P(H^c\mid E)} = \frac{P(H)}{P(H^c)} \cdot \frac{P(E\mid H)}{P(E\mid H^c)}. \]Therefore,
\[ \boxed{ \text{posterior odds} = \text{prior odds} \times \text{likelihood ratio} }. \]Why the Evidence Probability Cancels
In ordinary Bayes' theorem, both posterior calculations contain the same denominator:
\[ P(E). \]When posterior odds are formed, that denominator appears in both the numerator and denominator and cancels.
This is one reason odds form can be computationally convenient.
A First Odds Update
Suppose the prior probability of \(H\) is
\[ P(H)=0.20. \]Then
\[ P(H^c)=0.80, \]so the prior odds are
\[ \frac{0.20}{0.80} = \frac{1}{4}. \]The prior odds are therefore
\[ \boxed{1:4}. \]Apply a Likelihood Ratio
Suppose the observed evidence has likelihood ratio
\[ LR=3. \]Then the posterior odds are
\[ \frac{1}{4}\cdot3 = \frac{3}{4}. \]Thus the posterior odds are
\[ \boxed{3:4}. \]Convert the Posterior Odds Back to Probability
Odds of
\[ 3:4 \]correspond to probability
\[ \frac{3}{3+4}. \]Therefore,
\[ \boxed{ P(H\mid E)=\frac{3}{7}\approx0.4286 }. \]The evidence raises the probability from \(20\%\) to about \(42.86\%\).
Odds Updates Multiply Evidence Strength
If the likelihood ratio is \(3\), the evidence multiplies the prior odds by \(3\).
If the prior odds are
\[ 1:4, \]the posterior odds become
\[ 3:4. \]The likelihood ratio acts directly as an odds multiplier.
Example: Evidence That Lowers the Odds
Suppose the prior odds in favor of \(H\) are
\[ 2:1. \]Suppose the likelihood ratio is
\[ LR=0.25. \]Then the posterior odds are
\[ 2(0.25)=0.50. \]Therefore, the posterior odds are
\[ \boxed{1:2}. \]Convert the Lowered Odds to Probability
Odds of
\[ 1:2 \]correspond to probability
\[ \frac{1}{1+2}. \]Therefore,
\[ \boxed{ P(H\mid E)=\frac{1}{3} }. \]The evidence has shifted the balance against \(H\).
Example: Evidence with No Effect
Suppose the prior odds are
\[ 3:2 \]and the likelihood ratio is
\[ 1. \]Then
\[ 3:2\times1=3:2. \]The posterior odds equal the prior odds because the evidence is equally likely under both hypotheses.
Likelihood Ratio from Conditional Probabilities
Suppose
\[ P(E\mid H)=0.80 \]and
\[ P(E\mid H^c)=0.20. \]Then
\[ LR = \frac{0.80}{0.20} = \boxed{4}. \]The evidence is four times as likely under \(H\) as under \(H^c\).
Likelihood Ratio Is Relative Evidence
A likelihood value by itself, such as
\[ P(E\mid H)=0.80, \]does not tell us how strongly \(E\) distinguishes \(H\) from alternatives.
We also need the evidence probability under the competing hypothesis.
The likelihood ratio performs that comparison directly.
Example: High Likelihood but Weak Evidence
Suppose
\[ P(E\mid H)=0.90 \]and
\[ P(E\mid H^c)=0.85. \]Then
\[ LR = \frac{0.90}{0.85} \approx \boxed{1.0588}. \]Although \(E\) is very likely under \(H\), it is almost as likely under \(H^c\), so it provides only modest evidence favoring \(H\).
Example: Moderate Likelihood but Strong Evidence
Suppose
\[ P(E\mid H)=0.40 \]and
\[ P(E\mid H^c)=0.02. \]Then
\[ LR = \frac{0.40}{0.02} = \boxed{20}. \]The evidence is not especially common under \(H\), but it is far rarer under \(H^c\). Relative to the alternative, it strongly favors \(H\).
Prior Odds Can Counter Strong Evidence
Suppose the prior probability of \(H\) is only
\[ 0.01. \]Then the prior odds are
\[ \frac{0.01}{0.99} \approx 0.0101. \]Even a large likelihood ratio may not make the posterior probability close to \(1\) when the prior odds are extremely small.
