Maximum A Posteriori Ratio Test (MAPRT)

• MAPRT is a Bayesian decision rule that compares posterior odds between competing hypotheses, integrating prior information to achieve optimal decision-making.
• It extends beyond simple binary tests to applications such as model order selection in HAADF STEM, distributed radar detection, and statistical inverse problems.
• The methodology translates complex Bayesian formulations into practical linear discriminant forms, offering insights into error probabilities, CFAR properties, and networked sensing performance.

Maximum A Posteriori Ratio Test (MAPRT) denotes a Bayesian decision rule that compares posterior support between competing hypotheses or models and selects the alternative whose posterior odds, posterior density ratio, or analogous posterior quantity exceeds a prescribed threshold. In the canonical binary form,

$$\Lambda_{\mathrm{MAP}}(y)=\frac{p(y\mid H_1)\,P(H_1)}{p(y\mid H_0)\,P(H_0)} \underset{H_0}{\overset{H_1}{\gtrless}} \eta,$$

with $\eta=1$ for standard MAP under equal priors and costs. The same Bayesian logic reappears in model-order selection through posterior odds between $M_a$ and $M_b$, in adaptive radar through EM-estimated posterior responsibilities, and in inverse problems through posterior mass or posterior-density comparisons over constrained parameter sets (Anguluri et al., 2020, Fatermans et al., 2019, Yin et al., 4 Mar 2025, Kretschmann et al., 2024, Abdi et al., 2019).

1. Decision-theoretic formulation

For binary hypotheses, MAPRT is a posterior-odds test. Using Bayes’ rule, the detector multiplies the likelihood ratio by the prior odds and compares the result with a threshold. In this form, MAPRT differs from the pure likelihood ratio test, which compares $p(y\mid H_1)/p(y\mid H_0)$ alone, and from maximum likelihood, which ignores priors entirely. In the Gaussian case, the log-MAP ratio is the difference of two Gaussian log-likelihoods plus the log prior odds, so the decision boundary is quadratic when the covariances differ and linear when they coincide (Anguluri et al., 2020).

The same structure extends beyond simple binary testing. In model-order selection for HAADF STEM, one compares posterior odds

$$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$

and with a uniform prior over model orders the decision reduces to comparing marginal likelihoods. The selected model is

$M^*=\arg\max_k p(M_k\mid y).$

This makes MAPRT a natural bridge between hypothesis testing and Bayesian evidence maximization (Fatermans et al., 2019).

A distinct but related formulation arises in statistical inverse problems. There, the paper’s primary object is a MAP mass test that compares posterior probabilities of half-spaces,

$$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$

where $X_H=\{x:\langle \phi,x\rangle\le 0\}$ and $X_K=\{x:\langle \phi,x\rangle>0\}$. A posterior-density MAPRT can also be defined through

$\eta=1$0

but this is explicitly distinguished from both Bayes factors and the posterior-mass MAP test. A recurrent misconception is to identify all Bayesian tests with Bayes factors; the inverse-problem formulation shows that posterior set comparison, posterior density comparison, and model-evidence comparison are different constructions (Kretschmann et al., 2024).

2. Gaussian discriminants in networked sensing

In networked sensing, MAPRT is instantiated on a discrete-time linear time-invariant network

$\eta=1$1

with measurements

$\eta=1$2

and stacked observation vector $\eta=1$3. Under simple Gaussian input hypotheses

$\eta=1$4

the measurement mean and covariance are

$\eta=1$5

The matrices $\eta=1$6 and $\eta=1$7 encode, respectively, observability and impulse-response structure, so the network filtering induced by $\eta=1$8 is embedded directly into the MAPRT statistic (Anguluri et al., 2020).

For the mean-shift model, $\eta=1$9 and $M_a$0, the MAP detector reduces to a linear discriminant with sufficient statistic

$M_a$1

and a threshold shifted by $M_a$2. For the covariance-shift model, $M_a$3 and $M_a$4, the paper uses a one-dimensional linear discriminant $M_a$5 chosen to maximize the $M_a$6-divergence. The decision rule becomes

$M_a$7

so the statistic is the centered energy $M_a$8, with $M_a$9 selected through a generalized eigenproblem or, equivalently, as a principal eigenvector of $M_b$0 (Anguluri et al., 2020).

The central asymptotic characterization is expressed through the transfer function

$M_b$1

For mean shifts, the asymptotic signal-to-noise ratio is

$M_b$2

with $M_b$3. For covariance shifts, the key quantity is

$M_b$4

The paper proves that the corresponding error probabilities decrease monotonically with $M_b$5 or $M_b$6, respectively. This yields a network-theoretic interpretation of detection performance in terms of input-output gain rather than only graph distance (Anguluri et al., 2020).

