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Coauthor of research on proper scoring rules with Tilmann Gneiting and on Bayes factors with Robert E. Kass. These works concern evaluating probabilistic predictions and comparing statistical models.^[ref]

A Bayes factor measures how data change the relative support for two specified statistical models. Write $m_j(D)=p(D\mid H_j)$ for their Bayesian model evidence and define the direction explicitly:For positive finite $m_0(D),m_1(D)$, a value above one favors $H_1$ over $H_0$; a value below one favors $H_0$ over $H_1$. Reversing the comparison gives $B_{01}={\mathrm{BF}}_{01}=1/{\mathrm{BF}}_{10}$, the opposite direction used in some formulas below.

For mutually exclusive, exhaustive $H_0,H_1$ with positive prior probabilities, divide their two Bayes' theorem formulas. Their common data normalizer cancels, leavingThe Bayes factor therefore multiplies prior odds into posterior odds. It is evidence conditional on the specified statistical models and their within-model prior distributions; it is not $\mathbb{P}(H_1\mid D)$. Model prior probabilities supply a separate input.^[ref]

For an original stipulated example, consider ten independent and identically distributed Bernoulli trials, with heads as success. Compare $H_0:p=0.4$ with $H_1:p=0.7$, both fixed before seeing the data. Take $D$ to be one particular ordered sequence containing eight heads and two tails. The likelihood function under either point model is $p^8(1-p{)}^2$, soIf $\mathbb{P}(H_1)=0.1$, the prior odds for $H_1$ against $H_0$ are $1/9$. Hence the posterior odds are $21.9909706/9\approx 2.4434412$, andEqual model prior probabilities would instead give ${\mathrm{BF}}_{10}/(1+{\mathrm{BF}}_{10})\approx 0.956505$. The same Bayes factor can thus accompany different posterior probabilities. These statements condition on the two-model comparison; they do not establish that either statistical model describes a real coin.

Using the heads count $K=8$ instead of the ordered sequence changes the sample space. The binomial distribution assigns it mass $\left(\genfrac{}{}{0ex}{}{10}{8}\right)p^8(1-p{)}^2$. The same binomial coefficient multiplies both models' Bayesian model evidence and cancels in their Bayes factor. Mixing an ordered-sequence numerator with a count denominator would introduce an erroneous factor of $45$.

Fix the candidate statistical models, their parameter prior distributions, and model prior probabilities before using the current data for this update. The Bayes factor is useful because it separates evidence from model prior odds, but it still depends on the parameter prior distributions. An improper prior has an arbitrary multiplicative normalization: multiplying its kernel by $c$ multiplies its formal marginal likelihood by $c$, even if the within-model posterior distribution is normalizable. Without a separately justified convention, the ordinary Bayes factor is then undefined up to arbitrary constants.^[ref]

For a random sample $X_i\mid \theta \sim N(\theta ,{\sigma }^2)$ with known $\sigma >0$ and prior distribution $\theta \sim N(0,{\tau }^2)$ with $\tau >0$, compare $H_0:\theta <0$ to $H_1:\theta \ge 0$. The prior odds are one by symmetry. Normal-normal conjugacy gives a normal posterior distribution withIts negative and nonnegative masses are $\Phi (-z)$ and $\Phi (z)$, whereSince the Bayes factor is the posterior odds divided by the prior odds,Here $\Phi$ is the cumulative distribution function of the standard normal distribution. The ratio decreases with the sample mean. For any threshold $b>0$, the inequality $B_{01}<b$ is equivalent to $\Phi (z)>1/(1+b)$ and hence toFor $b=1/10$, the quantile is ${\Phi }^{-1}(10/11)\approx 1.33518$, and the equivalent posterior probability threshold for $H_0$ is $1/11$. The symmetry and proper normalization of the two hypothesis-conditional priors are essential to replacing the Bayes factor by the posterior odds directly.

The Bayesian model evidence is the prior predictive distribution evaluated at the data $D$. For a statistical model $H_j$ with a normalized prior density ${\pi }_j$, its marginal likelihood isThis is an expected value of the likelihood function under the prior distribution; for discrete statistical parameters, replace the integral by a sum. For continuous observations, $m_j(D)$ is a probability density function value with respect to a specified data reference measure, rather than a probability of one exact observation. Comparisons must use the same data and reference measure. The same $m_j(D)$ normalizes the within-model Bayesian posterior and supplies the between-model comparison in a Bayes factor.^[ref]

A normalized prior distribution makes the Bayesian model evidence an average, so it cannot exceed the largest likelihood function value on its support. A broad prior distribution can assign little mass to values that predict the data well, even when one value gives an excellent fit. The worked uniform-alternative binomial Bayes factor below makes the difference between averaging and maximum likelihood estimation explicit.

