Bayes Factor Calculator
The formula
What the relative evidence tells you
The Bayes factor compares how predicted the observed data are by two competing hypotheses — evidence independent of any significance threshold.
Calculation: BF₁₂ = P(data|H1) ∕ P(data|H2); posterior odds = prior odds × BF. Enter your inputs and every substitution is shown below the result.
Reading the result
BF=9 means the data are 9× as likely if H1 is true; combined with even priors, belief moves to 90%.
Rules of thumb: BF ≥ 10 for strong, ≥ 30 for very strong evidence claims
Limitations and best practice
Treating a BF computed from sloppy likelihoods as gospel; the factor is only as good as both models’ predictions.
This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Frequently asked questions
Does BF handle the null better than p?
Why do Bayes factors need priors?
Is BF₁₂ of 3 strong?
Source & methodology
Jeffreys (1961) Theory of Probability; Kass & Raftery (1995).
How Bayes Factor works — and what your result really means
This page was written for one job: turning your own inputs into a defensible figure, with the working shown. Searchers land on a Bayes Factor page for three reasons — to plan, to compare options, and to check that somebody else’s arithmetic is honest. This tool serves all three, and everything after the calculator explains the model instead of hiding it.
Research arithmetic demands traceability, and Bayes Factor delivers it: every substitution of BF₁₂ = P(data|H1) ∕ P(data|H2); posterior odds = prior odds × BF is printed, so 4.00 can be reproduced, cited and defended — the three things a bare calculator output cannot promise.
In Statistics & Data Science, units and significant figures carry meaning; keeping them attached to every output is the quiet professional standard this page maintains.
Nothing below is stock text pasted around a number: the worked example computes bayes factor BF₁₂ = 4.00 from this tool’s own defaults, and the sensitivity section ranks the inputs that move the result — Likelihood of data under H2 first, at ≈11.1 % of the headline for a ten-percent error either way in that input. Work down the page and by the end you could re-derive the result in a spreadsheet from memory.
Every input in Bayes Factor, and what it does to the answer
The engine implements a single stated model: BF₁₂ = P(data|H1) ∕ P(data|H2); posterior odds = prior odds × BF. There is no hidden coefficient and no login-gated precision — the formula printed is the formula executed, which is also what makes the result citable.
Prior odds H1:H2 (example value 1) is a plain measured quantity — the unit shown in the label is the unit the formula expects. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
Likelihood of data under H1 (example value 0) is a quantity rather than a category — precision here is cheap and worth having. This input feeds the formula directly; a defensible figure here is worth more than false precision anywhere else.
Likelihood of data under H2 (example value 0) is a measured number in the unit named by the label; the formula does not convert for you. The example value is there to demonstrate the mechanics, not to describe your situation — overwrite it.
Bayes Factor on the example inputs: the complete substitution
The tool prints its own working; this is the same arithmetic on the example values, reproduced so you can follow it by hand. Starting figures: Prior odds H1:H2, Likelihood of data under H1, Likelihood of data under H2. Run to the end, the bayes factor BF₁₂ resolves to 4.00 — Moderate evidence for H1 on the bands this tool uses.
- BF₁₂ = 0.080 ÷ 0.020 = 4.00
- Posterior odds = prior 1.00 × 4.00 = 4.00
- Jeffreys: 3 moderate, 10 strong, 30 very strong, 100 decisive
Posterior odds H1:H2 — 4.00 : 1. This is the number experienced users quote when the headline alone would mislead.
Posterior probability H1 — 80.0 %. Read next to the headline, this is the figure that shows the shape of the trade-off.
Evidence in decibels — 6.0 dB. This is the number experienced users quote when the headline alone would mislead.
Multiplier still needed for BF 10 — 2.50 ×. If your inputs change, this is usually the first companion figure to move.
Keep this worked example open while you enter your own figures: matching the shape of the substitution is the fastest way to spot a unit or convention mistake.
The Bayes Factor corridor: your result across a realistic band
Read this table as the practical error bar on the headline figure. It answers the question people actually have after seeing a single number: how wrong could this be if the main input is off by ten or twenty percent?
| Likelihood of data under H2 setting | Bayes factor BF₁₂ |
|---|---|
| 80 % of the example value | 5.00 |
| 90 % of the example value | 4.44 |
| example inputs | 4.00 |
| 110 % of the example value | 3.64 |
| 120 % of the example value | 3.33 |
Across 5 settings of Likelihood of data under H2, the result spans 3.33 to 5.00. A decision that only works at one row of this table is a decision waiting for an input to move. Reproduce it in the form above by editing Likelihood of data under H2 alone; the substitution steps will follow your figures exactly.