Example: Rare Hypothesis with Strong Evidence
Suppose
\[ P(H)=0.01 \]and
\[ LR=20. \]Prior odds:
\[ \frac{0.01}{0.99}\approx0.010101. \]Posterior odds:
\[ 0.010101(20)\approx0.20202. \]Convert the Rare-Hypothesis Posterior Odds
Convert odds \(0.20202:1\) to probability:
\[ P(H\mid E) = \frac{0.20202}{1+0.20202}. \]Therefore,
\[ \boxed{ P(H\mid E)\approx0.1681 }. \]A likelihood ratio of \(20\) raises the probability substantially, but the posterior remains well below \(50\%\) because the prior was only \(1\%\).
Posterior Odds Can Be Converted Directly
If the posterior odds are represented by a positive number \(O\), then
\[ \boxed{ P(H\mid E)=\frac{O}{1+O} }. \]Conversely,
\[ \boxed{ O=\frac{P(H\mid E)}{1-P(H\mid E)} }. \]Example: Posterior Odds of 5
Suppose
\[ O_{\text{post}}=5. \]Then
\[ P(H\mid E) = \frac{5}{1+5} = \boxed{\frac{5}{6}} \approx \boxed{0.8333}. \]Sequential Evidence in Odds Form
Odds form is especially useful when evidence arrives in stages.
If \(E_1\) produces likelihood ratio \(LR_1\) and \(E_2\) later produces likelihood ratio \(LR_2\), then under an appropriate model,
\[ O_{\text{post}} = O_{\text{prior}} \cdot LR_1 \cdot LR_2. \]Additional evidence can continue multiplying the current odds.
Conditional Independence of Evidence Matters
Multiplying separate likelihood ratios requires a model that justifies the factorization of the evidence probabilities.
If pieces of evidence are conditionally independent given each competing hypothesis, then their combined likelihood ratio factors naturally.
If they are dependent, multiplying isolated likelihood ratios without adjustment can double count information.
Example: Two Sequential Likelihood Ratios
Suppose prior odds are
\[ 1:3. \]Evidence \(E_1\) has
\[ LR_1=4, \]and later evidence \(E_2\) has
\[ LR_2=2. \]Then the posterior odds are
\[ \frac{1}{3}\cdot4\cdot2 = \frac{8}{3}. \]Thus the final odds are
\[ \boxed{8:3}. \]Convert the Sequential Posterior Odds
Odds of
\[ 8:3 \]correspond to probability
\[ \frac{8}{8+3}. \]Therefore,
\[ \boxed{ P(H\mid E_1,E_2)=\frac{8}{11}\approx0.7273 }. \]Order of Independent Evidence Updates
When likelihood-ratio multiplication is valid,
\[ LR_1LR_2=LR_2LR_1. \]Therefore, mathematically, the final odds are the same regardless of the order in which those likelihood ratios are multiplied.
The modeling assumptions governing the evidence remain essential.
Comparing Two Explicit Hypotheses
Odds need not always compare \(H\) with \(H^c\).
For two explicit hypotheses \(H_1\) and \(H_2\), prior odds are
\[ \frac{P(H_1)}{P(H_2)}. \]The likelihood ratio is
\[ \frac{P(E\mid H_1)}{P(E\mid H_2)}. \]Posterior odds are
\[ \boxed{ \frac{P(H_1\mid E)}{P(H_2\mid E)} = \frac{P(H_1)}{P(H_2)} \cdot \frac{P(E\mid H_1)}{P(E\mid H_2)} }. \]Example: Comparing Two Sources
Suppose
\[ P(H_1)=0.60, \qquad P(H_2)=0.40. \]The prior odds favoring \(H_1\) are
\[ \frac{0.60}{0.40}=1.5. \]Suppose
\[ P(E\mid H_1)=0.20, \qquad P(E\mid H_2)=0.50. \]Then
\[ LR = \frac{0.20}{0.50} = 0.40. \]Update the Source Odds
Posterior odds favoring \(H_1\) are
\[ 1.5(0.40)=0.60. \]Thus the posterior odds are
\[ 0.60:1 \]or
\[ \boxed{3:5}. \]The evidence reverses the initial advantage and now favors \(H_2\).
