Explore MAP estimates across common Bayesian model families. Enter priors, evidence, and assumptions for analysis. See posterior modes, intervals, and trends without extra complexity.
Select a conjugate model, enter prior settings and observed evidence, then calculate the posterior mode and related Bayesian summaries.
| Model | Prior | Observed data | Posterior | MAP |
|---|---|---|---|---|
| Beta-Binomial | α = 2, β = 3 | 12 successes from 20 trials | α′ = 14, β′ = 11 | 0.565217 |
| Gamma-Poisson | α = 2, β = 1 | 18 events over exposure 6 | α′ = 20, β′ = 7 | 2.714286 |
| Normal-Normal | μ₀ = 0, τ = 2 | x̄ = 1.8, n = 25, σ = 1.2 | μ′ ≈ 1.775568, SD′ ≈ 0.238328 | 1.775568 |
p(θ | D) ∝ p(D | θ) × p(θ)
Prior: θ ~ Beta(α, β)
Likelihood: X | θ ~ Binomial(n, θ)
Posterior: θ | X ~ Beta(α + x, β + n - x)
MAP:
(α + x - 1) / (α + β + n - 2) when both posterior shape terms exceed 1.
Prior: λ ~ Gamma(α, β) using shape-rate form
Likelihood: Y | λ ~ Poisson(λ × exposure)
Posterior: λ | Y ~ Gamma(α + y, β + exposure)
MAP:
(α + y - 1) / (β + exposure) when the posterior shape exceeds 1.
Prior: θ ~ Normal(μ₀, τ²)
Data model: x̄ | θ ~ Normal(θ, σ² / n)
Posterior variance:
1 / (1/τ² + n/σ²)
Posterior mean:
[(μ₀/τ²) + (n x̄/σ²)] / [(1/τ²) + (n/σ²)]
Because the posterior is normal, the MAP equals the posterior mean.
The MAP estimate is the parameter value with the highest posterior density after combining prior beliefs with observed evidence. It is a Bayesian point estimate focused on the most plausible updated value.
Maximum likelihood uses only observed data. MAP uses both the likelihood and a prior distribution. When the prior is weak or flat, MAP can be close to the maximum likelihood estimate.
Different data structures need different likelihood-prior pairs. Binomial data fits Beta-Binomial, count rates fit Gamma-Poisson, and continuous means with known sampling spread fit Normal-Normal.
Exposure is the amount of opportunity for events to occur. It can represent time, distance, area, machine hours, patient years, or any unit where event counts accumulate.
If posterior shape parameters are not both above one, the density may peak at an edge instead of the interior. That usually happens with sparse data or very strong asymmetric priors.
The Normal-Normal interval is exact from the posterior standard deviation. The Beta-Binomial and Gamma-Poisson intervals here use a quick normal approximation for practical reporting and screening.
Stronger priors pull the posterior peak toward the prior center and reduce movement from new data. Larger samples or exposure usually weaken that pull by increasing data precision.
For symmetric unimodal posteriors such as the normal posterior in the Normal-Normal model, the mean and mode match. In skewed posteriors, the posterior mean and MAP often differ.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.