For Bayesian deviance $D(\theta )=-2\log L(\theta )$ under a common likelihood convention, define $\overline{D}=\mathbb{E}[D(\theta )\mid y]$ and $p_D=\overline{D}-D(\mathbb{E}[\theta \mid y])$. Then $\mathrm{DIC}=\overline{D}+p_D$ trades average fit against effective complexity. Lower values suggest better penalized predictive fit in comparable models; differences are not Bayes factors or model posterior probabilities. The choice of stochastic parameters and latent-variable representation matters, especially in a hierarchical Bayesian model.

For two models with finite positive prior odds, a sequence of data with identical conditional likelihoods under both models has Bayes factor one. The posterior odds therefore equal the prior odds. This requires equality of the probability model for all the data considered, not merely that both models fit a selected data set.

A Gaussian practical-null mixture compares a narrow centered normal distribution prior with a wider centered normal distribution prior. Neither component is a point mass. With observed mean $y\mid \beta \sim N(\beta ,1/n)$ and prior precisions $q_0>q_1>0$, its model predictive variances are $V_i=1/n+1/q_i$, and the Bayes factor isThe overall posterior density is a Bayesian model averaging mixture with component means $ny/(n+q_i)$. A wide-prior penalty can favor the practical null near zero; its dominance depends quantitatively on both prior scales and the observation, not merely on the observation being of order $n^{-1/2}$.

For $n$ studies conditionally independent given the truth of one relationship, the event that at least one is positive has probability $1-{\beta }^n$ under the alternative and $1-(1-\alpha {)}^n$ under the null. Its Bayes factor therefore gives the displayed posterior odds. For interior error probabilities these tend back to the prior odds $R$. If power exceeds size, the Bayes factor decreases to one: the quotient of geometric sums is a weighted average of decreasing powers $[\beta /(1-\alpha ){]}^k$.

This result concerns the coarsened selection event. Full observed results instead multiply their individual Bayes factors, and concordant evidence can accumulate without this bound. It is a useful example of information discarded by publication bias.

Within-model inference. The data $\mathbf{d}$ are the measurements, $H$ specifies a statistical model including its sampling assumptions, and $\theta$ is its unknown parameter or parameter vector. The posterior $p(\theta \mid \mathbf{d},H)$ describes parameter uncertainty after the observation. The likelihood $\mathcal{L}(\theta )=p(\mathbf{d}\mid \theta ,H)$ describes how that observed data set depends on a proposed parameter. The parameter prior $\pi (\theta \mid H)=p(\theta \mid H)$ encodes information before these data. The evidence $Z_H=p(\mathbf{d}\mid H)$ is the prior predictive probability density of the data and normalizes the posterior:For continuous observations the likelihood and evidence are densities in data space, rather than probabilities of one exact observation. The evidence compares different models only when they use the same data and reference measure.

Uniform parameter priors. Normalize each model's top-hat prior, so ${\pi }_H=1/({\theta }_2-{\theta }_1)$ inside its allowed interval and zero elsewhere. ThenThus evidence is the average likelihood over the allowed parameter range, not its maximum and not the unnormalized area alone. Interpret the supplied measured distribution as proportional to a common likelihood, $\mathcal{L}(\theta )=C\exp [-|100u-\theta |/(50u)]$, where the common constant $C$ cancels in model comparisons. If that distribution were instead a posterior calculated using a nonuniform reference prior, it would first need to be divided by that prior. No such alternative prior is specified here.

Evidence calculation. Put $x=\theta /u$. The three integrals, divided by the same $C$, areConsequently the three pairwise Bayes factors may be given asB has the largest evidence. Model A is narrow but lies away from the likelihood peak. Model C contains that peak but spreads much of its normalized prior over less well-fitting values. Model B concentrates its parameter prior near the peak and achieves the best average. Width is therefore not a universal penalty by itself: changing an interval can also improve its location. At fixed centre near a localized likelihood peak, extending the interval into poorly fitting values dilutes the evidence.