Sensor-placement consequences are formulated through node cutsets and downstream partitions. In the noiseless case, measuring the cutset is always at least as good as measuring the downstream partition. In the noisy case, however, the subnetwork transfer

$M_b$7

controls whether cutset or downstream sensors perform better. If the gain from cutset to partition is at most one, cutset sensing is superior; if it exceeds one, downstream sensing can outperform cutset sensing because the downstream subnetwork can amplify the signal relative to noise. The numerical examples make this reversal explicit: in a 10-node synthetic network with $M_b$8, partition nodes achieved lower error probabilities than the cutset node when the “row sums $M_b$9” condition held, while in a 50-node random network the cutset uniformly outperformed every other 3-node subset when $p(y\mid H_1)/p(y\mid H_0)$0 (Anguluri et al., 2020).

3. Distributed hypothesis testing over graphs

In distributed detection over a pairwise Markov random field, MAPRT appears as a per-node posterior-odds test derived from max-product beliefs. The posterior factorization is

$p(y\mid H_1)/p(y\mid H_0)$1

with $p(y\mid H_1)/p(y\mid H_0)$2 and $p(y\mid H_1)/p(y\mid H_0)$3. At node $p(y\mid H_1)/p(y\mid H_0)$4, the log-ratio of max-product beliefs is

$p(y\mid H_1)/p(y\mid H_0)$5

which is the nodewise MAPRT in log form. The local observation contribution is

$p(y\mid H_1)/p(y\mid H_0)$6

and for coherent Gaussian detection

$p(y\mid H_1)/p(y\mid H_0)$7

The paper’s key result is that the max-product decision variable is a linear combination of local log-likelihood ratios: $p(y\mid H_1)/p(y\mid H_0)$8 after enough iterations (Abdi et al., 2019).

This linearity is established by showing that the maximizers inside the message updates are affine functions of neighboring states, which yields linear message-LLR recursions. A one-hop approximation gives

$p(y\mid H_1)/p(y\mid H_0)$9

Thus, in the binary-Gaussian setting studied, max-product behaves as linear data fusion rather than as an intrinsically nonlinear detector. The paper reaches the same conclusion for a linearized sum-product recursion, where the coefficients become $$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$0, and reports very similar behavior of max-product and sum-product under the distributed hypothesis test (Abdi et al., 2019).

Performance analysis follows because, conditioned on a fixed network state, each $$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$1 is Gaussian or approximately Gaussian, so any linear fusion $$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$2 is Gaussian as well. This permits explicit false-alarm and detection probabilities

$$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$3

with thresholds selected either by Bayesian tuning or, as in the paper, by Neyman–Pearson constraints. A convergence condition is given for the sum-product linearization,

$$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$4

which ensures contraction. Simulations on a five-node network with $$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$5 samples per node and a training window $$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$6 show that the optimal centralized linear fusion (“linOpt”) achieves the best $$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$7, the decentralized one-hop optimization (“linProp” and “linPropB”) is close to it, and max-product and sum-product exhibit similar performance while remaining sensitive to the coupling-learning factor $$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$8 (Abdi et al., 2019).

4. Evidence-based MAPRT in HAADF STEM

In HAADF STEM, MAPRT is used for atom-column detection and model-order selection. Competing hypotheses $$PO_{a,b}=\frac{p(M_a\mid y)}{p(M_b\mid y)} =\frac{p(y\mid M_a)\,p(M_a)}{p(y\mid M_b)\,p(M_b)},$$9 represent images containing $M^*=\arg\max_k p(M_k\mid y).$0 projected atom columns. With a uniform prior over $M^*=\arg\max_k p(M_k\mid y).$1, the posterior satisfies

$M^*=\arg\max_k p(M_k\mid y).$2

so selection reduces to maximizing the marginal likelihood $M^*=\arg\max_k p(M_k\mid y).$3. Pairwise MAPRT between two model orders is therefore a posterior-odds test, with threshold $M^*=\arg\max_k p(M_k\mid y).$4 for pure MAP selection (Fatermans et al., 2019).