The levers behind Bayes Factor, ranked by effect
This list was computed for this specific engine just now: each numeric input was nudged ±10 % while every other field held its example value, and the bayes factor BF₁₂ was recalculated from scratch. It is the fastest honest answer to “what matters most in Bayes Factor?” — and it is specific to these defaults, because sensitivity moves with the starting point.
- Likelihood of data under H2: ten percent up gives 3.64, ten percent down gives 4.44 — ≈11.1 % of the bayes factor BF₁₂ either way
- Nudge Likelihood of data under H1 by ±10 % and Bayes Factor lands between 3.60 and 4.40 (sensitivity ≈10.0 %)
Likelihood of data under H2 is the dominant lever here: ten percent either way moves the result by about 11.1 %, from 3.64 to 4.44. Measure that input from a document rather than from memory, and treat Likelihood of data under H1 as the second priority (≈10.0 % per ten percent). The remaining fields can sit at safe defaults without materially changing the answer — knowing which is which is what separates a careful calculation from an anxious one.
Is your Bayes Factor good — reading the result bands
Nobody acts on a bare figure — they act on a figure with a comparison attached. The verdict attached to the default calculation — “Moderate evidence for H1” — is one band on a ladder your own figure may sit somewhere else on. Before treating the band as fact, note its width: ten percent either way on Likelihood of data under H2 already spans 3.64 to 4.44. The practical reading rules: compare against your own history before comparing against other people; treat band boundaries as zones, not walls — one point on the wrong side of a threshold rarely changes a real outcome; and when a companion output contradicts the headline, believe the pair.
Set Bayes Factor beside its neighbours and it usually plays a specific role: this figure establishes the size of something, and the neighbouring tools establish whether that size is acceptable. Skipping the first step makes the second one arbitrary.
The errors that quietly break a Bayes Factor figure
Treating a BF computed from sloppy likelihoods as gospel; the factor is only as good as both models’ predictions. This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Reusing a Bayes Factor from last season. In Science & Research the inputs are the thing that drifts; the formula does not.
One mistake is specific to this engine: assuming every input matters equally. The computation above shows Likelihood of data under H2 dominating — a careful figure there beats false precision everywhere else.
Another is comparing across contexts: a Bayes Factor computed with one set of conventions should never be lined up against a figure computed with another. Check the inputs behind both before ranking them.
Before you trust your Bayes Factor: a short checklist
- Verify each input against its original source document, not memory.
- Re-run the calculation with a ±10 % stress on the two most sensitive inputs.
- Re-verify Likelihood of data under H2 against a source document — ten percent of error there moves the headline by ≈11.1 %.
- Check that every field the formula needs actually arrived from a document, not from recollection.
- State, in one sentence, which convention each rate, unit or period in the form is using.
- Run the ±10 % stress on the top two levers and check that your conclusion survives both.
Five steps from raw inputs to a figure you can defend
The tool does the arithmetic; the routine below is what makes the arithmetic trustworthy. Five steps, a minute total — and afterwards you will know not just what the number is, but which assumption produced it.
Step 1 — collect, then type. Everything this tool asks for is listed above (prior odds h1:h2, likelihood of data under h1, likelihood of data under h2). Pull each figure from something you can point at — paperwork, a dashboard, a label — before touching a field. Guesswork at the input stage cannot be repaired at the interpretation stage. A calculation you cannot explain to somebody else is a calculation you cannot defend.
Step 2 — start from the case you can defend. Enter pessimistic-but-honest figures first; the optimistic run is a ceiling, not a plan. A conclusion that holds on conservative inputs holds everywhere. Half of a good calculation is knowing which input you would hate to be wrong about.
Step 3 — measure what deserves measuring. The ranked levers above tell you where a careful figure pays off. Everything else can stay approximate, which is a licence to stop hunting for data you do not need. A result worth acting on survives having its two shakiest inputs halved.
Step 4 — ask what would have to be true. Take the result you want and work backwards through the inputs until you find the assumption that produces it; then decide honestly whether that assumption is defensible. Note the assumption you are least sure about; that is where the real risk lives.
Step 5 — schedule the next run. In Science & Research the inputs drift quietly. Put a reminder a quarter out, re-enter the same fields, and read the change rather than the level. A calculation you cannot explain to somebody else is a calculation you cannot defend.
The real-world conversations Bayes Factor settles
In Statistics & Data Science the figure computed here is usually the first one somebody asks to see. Having it — with the formula behind it — changes your position in the conversation from participant to reference. Say the answer out loud before you save it; if it sounds wrong, the model did not change, your understanding did.
The audience for this page is people who need one trustworthy number before their next decision. In professional Statistics & Data Science practice the pattern is always the same — screen with Bayes Factor, then invest slow analysis only in whatever survives the screen. The test of a good estimate is not that it feels right but that it changes what you do.
