Statistical Intervals

UniDist.jl provides functions for computing statistical intervals from distributions, useful for uncertainty quantification, Bayesian inference, and hypothesis testing.

Overview

Equal-Tailed Interval

interval - Symmetric Probability Interval

Computes an equal-tailed interval that contains (1 - α) × 100% of the probability mass.

interval(dist, alpha=0.05)

Arguments:

• dist: A distribution object
• alpha: Significance level (default: 0.05 for 95% interval)

Returns: Tuple (lower, upper) where:

• lower = quantile(dist, α/2)
• upper = quantile(dist, 1 - α/2)

Examples:

# 95% interval for standard normal
d = Normal(0, 1)
interval(d, 0.05)     # (-1.96, 1.96)

# 90% interval
interval(d, 0.10)     # (-1.645, 1.645)

# 99% interval
interval(d, 0.01)     # (-2.576, 2.576)

Properties:

• Equal probability in each tail: P(X < lower) = P(X > upper) = α/2
• Always symmetric in probability, not necessarily in value
• Simple and widely understood

Highest Density Interval

hdi - Highest Density Interval (HDI/HPD)

Computes the Highest Density Interval - the narrowest interval containing the specified probability mass.

hdi(dist, mass=0.9; grid=2048, eps=1e-6)  # Continuous
hdi(dist, mass=0.9)                        # Discrete

Arguments:

• dist: A distribution object
• mass: Probability mass to contain (default: 0.9 for 90% HDI)
• grid: Grid resolution for continuous distributions (default: 2048)
• eps: Epsilon for tail trimming (default: 1e-6)

Returns: Tuple (lower, upper)

Examples:

# HDI for symmetric distribution (same as equal-tailed)
d = Normal(0, 1)
hdi(d, 0.95)          # ≈ (-1.96, 1.96)

# HDI for skewed distribution (narrower than equal-tailed)
d = Exponential(1.0)
hdi(d, 0.95)          # Narrower interval
interval(d, 0.05)     # Wider interval (equal-tailed)

# HDI for discrete distribution
d = Binomial(10, 0.3)
hdi(d, 0.9)           # Smallest set of values containing 90%

Comparing Interval Types

Symmetric Distributions

For symmetric distributions (Normal, Cauchy, Logistic, etc.), both methods give the same result:

d = Normal(0, 1)

interval(d, 0.05)     # (-1.96, 1.96)
hdi(d, 0.95)          # (-1.96, 1.96) - same!

Skewed Distributions

For skewed distributions, HDI is narrower:

d = Exponential(1.0)

# Equal-tailed: wider
lo_eq, hi_eq = interval(d, 0.05)
width_eq = hi_eq - lo_eq

# HDI: narrower
lo_hdi, hi_hdi = hdi(d, 0.95)
width_hdi = hi_hdi - lo_hdi

println("Equal-tailed width: $width_eq")
println("HDI width: $width_hdi")
# HDI is always ≤ equal-tailed width

Visual Comparison

using UniDist

# Gamma distribution (skewed)
d = Gamma(2, 2)

# Equal-tailed 90% interval
eq_lo, eq_hi = interval(d, 0.10)
println("Equal-tailed: [$eq_lo, $eq_hi], width = $(eq_hi - eq_lo)")

# HDI 90% interval
hdi_lo, hdi_hi = hdi(d, 0.90)
println("HDI: [$hdi_lo, $hdi_hi], width = $(hdi_hi - hdi_lo)")

# The HDI is narrower and shifted toward the mode

When to Use Each Method

Use Equal-Tailed Intervals When:

• Distribution is symmetric
• You want equal error probability in both tails
• Communicating with audiences familiar with confidence intervals
• Computing p-values (two-tailed tests)

Use HDI When:

• Distribution is skewed
• You want the shortest possible interval
• Doing Bayesian inference
• The mode is more relevant than the median
• Distribution may be multimodal

Practical Applications

Confidence Intervals for Parameters

# Estimating population mean
# After Bayesian analysis, posterior is Normal(5.2, 0.8)
posterior = Normal(5.2, 0.8)

# 95% credible interval
lo, hi = interval(posterior, 0.05)
println("95% CI for mean: [$lo, $hi]")

Bayesian Inference

# Beta posterior for success probability
# Prior: Beta(1, 1), Data: 7 successes, 3 failures
posterior = Beta(1 + 7, 1 + 3)  # Beta(8, 4)

# 95% HDI
lo, hi = hdi(posterior, 0.95)
println("95% HDI for p: [$lo, $hi]")

# Compare with equal-tailed
lo_eq, hi_eq = interval(posterior, 0.05)
println("95% Equal-tailed: [$lo_eq, $hi_eq]")

Predictive Intervals

# Predict next observation from Poisson(λ=5)
d = Poisson(5.0)

# 90% prediction interval
lo, hi = hdi(d, 0.90)
println("90% of observations will be in [$lo, $hi]")

Comparing Groups

# Effect size has posterior Normal(0.5, 0.2)
effect = Normal(0.5, 0.2)

# 95% HDI
lo, hi = hdi(effect, 0.95)

# Check if interval excludes zero (evidence of effect)
if lo > 0 || hi < 0
    println("95% HDI excludes zero: evidence of effect")
else
    println("95% HDI includes zero: no clear evidence")
end

Discrete Distribution HDI

For discrete distributions, HDI finds the smallest set of values containing the desired mass:

d = Binomial(20, 0.3)

# Find 90% HDI
lo, hi = hdi(d, 0.90)
println("90% HDI: {$lo, ..., $hi}")

# Verify the coverage
coverage = sum(pdf(d, k) for k in lo:hi)
println("Actual coverage: $(round(coverage * 100, digits=1))%")

Note: For discrete distributions, the actual coverage may exceed the requested mass because probability is concentrated at discrete points.

Algorithm Details

Continuous HDI Algorithm

1. Create a fine grid over the distribution support
2. Compute PDF at each grid point
3. Sort points by density (highest first)
4. Include points until cumulative mass ≥ target
5. Return (min, max) of included points

Discrete HDI Algorithm

1. Enumerate support of distribution
2. Compute PMF at each point
3. Sort by probability (highest first)
4. Include points until cumulative mass ≥ target
5. Return (min, max) of included points

Limitations

• Multimodal distributions: HDI returns a single interval (min to max of high-density points), which may include low-density regions between modes
• Discrete distributions: Require finite, enumerable support
• Numerical precision: Continuous HDI depends on grid resolution

API Reference

interval(dist, alpha=0.05)

hdi(dist, mass=0.9; grid=2048, eps=1e-6)