The image model represents $M^*=\arg\max_k p(M_k\mid y).$5 columns by Gaussian peaks on a constant background: $M^*=\arg\max_k p(M_k\mid y).$6 with reduced-parameter variants for equal widths or equal widths and equal heights. The observation noise is physically Poisson, but the implementation adopts a normal approximation to Poisson with plug-in variance $M^*=\arg\max_k p(M_k\mid y).$7, which yields

$M^*=\arg\max_k p(M_k\mid y).$8

Maximum likelihood estimation therefore becomes weighted least squares, and the marginal likelihood is evaluated with a Laplace approximation around the ML estimate $M^*=\arg\max_k p(M_k\mid y).$9 (Fatermans et al., 2019).

For the general unequal-width model, the practical posterior expression is

$$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$0

with analogous forms for the equal-width and equal-width/equal-height cases. The factors $$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$1, the Hessian determinant, and the prior-volume terms are the characteristic finite-sample corrections of this MAPRT. In “information criterion” form,

$$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$2

and the paper shows that as $$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$3, the leading terms reduce to $$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$4, so the MAP rule is asymptotically equivalent to BIC while retaining curvature, labeling, and prior-volume penalties in finite samples (Fatermans et al., 2019).

Algorithmically, the method performs forward selection from $$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$5 to $$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$6, adds peaks incrementally, tests many initial positions for each added peak, minimizes $$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$7 under prior-range constraints, computes the Hessian determinant, and then evaluates $$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$8. In simulations of 1000 images of size $$\Pi(X_K\mid Y=y)\quad \text{versus}\quad \Pi(X_H\mid Y=y),$$9 with pixel size $X_H=\{x:\langle \phi,x\rangle\le 0\}$0, 1–5 randomly positioned Au atoms, dose uniformly $X_H=\{x:\langle \phi,x\rangle\le 0\}$1–$X_H=\{x:\langle \phi,x\rangle\le 0\}$2, and a Gaussian model with equal widths and equal heights, the MAP rule detected the correct number most frequently at $X_H=\{x:\langle \phi,x\rangle\le 0\}$3; AIC, GIC, and HQC tended to overfit, whereas BIC tended to underfit. At $X_H=\{x:\langle \phi,x\rangle\le 0\}$4 and $X_H=\{x:\langle \phi,x\rangle\le 0\}$5, all criteria improved and the differences narrowed (Fatermans et al., 2019).

A major by-product of this work is the integrated CNR,

$X_H=\{x:\langle \phi,x\rangle\le 0\}$6

defined from the integrated signal and integrated background over a footprint of radius $X_H=\{x:\langle \phi,x\rangle\le 0\}$7, containing $X_H=\{x:\langle \phi,x\rangle\le 0\}$8 of the Gaussian volume. The paper reports that ICNR correlates with atom detectability better than conventional SNR and CNR, and that columns with ICNR less than around $X_H=\{x:\langle \phi,x\rangle\le 0\}$9 become challenging, with detection rate rapidly dropping below this value. This distinguishes the MAPRT from a purely formal model-selection device: it also supports quantitative detectability analysis in low-dose imaging (Fatermans et al., 2019).

5. EM-based adaptive radar detection

In adaptive radar, MAPRT is instantiated through a hybrid ML-Bayesian detector for a cell under test $X_K=\{x:\langle \phi,x\rangle>0\}$0 and secondary target-free data $X_K=\{x:\langle \phi,x\rangle>0\}$1. The hypotheses are

$X_K=\{x:\langle \phi,x\rangle>0\}$2

where $X_K=\{x:\langle \phi,x\rangle>0\}$3, $X_K=\{x:\langle \phi,x\rangle>0\}$4 is a known steering vector, $X_K=\{x:\langle \phi,x\rangle>0\}$5 is an unknown target amplitude, and $X_K=\{x:\langle \phi,x\rangle>0\}$6 is an unknown Hermitian positive definite covariance matrix. Defining

$X_K=\{x:\langle \phi,x\rangle>0\}$7

the joint likelihoods are

$X_K=\{x:\langle \phi,x\rangle>0\}$8

$X_K=\{x:\langle \phi,x\rangle>0\}$9

A binary latent variable $\eta=1$00 indexes the hypothesis, with priors $\eta=1$01 estimated inside the EM procedure rather than fixed externally (Yin et al., 4 Mar 2025).

The MAPRT statistic is the posterior ratio

$\eta=1$02

Because EM computes the posterior responsibilities

$\eta=1$03

the same detector can be written as

$\eta=1$04

Its log form is

$\eta=1$05

or equivalently

$\eta=1$06

This makes the radar detector a MAPRT whose posterior terms are estimated jointly with nuisance parameters (Yin et al., 4 Mar 2025).