Convert the Source Odds to Posterior Probabilities
Odds of
\[ 3:5 \]imply
\[ P(H_1\mid E)=\frac{3}{8} \]and
\[ P(H_2\mid E)=\frac{5}{8}. \]Therefore,
\[ \boxed{ P(H_1\mid E)=0.375, \qquad P(H_2\mid E)=0.625 }. \]Likelihood Ratios and Evidence Strength
A likelihood ratio tells us how many times more likely the evidence is under one hypothesis than another.
For example:
\[ LR=5 \]means the evidence is five times as likely under the numerator hypothesis as under the denominator hypothesis.
This statement is about the evidence model, not directly about posterior probability.
Reciprocal Likelihood Ratios
If the likelihood ratio favoring \(H_1\) over \(H_2\) is
\[ LR_{12} = \frac{P(E\mid H_1)}{P(E\mid H_2)}, \]then the likelihood ratio favoring \(H_2\) over \(H_1\) is
\[ LR_{21} = \frac{P(E\mid H_2)}{P(E\mid H_1)} = \frac{1}{LR_{12}}. \]Reversing the comparison takes the reciprocal.
Example: Reciprocal Likelihood Ratios
If
\[ LR_{12}=4, \]then
\[ LR_{21}=\frac{1}{4}=0.25. \]The same evidence can be described as four times more likely under \(H_1\), or one-quarter as likely under \(H_2\) relative to \(H_1\).
Likelihood Ratios Do Not Depend on Prior Odds
The likelihood ratio
\[ \frac{P(E\mid H)}{P(E\mid H^c)} \]depends on the evidence model.
It does not depend on how common \(H\) was before the evidence.
Prior odds and likelihood ratio play separate roles in the update.
The Same Evidence Can Produce Different Posteriors
Suppose two populations use the same evidence model, so both have
\[ LR=5. \]If the prior odds differ, the posterior odds will also differ:
\[ O_{\text{post}}=O_{\text{prior}}\cdot5. \]The same evidence strength can therefore lead to different posterior probabilities in populations with different base rates.
Example: Same Evidence, Different Priors
Population 1 has prior probability
\[ P(H)=0.50. \]Its prior odds are
\[ 1:1. \]With
\[ LR=5, \]the posterior odds are
\[ 5:1. \]Therefore,
\[ P(H\mid E)=\frac{5}{6}\approx0.8333. \]Second Population with the Same Evidence
Population 2 has prior probability
\[ P(H)=0.10. \]The prior odds are
\[ 1:9. \]After multiplying by
\[ LR=5, \]the posterior odds become
\[ 5:9. \]Therefore,
\[ P(H\mid E)=\frac{5}{14}\approx0.3571. \]The evidence is identical, but the posterior differs because the prior odds differ.
Odds Form Avoids Recomputing the Evidence Probability
In probability form, Bayes' theorem often requires calculating
\[ P(E). \]In odds form, the common evidence probability cancels.
This can make comparisons between two hypotheses more efficient.
Odds Are Multiplicative
Probability updates can appear nonlinear when written directly in terms of probabilities.
In odds form, Bayesian evidence updates are multiplicative:
\[ O_{\text{post}} = O_{\text{prior}}LR. \]This structure is especially useful for repeated evidence updates.
Likelihood Ratios Near One
A likelihood ratio near
\[ 1 \]changes the odds only slightly.
For example,
\[ LR=1.05 \]increases the odds by only \(5\%\).
Evidence may be common without being strongly discriminating.
Very Large or Very Small Likelihood Ratios
A large likelihood ratio strongly increases the odds for the numerator hypothesis.
A very small likelihood ratio strongly decreases them.
For example,
\[ LR=50 \]multiplies the prior odds by \(50\), while
\[ LR=0.02 \]multiplies them by only \(2\%\) of their original value.
Likelihood Ratio of Zero
If
\[ P(E\mid H)=0 \]while
\[ P(E\mid H^c)>0, \]then
\[ LR=0. \]Under the model, observing \(E\) eliminates \(H\) because \(E\) is impossible if \(H\) is true.
Extremely Strong Evidence Still Depends on the Model
A large likelihood ratio is meaningful only to the extent that the conditional probabilities
\[ P(E\mid H) \]and
\[ P(E\mid H^c) \]accurately describe how the evidence is generated.
Model error can produce misleading likelihood ratios.
Common Error: Treating Odds as Probability
Odds of
\[ 3:1 \]do not mean probability \(3\).