Between-model prior information. The supplied $R_A:R_B:R_C=1:1:0.3$ are model prior odds, distinct from each model's uniform parameter prior. They multiply the evidences to give posterior model weights. The corresponding posterior odds areNormalizing these weights gives $p(A\mid \mathbf{d})=0.270915$, $p(B\mid \mathbf{d})=0.602933$ and $p(C\mid \mathbf{d})=0.126152$. B is preferred with either equal model priors or the stated theoretical prior odds. Multiplying by $R$ changes posterior odds; it does not redefine the Bayes factors.

Euclidean luminosity selection. In a flux-limited catalogue, a source with isotropic luminosity $L$ is detected out toA homogeneous Euclidean population with intrinsic luminosity function $\phi (L)$ is sampled approximately as $d{N}_{\det }/dL\propto \phi (L)L^{3/2}$. Brighter objects occupy a much larger accessible volume and are overrepresented; at a fixed large distance the faint tail is missing entirely. This is Malmquist bias. Using the selected population as if it were representative can bias luminosity calibrations or inferred correlations with $\theta$. A selection function, a volume-limited sample, or appropriate accessible-volume weighting can address the sampling effect, subject to completeness assumptions.

The Euclidean derivation concerns flux-limited luminous objects or selected tracers associated with the structures. A CMB temperature map is a diffuse surface-brightness field on a cosmological last-scattering surface: it is not automatically a Euclidean catalogue of identical inverse-square point sources. For a particular CMB-structure catalogue, its actual detection statistic and selection function must be specified before applying this luminosity-weighting law quantitatively.

For data $D$, parameters $\theta$ and model $M$, conditional probability givesThe likelihood describes the measurement process, the prior encodes parameter information before these data, and the Bayesian evidence normalizes the posterior. A prior is a probability density with respect to a specified measure; a uniform prior in distance is not uniform in parallax. Bayes' theorem combines conditional probabilities and does not itself select an appropriate likelihood or prior.

An astronomical application starts with a forward model. For measured fluxes $d_i$ with predictions $f_i(\theta )$ and known Gaussian covariance $C$, $L\propto |C{|}^{-1/2}\exp [-(d-f{)}^{\mathsf{T}}{C}^{-1}(d-f)/2]$. Correlated calibration errors belong in $C$ or in explicit calibration nuisance parameters; assuming every filter or epoch is independent can give falsely narrow uncertainties. For a faint photon-counting source, the more appropriate likelihood can instead be $p(N\mid s,b)=e^{-(s+b)}(s+b{)}^N/N!$, with nonnegative expected source and background counts. Background measurements constrain $b$, which is then marginalized rather than set to an exact subtracted number. The observed background-subtracted count can be negative even though the physical expected source count is nonnegative.

For a simple flux example with $d\mid F\sim \mathcal{N}(F,{\sigma }^2)$ and a Gaussian calibration prior $F\sim \mathcal{N}(F_0,{\sigma }_0^2)$, completing the square givesThis explicitly shows how independent information combines. A physical nonnegative-flux prior would truncate or replace that illustrative Gaussian prior. Photometric-redshift fitting similarly compares predicted galaxy colours with measured fluxes, then integrates over luminosity, dust and other nuisance parameters to obtain $p(z\mid D)$. Multimodal posteriors should be retained when different spectral features produce similar colours; a best fit alone hides that ambiguity. The same framework applies to orbit parameters, stellar temperatures, cosmological parameters and cluster SZ amplitudes.

credible intervals state posterior probability conditional on the model and prior. posterior predictive checks compare replicated observations with the data and can reveal a deficient noise or source model. Sensitivity to plausible priors is especially important for weak signals and inverse problems. For competing models, posterior odds equal prior odds times the Bayes factor $Z_1/Z_2$; the evidence averages fit quality over the prior volume and therefore depends on proper, physically justified prior ranges. Detection thresholds and catalogue selection must enter the sampling model when inferring a population, otherwise a carefully calculated posterior can still describe a biased sample. The practical gain is a joint uncertainty distribution with nuisance parameters propagated, not merely a single fitted number.

