Decision Lab
HDI Β· ETI Β· ROPE β€” From posterior to decision Β· bayestestR Β· Kruschke Β· McElreath
Β© Dr. Rainer DΓΌsing Β· Interactive Tools by Claude
Three decision rules for the same posterior β€” HDI, ETI and ROPE β€” directly comparable, without any data of your own.

Posterior distribution

Mean ΞΌ105
Spread Οƒ8

ROPE

ROPE lower bound97
ROPE upper bound113
ROPE = Region of Practical Equivalence. Ideally determined by content (smallest effect of interest). Without prior knowledge: Β±0.1Β·SD(y) following Kruschke.

Interval width

95% β€” Kruschke's standard

ROPE decision

100% β€” Full ROPE: entire posterior mass (rope_ci = 1 in bayestestR)

Interval type

HDI = shortest interval containing X% of the mass. Identical to ETI for normal distributions.

pd reference value

Reference value for pd (default: 0). E.g. 1 for odds ratios or 100 for IQ scale.
Posterior distribution Β· ROPE Β· Intervals
Posterior Median HDI ROPE
Value range
Min 50
Max 160
Default: IQ scale.
For Gamma: from 0.
Approach A β€” Kruschke
HDI vs. ROPE
β€”
β€”
Trichotomous decision: HDI outside, inside, or overlapping ROPE.
Approach B β€” Full ROPE
% in ROPE
β€”
β€”
Gradual: proportion of posterior mass in the ROPE. <2.5% = effect, >97.5% = equivalence (Makowski et al. 2019).
Approach C β€” ETI
ETI vs. ROPE
β€”
β€”
Invariant under transformations. Equal to HDI for normal distributions. Differences appear with skewed posteriors.
Metric D β€” bayestestR
Probability of Direction
β€”
β€”
Proportion of posterior mass in the more probable direction. pd β‰₯ 97.5% β‰ˆ p ≀ .05 (two-sided).
Explanations & Backgroundβ–Ά

HDI (Highest Density Interval) β€” The shortest interval containing X% of the posterior mass. The key property: no value outside the HDI has a higher density than any value inside. This means the HDI contains exactly the "most plausible" parameter values β€” those most strongly supported by the posterior. Density is proportional to probability, so a value with higher density is more credible than one with lower density. For a unimodal distribution the HDI is always a contiguous interval around the mode (densest point). For symmetric posteriors (Normal, t) HDI = ETI.

ETI (Equal-Tailed Interval) β€” Cuts equal probability mass from both tails (at 95%: 2.5% each). The ETI corresponds to what frequentist confidence intervals approximate. Key property: invariant under monotone transformations. If you transform a parameter (e.g. log(Οƒ) instead of Οƒ), the ETI on the original scale gives the same decision as on the transformed scale. The HDI can lead to different decisions depending on parameterisation (HDI on log scale β‰  exp(HDI on log scale)). For Normal and t distributions: HDI = ETI, since both distributions are symmetric. Differences arise only with skewed posteriors (here: Gamma).

Kruschke's decision rule (Approach A) β€” Trichotomous using the chosen interval (HDI or ETI): Interval completely outside ROPE β†’ accept effect as real Interval completely inside ROPE β†’ accept practical equivalence Interval overlaps ROPE β†’ withhold judgment

Full-ROPE approach (Approach B, bayestestR) β€” Computes the proportion of the entire posterior mass that lies within the ROPE. This is a gradual, continuous statement: "37% of the posterior mass lies in the ROPE." Decision rule following Makowski et al. (2019): <2.5% = effect (negligible proportion in ROPE); >97.5% = practical equivalence; in between = undecided.

Setting the ROPE: SESOI β€” The ROPE should always be determined by content. The key concept is SESOI (Smallest Effect Size Of Interest): Which effect would be so small that it has no practical relevance for your research question? This is a scientific, not a statistical question. Example therapy study: An IQ gain of less than 3 points is indistinguishable in practice β€” so ROPE = [βˆ’3, +3]. Methods for determining SESOI: (1) pilot data / expert knowledge, (2) standardised effect size (e.g. d=0.2 as "small effect"), (3) what clinicians specify as the minimal clinically important difference (MCID). Lakens et al. (2018) provide a detailed introduction: Lakens, D., Scheel, A. M., & Isager, P. M. (2018). Equivalence testing for psychological research: A tutorial. Advances in Methods and Practices in Psychological Science, 1(2), 259–301. β€” Without any prior knowledge: Kruschke suggests Β±0.1Β·SD(y).