They correspond to probability
\[ \frac{3}{3+1} = \boxed{0.75}. \]Common Error: Converting Odds Incorrectly
If odds are
\[ a:b, \]the probability is
\[ \frac{a}{a+b}, \]not
\[ \frac{a}{b}. \]The ratio \(a/b\) is the odds value itself.
Common Error: Reversing the Likelihood Ratio
The ratio
\[ \frac{P(E\mid H)}{P(E\mid H^c)} \]favors \(H\) over \(H^c\).
Reversing numerator and denominator changes the direction of comparison and replaces the ratio by its reciprocal.
Common Error: Calling a Likelihood Ratio a Posterior Probability
A likelihood ratio can be greater than \(1\).
A posterior probability cannot.
For example,
\[ LR=8 \]means the evidence is eight times as likely under one hypothesis as another. It does not mean the posterior probability is \(8\).
Common Error: Ignoring Prior Odds
A likelihood ratio alone does not determine the posterior odds.
The correct update is
\[ O_{\text{post}} = O_{\text{prior}}LR. \]The same likelihood ratio can produce very different posterior probabilities from different starting odds.
Common Error: Multiplying Evidence Ratios Without Checking Dependence
Separate likelihood ratios can be multiplied only when the probability model supports the required factorization.
Correlated evidence may contain overlapping information.
Treating correlated evidence as conditionally independent can exaggerate the strength of the update.
Common Error: Interpreting Evidence Strength as Causation
A likelihood ratio compares probabilities of evidence under competing hypotheses.
It does not by itself establish that a hypothesis caused the evidence.
Causal interpretation requires additional assumptions and structure.
A Reliable Procedure
- Identify the hypothesis being favored in the odds numerator.
- Convert the prior probability to prior odds if necessary.
- Identify the observed evidence.
- Compute the likelihood under the numerator hypothesis.
- Compute the likelihood under the competing hypothesis.
- Form the likelihood ratio in the correct direction.
- Multiply prior odds by the likelihood ratio.
- Convert posterior odds back to probability if required.
- Interpret whether the evidence increased, decreased, or preserved the odds.
Example: Full Odds Workflow
Suppose
\[ P(H)=0.25. \]Then
\[ P(H^c)=0.75 \]and the prior odds are
\[ \frac{0.25}{0.75} = \frac{1}{3}. \]Suppose
\[ P(E\mid H)=0.60 \]and
\[ P(E\mid H^c)=0.20. \]The likelihood ratio is
\[ LR=\frac{0.60}{0.20}=3. \]Posterior odds:
\[ \frac{1}{3}\cdot3=1. \]Thus the posterior odds are
\[ 1:1, \]which corresponds to
\[ \boxed{P(H\mid E)=0.50}. \]Example: Full Two-Hypothesis Comparison
Suppose
\[ P(H_1)=0.70, \qquad P(H_2)=0.30. \]Prior odds favoring \(H_1\) are
\[ \frac{0.70}{0.30}=\frac{7}{3}. \]Suppose the evidence satisfies
\[ P(E\mid H_1)=0.15, \qquad P(E\mid H_2)=0.45. \]The likelihood ratio favoring \(H_1\) is
\[ \frac{0.15}{0.45}=\frac{1}{3}. \]Therefore, posterior odds are
\[ \frac{7}{3}\cdot\frac{1}{3} = \frac{7}{9}. \]Thus
\[ P(H_1\mid E) = \frac{7}{7+9} = \boxed{\frac{7}{16}} = \boxed{0.4375}. \]The evidence shifts the probability of \(H_1\) from \(70\%\) to \(43.75\%\).
Practice 1: Probability to Odds
Convert
\[ P(A)=0.80 \]to odds in favor of \(A\).
Practice 2: Odds to Probability
Convert odds
\[ 3:2 \]in favor of \(A\) to \(P(A)\).
Practice 3: Likelihood Ratio
Suppose
\[ P(E\mid H)=0.75 \]and
\[ P(E\mid H^c)=0.25. \]Find the likelihood ratio favoring \(H\).
Practice 4: Odds Update
Prior odds in favor of \(H\) are
\[ 1:4. \]The likelihood ratio is
\[ 6. \]Find the posterior odds.
Practice 5: Convert Posterior Odds
Convert posterior odds
\[ 3:2 \]to posterior probability.