For each observation time, use only the currently susceptible risk set $S_t$, with new-infection indicators $Y_{it}$. PutThis is the spatial Bernoulli infection model, under conditional independence of the indicators. Distances must be positive. Already infected trees should not be included as new susceptible trials. The minus sign in the power kernel is present in the original PDF, although it is absent in the converted TeX.

The parameter $\alpha$ must be nonnegative: for $\alpha <0$ and positive exposure, $1-e^{-\alpha A_{it}}<0$ is not a probability. Therefore the stated unrestricted normal prior distribution cannot literally define the model. A coherent interpretation is a normal prior distribution restricted to $\alpha >0$,Here $\Phi$ is the standard normal distribution function. Use the same proper prior distribution in both kernel models. The given Gaussian prior distributions for $\beta$ and $\gamma$ are mathematically valid on the real line for finite positive distances, though restricting them to positive values would impose decaying kernels and would require the corresponding normalized prior distributions.

The full conditional kernels areNeither is generally normal or another elementary conjugate family: the infection factors contain $1-e^{-\alpha A}$, and $A$ depends nonlinearly on $\beta$. Thus a simple conjugate Gibbs sampler is unavailable. This does not make Gibbs sampling logically impossible; exact rejection or slice conditional draws could be used if implemented. A straightforward alternative is Metropolis–Hastings algorithm updates within each coordinate.

For example, put $\eta =\log \alpha$ and use symmetric Gaussian random-walk proposals for $(\eta ,\beta )$. The target in these coordinates isThe factor $e^{\eta }$ is the Jacobian determinant and must be retained. Accept according to the ratio of this target, either for a block or one coordinate at a time. Posterior averages and quantiles of $e^{\eta }$ and $\beta$ estimate the requested marginal posterior distributions.

To compare kernels, introduce $M\in \{1,2\}$ with positive model prior distribution probabilities $w_1,w_2$. Under model two replace $A_{it}$ by ${\sum }_{j\in I_t}{e}^{-\gamma d_{ij}}$, defining $L_2(\alpha ,\gamma )$. The joint model/parameter target iswhere $\zeta =\beta$ or $\gamma$ and $p_M$ is its normalized proper prior distribution. Within each model use the preceding random-walk updates. For a concrete model switch, keep $\alpha$, propose the other model's kernel parameter independently from its prior distribution, and use equal forward and reverse switch-selection probabilities. A move from $(1,\alpha ,\beta )$ draws $\gamma \sim p_{\gamma }$ and has ratioAccept with the minimum of this ratio and one. The auxiliary-coordinate swap $(\alpha ,\beta ,\gamma )\mapsto (\alpha ,\gamma ,\beta )$ has absolute Jacobian determinant one, so no additional factor occurs; unequal switch-selection probabilities would require their reverse/forward ratio. This is a valid model-switching Markov chain Monte Carlo construction even though the two active models have equal parameter dimension. Prior distribution proposals may be inefficient, so fitted proposals can be substituted provided their probability densities remain in the acceptance ratio.

After checking mixing, estimate posterior model probabilities by the fractions of retained iterations in each model. The Bayes factor in favor of the first kernel isUse the posterior distributions within each model as well as these odds to assess the competing distance kernels. Proper prior distributions and their model-dependent normalizing constants are essential for this comparison.

For these two simple hypotheses the Bayes factor is the ratio of their normal probability densities:Therefore $\boxed{B_{10}(y)=e^{2.49(y-1.245)}}$, and the posterior odds are $R{B}_{10}(y)$.

Yes: replacing the observed statistic by a significance label loses information. The binary positive event has Bayes factor $(1-\beta )/\alpha \simeq 16$, but the observed-statistic Bayes factor varies continuously with $y$. A result just at the threshold has $B_{10}(1.65)\simeq 2.74$, while $B_{10}(3)\simeq 79.0$. Both would receive the same positive label.

The binary calculation is correct for that deliberately coarsened observation; its problem is treating it as the whole evidence when the statistic is available. This is the point developed in Goodman and Greenland's response.

Let $A_n$ denote the event that at least one study is positive. Conditional independence gives $\mathbb{P}(A_n\mid T)=1-{\beta }^n$ and $\mathbb{P}(A_n\mid F)=1-(1-\alpha {)}^n$. Updating the prior odds by this event's Bayes factor givesThese are the odds under at-least-one-positive selection, not the odds using every study's actual result.