Interval width 89% vs. 95% β€” McElreath prefers 89% because there is no special privilege for the number 95 β€” this convention comes from the frequentist world. 89 is the largest prime number below 90. The 95% interval is widely used and facilitates communication. Choose deliberately and justify the choice.

R Code (bayestestR)β–Ά

Install bayestestR: install.packages("bayestestR")

library(bayestestR) # Generate posterior samples (here: normal distribution) posterior <- rnorm(10000, mean = 105, sd = 8) # Compute HDI hdi(posterior, ci = 0.95) # Compute ETI eti(posterior, ci = 0.95) # ROPE analysis rope(posterior, range = c(97, 113), ci = 0.95) # Full decision p_direction(posterior) equivalence_test(posterior, range = c(97, 113), ci = 0.95)
β„Ή Decision Lab β€” Help
What will I learn here?
A posterior alone does not make a decision β€” that requires a decision framework. This tool shows three complementary approaches: HDI describes where the most plausible values lie; ETI provides symmetric quantile bounds; ROPE tests whether an effect is practically meaningful. All three come from the Bayesian toolkit (Kruschke, McElreath) and are implemented in bayestestR.
Recommended exploration
HDI β€” explained in detail
Picture the posterior as a mountain range. For the HDI, you draw a horizontal cutting line and lower it until exactly 95% of the mass lies above the line. The range where the mountain stands above the line is the HDI. Three properties follow:
1. Density criterion: every value inside the HDI has a higher density than every value outside. So the HDI always contains the "most plausible" values β€” and always the mode.
2. Shortest interval: no other interval with the same mass is shorter.
3. Not transformation-invariant: the HDI of a parameter on the log scale, back-transformed with exp(), is not the HDI on the original scale β€” because transforming changes the density (Jacobian term), but not the mass.
For multimodal posteriors the HDI can consist of several separate sub-intervals. From MCMC draws it is computed as the shortest window containing the desired share of draws β€” at the edges it is therefore somewhat noisier than a quantile.
ETI β€” explained in detail
The ETI cuts off equal mass at both ends β€” at 95% that is 2.5% each. Its limits are simply the 2.5% and the 97.5% quantile:
1. Simple and stable: quantiles can be estimated reliably from draws. The brms summary (summary()) shows such quantile-based intervals.
2. Transformation-invariant: the quantile of exp(ΞΈ) is exp(quantile of ΞΈ). Whether you compute on the log scale or the original scale β€” the ETI gives the same statement.
3. Always contiguous β€” but not always the most plausible: for skewed distributions it includes, on the long side, values that are less dense than values it excludes on the short side.
When do they differ?
Symmetric (Normal, t): HDI = ETI β€” the choice does not matter.
Skewed (Gamma): the ETI is longer and shifted toward the long side; the HDI is shorter and follows the peak. Try it: choose Gamma, decrease the shape Ξ± β€” the skewness grows and the two intervals (red and blue lines) drift apart. Then place a ROPE limit between the two edges: the decision can flip.
Multimodal (not adjustable here, but e.g. in stage 2 of the MCMC Visualizer): the HDI falls apart into pieces, the ETI also covers the valley in between.
HDI or ETI β€” which one?
HDI if you want to know where the most plausible values lie β€” especially for skewed posteriors.
ETI if you want to transform results (e.g. odds ratio = exp(b)) or compare with quantile-based standard output.
What matters is to state the interval type and the width β€” and to fix the rule beforehand. If the two intervals decide differently, that is information in itself: the finding then depends on the choice, and this belongs in the report. For the implementation in R see bayestestR (hdi(), eti(), rope()); basics on the credible interval and the ROPE (chosen via SESOI) are in the glossary; certainty about the direction is given by the probability of direction.
The three decision approaches
HDI (red): Highest Density Interval β€” shortest region containing the desired posterior mass. Optimal for skewed distributions.

ETI (blue): Equal-Tailed Interval β€” symmetric quantiles (e.g. 5 % / 95 %). Follows the same procedure as frequentist confidence intervals, but with a Bayesian interpretation based on the posterior distribution.

ROPE (green): Region of Practical Equivalence β€” proportion of the posterior within the Β«practically zeroΒ» range. Kruschke's decision rule: > 97.5 % β†’ accept Hβ‚€; < 2.5 % β†’ reject; in between β†’ withhold judgement.
Further Reading
Kruschke (2014) β€” HDI/ROPE decision logic
McElreath (2020) β€” 89% CrI recommendation
Makowski et al. (2019) β€” Full-ROPE decision rule
Lakens et al. (2018) β€” SESOI, equivalence bounds