Practice 6: Evidence Against a Hypothesis
Prior odds in favor of \(H\) are
\[ 4:1. \]The likelihood ratio is
\[ 0.20. \]Find the posterior odds and posterior probability.
Practice 7: Same Evidence, Different Priors
A likelihood ratio equals \(4\).
Population A has prior odds
\[ 1:1, \]while Population B has prior odds
\[ 1:9. \]Find the posterior odds for both populations.
Practice 8: Sequential Evidence
Prior odds are
\[ 1:2. \]Two conditionally independent pieces of evidence have likelihood ratios
\[ 3 \]and
\[ 2. \]Find the final posterior odds.
Practice 9: Reciprocal Likelihood Ratio
If the likelihood ratio favoring \(H_1\) over \(H_2\) is
\[ 5, \]find the likelihood ratio favoring \(H_2\) over \(H_1\).
Practice 10: Identify the Error
A student calculates
\[ LR=7 \]and concludes
\[ P(H\mid E)=0.70. \]Explain why the conclusion does not follow from the likelihood ratio alone.
Practice Answers
Practice 1
The complement probability is
\[ 0.20. \]Therefore,
\[ O(A)=\frac{0.80}{0.20}=4. \]The odds are
\[ \boxed{4:1}. \]Practice 2
\[ P(A)=\frac{3}{3+2}=\boxed{0.60}. \]Practice 3
\[ LR = \frac{0.75}{0.25} = \boxed{3}. \]Practice 4
The prior odds value is
\[ \frac{1}{4}. \]Multiply by \(6\):
\[ \frac{1}{4}\cdot6=\frac{3}{2}. \]Therefore, the posterior odds are
\[ \boxed{3:2}. \]Practice 5
\[ P(H\mid E) = \frac{3}{3+2} = \boxed{0.60}. \]Practice 6
Prior odds value:
\[ 4. \]Posterior odds:
\[ 4(0.20)=0.80. \]Thus the posterior odds are
\[ \boxed{4:5}. \]The posterior probability is
\[ \frac{4}{4+5} = \boxed{\frac{4}{9}} \approx \boxed{0.4444}. \]Practice 7
Population A:
\[ 1\cdot4=4, \]so the posterior odds are
\[ \boxed{4:1}. \]Population B:
\[ \frac{1}{9}\cdot4=\frac{4}{9}, \]so the posterior odds are
\[ \boxed{4:9}. \]Practice 8
\[ \frac{1}{2}\cdot3\cdot2=3. \]Therefore, the final posterior odds are
\[ \boxed{3:1}. \]Practice 9
\[ LR_{21}=\frac{1}{5}=\boxed{0.20}. \]Practice 10
A likelihood ratio measures how strongly the evidence favors one hypothesis over another. It is an odds multiplier, not a posterior probability.
The prior odds are also required:
\[ O_{\text{post}} = O_{\text{prior}}LR. \]Only after finding the posterior odds can they be converted into a posterior probability.
What to Remember
Convert probability \(p\) to odds with
\[ \boxed{ O=\frac{p}{1-p} }. \]Convert odds \(O\) back to probability with
\[ \boxed{ p=\frac{O}{1+O} }. \]The likelihood ratio comparing \(H\) with \(H^c\) is
\[ \boxed{ LR= \frac{P(E\mid H)} {P(E\mid H^c)} }. \]Bayesian updating in odds form is
\[ \boxed{ O_{\text{post}} = O_{\text{prior}} \times LR }. \]- Odds compare an event directly with its alternative.
- A likelihood ratio compares how likely the same evidence is under competing hypotheses.
- \(LR>1\) favors the numerator hypothesis.
- \(LR<1\) favors the denominator hypothesis.
- \(LR=1\) leaves the odds unchanged.
- The same likelihood ratio can produce different posteriors from different prior odds.
- Sequential likelihood ratios can multiply when the model justifies the required conditional independence.
- A likelihood ratio is not itself a posterior probability.
The central habit is: treat evidence as an odds multiplier, then convert back to probability only when needed.
Where We Go Next
Odds and likelihood ratios provide a compact way to measure how strongly evidence shifts probability between competing possibilities. A major application occurs when evidence comes from a test or screening procedure.
The next lesson, Diagnostic Testing, applies conditional probability, Bayes' theorem, base rates, false positives, false negatives, sensitivity, specificity, and related quantities to structured testing problems.