For $0<\alpha ,\beta <1$, both event probabilities tend to one, so the posterior odds tend to $\boxed{R}$. If $1-\beta >\alpha$, the event's Bayes factor decreases toward one. To see monotonicity, put $a=1-\alpha >\beta$ and writeThe last quotient is a weighted average of the decreasing values $(\beta /a{)}^k$, with weights $a^k$. Adding the next, smaller value decreases the average.

Thus the selection rule becomes less informative as more teams try: eventually a positive study is almost certain even under the null. This explains the hotter-field interpretation in Ioannidis's model. It describes evidence retained by this particular selection event, rather than proving that every individual study becomes less reliable or that the full accumulated evidence gets weaker.

Consistent independent replications should strengthen confidence; contradictory results can weaken it. The change depends on the actual evidence. If $k$ of $n$ studies are positive and all labels are available, the posterior odds areFor the normal model, retaining the full independent statistics instead givesEach study multiplies the existing odds by its own Bayes factor. Reading a complete synthesis is therefore different from learning only that somebody obtained a positive result. Publication bias, dependence between studies and changes in study design must be accounted for before applying the independent-product formula.

A sufficient statistic $T$ has a conditional distribution of the data given $T$ that does not depend on the parameter. In a dominated family, the factorization criterion for sufficiency states that this is equivalent to $p_{\mu }(x)=h(x)g_{\mu }(T(x))$ for a nonnegative parameter-free $h$. Bayes' formula then gives a posterior distribution proportional to $\pi (\mu )g_{\mu }(T(x))$, because $h(x)$ cancels from numerator and normalizing denominator. Thus the posterior depends on the data only through $T$.

For the normal sample,The first term is parameter-free, so the factorization proves sufficiency of $\bar{X}$. Combining its likelihood with the normal prior and completing the square givesThe prior predictive distribution of the sample mean is obtained from $\bar{X}=M+\epsilon$, $\epsilon \sim N(0,1/n)$ independent of $M$:

Define the Bayes factor $B_{01}$ as the marginal likelihood under $H_0$ divided by that under $H_1$. The common residual factor cancels, so it is the ratio of the densities of $\bar{X}$ under these two hypotheses:The reverse Bayes factor is its reciprocal. At $|\bar{x}|=1.96/\sqrt{n}$,for a fixed positive $\tau$. Thus the frequentist test is at its rejection threshold while fixed prior odds give increasingly strong posterior support for the point null. This is the Jeffreys-Lindley paradox for a Gaussian point null: the standardized statistic stays fixed, but the observed effect tends to zero and the fixed-scale alternative spreads predictive mass much more widely. There is no logical contradiction, since the procedures compare hypotheses using different calibrations. For the degenerate $\tau =0$ the hypotheses coincide and $B_{01}=1$.

Under the common observation-level Bayesian deviance convention, DIC is $\overline{D}+p_D$, with the effective parameter count in DIC defined by $p_D=\overline{D}-D(\overline{\psi })$. The no-random-effect model's effective count is near three, corresponding to its three fitted regression coefficients. With plate random effects, the effective count rises to $13.6$: the local effects add flexibility, but their partial pooling means they do not count as eighteen fully independent unrestricted fitted parameters. The effective count need not be an integer or equal the raw parameter count; the scale parameter controls that shrinkage.

Adding the plate effects reduces the average Bayesian deviance by $28.6$ at an effective-complexity cost of $10.7$. Therefore its DIC is lower byThe reported criterion favours the overdispersed model. This is a fit-complexity comparison, not a posterior probability or Bayes factor. In a hierarchical Bayesian model, the interpretation depends on whether deviance is conditional on local effects or has marginalized them out; these figures use the supplied observation-level convention.

Compute the Bayesian model evidence for each model by integrating its sampling density against its parameter prior:The posterior odds are the prior odds multiplied by the Bayes factor:The parameter priors used for evidence must be normalized; arbitrary normalization constants in model-specific improper priors would make this comparison undefined.

Use the orientation that favours the null hypothesis:For a composite model the denominator is its Bayesian model evidence, obtained by integrating the likelihood against the model's prior distribution. This Bayes factor exceeds one when the observed data have greater marginal probability under $H_0$. By Bayes' theorem,Thus the posterior odds equal the prior odds multiplied by the Bayes factor. The reciprocal orientation instead gives $B_{10}=1/B_{01}$.