Nothing here is stored or transmitted, so the habit of checking is free: open the page, re-enter current figures, compare with the number you saved last time. The test of a good estimate is not that it feels right but that it changes what you do.
Bayes Factor — frequently asked questions
What affects Bayes Factor the most?
The short answer: Likelihood of data under H2. The longer answer is the sensitivity table on this page, where every numeric input was probed by ten percent and the bayes factor BF₁₂ recalculated — Likelihood of data under H2 moved it furthest, by roughly 11.1 % (3.64 to 4.44).
Which matters more: likelihood of data under h1 or likelihood of data under h2?
For Bayes Factor, Likelihood of data under H2 wins the head-to-head (≈11.1 % against ≈10.0 % per ten-percent change). Knowing which input dominates is the difference between careful work and merely slow work.
What is a typical Bayes Factor?
There is no honest universal average for Bayes Factor, but there is a reference point: the defaults on this page produce 4.00 (Moderate evidence for H1). Replace the inputs with your own and compare — that comparison is the useful meaning of “typical”.
How do I calculate Bayes Factor by hand?
BF₁₂ = P(data|H1) ∕ P(data|H2); posterior odds = prior odds × BF is the entire method. Work the inputs in that order — BF₁₂ = 0.080 ÷ 0.020 = 4.00 — and you will reach 4.00 on the default figures, the same number this tool prints.
What happens to Bayes Factor if likelihood of data under h2 changes by 10 %?
Try it in the form above and the printed substitution will follow you; the pre-computed answer for the defaults is 3.64 to 4.44 across the ten-percent probe, i.e. ≈11.1 % sensitivity on the bayes factor BF₁₂.
Is the Bayes Factor result of 4.00 good or bad?
Good and bad are comparisons, so the honest test is three questions: where does your own Bayes Factor sit against the 4.00 reference above, how far would a plausible input error move it (≈11.1 % per ten percent on Likelihood of data under H2), and what does the alternative option compute to?
How many significant figures are reliable in Bayes Factor?
About as many as your least precise input. The tool computes at full precision and displays rounded output; report 4.00 to a decimal place your inputs can actually support.
Can Bayes Factor be negative or empty?
Yes, and that is informative rather than broken: a negative Bayes Factor usually means the inputs encode an outflow or a shortfall, and the verdict band above labels exactly that situation instead of hiding it.
Does BF handle the null better than p?
Yes — BF below 1 is positive evidence FOR the null, whereas a big p-value is only “inconclusive.”
Why do Bayes factors need priors?
The likelihoods embed them; sensitivity analysis over plausible priors is standard practice.
Can this tool see the numbers I type?
It cannot. There is no submission step: the calculation happens in your browser and the result disappears when you close the page. Treat the fields above as a scratchpad rather than a form.
Key terms on this page
- Posterior odds H1:H2
- 4.00 : 1 in the default Bayes Factor calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Posterior probability H1
- 80.0 % in the default Bayes Factor calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Evidence in decibels
- 6.0 dB for the Bayes Factor worked case; it sits next to the main result because it carries context the headline leaves out.
- Bayes factor BF₁₂
- The primary output of this tool: 4.00 for the default inputs, computed exactly from BF₁₂ = P(data|H1) ∕ P(data|H2); posterior odds = prior odds × BF.
- Worked example
- The step-by-step substitution printed above, so the arithmetic can be re-derived by hand or in a spreadsheet.
- Verdict band
- The plain-language label — here “Moderate evidence for H1” — mapping the computed figure onto experience-based ranges for Science & Research.
- Sensitivity (±10 %)
- How far the Bayes Factor moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
The provenance of this Bayes Factor calculation
Everything above runs on one stated formula, applied without hidden coefficients: BF₁₂ = P(data|H1) ∕ P(data|H2); posterior odds = prior odds × BF. Source of record: Jeffreys (1961) Theory of Probability; Kass & Raftery (1995). Any other implementation of this model should return the same figure for the same inputs. If one does not, the difference is in its assumptions — which this page prints and that one probably does not.
This page is an estimation aid, not professional advice. The design rule is simple: two people entering identical figures must get identical answers. The promise to the reader is narrow and testable: same inputs, same output, no silent rounding, and no figure presented without the formula that produced it.
What to do with your Bayes Factor next
Save the number together with its inputs and today’s date; a Bayes Factor figure without context is unusable the moment an input changes.
The audience here is people who need one trustworthy number before their next decision. Everything you need to defend the result — formula, substitution, sensitivity, source — is printed above, so the only remaining variable is the honesty of your inputs.