The EM iteration alternates between an E-step, which computes $\eta=1$07 and $\eta=1$08, and an M-step, which updates

$\eta=1$09

$\eta=1$10

$\eta=1$11

The detector is named EM-BML-D in the paper and is explicitly described as a “generalized MAP” rule because it compares posterior probabilities with ML-estimated nuisance parameters. Initialization with $\eta=1$12, $\eta=1$13, and the generalized matched-filter estimate for $\eta=1$14 yields a CFAR statistic with respect to the unknown $\eta=1$15 (Yin et al., 4 Mar 2025).

The CFAR proof is based on whitening: $\eta=1$16 Under $\eta=1$17, the distribution of $\eta=1$18 depends only on whitened invariants and is therefore independent of $\eta=1$19. By induction over EM iterations, this invariance propagates to the final statistic, so a threshold $\eta=1$20 can be calibrated for a desired $\eta=1$21 without knowing the interference covariance. The paper contrasts this detector with the GLRT and Kelly’s detector, emphasizing that MAPRT incorporates hypothesis priors and fuses information from both hypotheses through the responsibilities, whereas the GLRT remains purely ML-based (Yin et al., 4 Mar 2025).

Numerically, for $\eta=1$22, $\eta=1$23, $\eta=1$24, $\eta=1$25, and $\eta=1$26, the proposed MAPRT achieved $\eta=1$27, versus approximately $\eta=1$28 for GLRT and approximately $\eta=1$29 for AMF. The paper also reports CFAR preservation across $\eta=1$30 from $\eta=1$31 to $\eta=1$32 and correlation $\eta=1$33 from $\eta=1$34 to $\eta=1$35, improved performance on MIT Lincoln Laboratory Phase One data, and robustness under mismatch similar to AMF. Empirically, about five EM iterations suffice, with average execution times for $\eta=1$36 over 1000 trials of approximately $\eta=1$37 for GLRT, $\eta=1$38 for AMF, $\eta=1$39 for EM-BML-D5, and $\eta=1$40 for EM-BML-D7 (Yin et al., 4 Mar 2025).

6. MAPRT and MAP testing in statistical inverse problems

For linear-Gaussian inverse problems on separable Hilbert spaces,

$\eta=1$41

with Gaussian prior $\eta=1$42, the posterior is Gaussian $\eta=1$43 with

$\eta=1$44

$\eta=1$45

The tested feature is $\eta=1$46, with hypotheses

$\eta=1$47

Since

$\eta=1$48

the paper’s MAP mass test rejects $\eta=1$49 iff $\eta=1$50, equivalently iff $\eta=1$51 (Kretschmann et al., 2024).

This mass test admits a linear-statistic representation. Defining

$\eta=1$52

the decision rule becomes

$\eta=1$53

The paper proves an exact Gaussian power formula,

$\eta=1$54

which yields both level and power once $\eta=1$55 or the threshold is calibrated. It also shows that, without a priori restrictions, the level is typically $\eta=1$56; under commuting assumptions and when $\eta=1$57 is an eigenvector of $\eta=1$58, one can obtain exact level $\eta=1$59, and the resulting test coincides with the uniformly most powerful unregularized test (Kretschmann et al., 2024).

A posterior-density MAPRT can also be defined in this setting. Let

$\eta=1$60

Because the posterior is Gaussian and the constrained maximizers are projections of $\eta=1$61 onto the half-space boundaries in the $\eta=1$62-norm, the ratio has the closed form

$\eta=1$63

Hence rejection of $\eta=1$64 at threshold $\eta=1$65 is equivalent to

$\eta=1$66

The paper stresses that this posterior-density MAPRT is distinct from the MAP mass test that forms the main object of the frequentist analysis (Kretschmann et al., 2024).

The regularization viewpoint is central. Under Assumption 2.1, the MAP mass test is a regularized test of the form

$\eta=1$67

with $\eta=1$68 and

$\eta=1$69

Moreover, $\eta=1$70 is the Tikhonov–Phillips regularized solution of $\eta=1$71, or equivalently the solution of

$\eta=1$72

Under the spectral source condition $\eta=1$73, $\eta=1$74, and priors $\eta=1$75 with $\eta=1$76, the paper derives explicit lower bounds on power and shows that for scalings $\eta=1$77, $\eta=1$78, the power tends to $\eta=1$79 as $\eta=1$80. Numerical experiments in deconvolution, numerical differentiation, and the backward heat equation show conservative level control and strong power even in severely ill-posed settings, where unregularized testing becomes numerically infeasible (Kretschmann et al., 2024).