Cosmology · Cosmological parameters and statistical inference
Likelihoods, priors and posterior distributions
Normalize a three-trial count under two explicit priors and distinguish density, range probability, mean and mode.
Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.
Back to the chapterAfter reading, you can
- Define a likelihood for a specified count event.
- Normalize two proper posterior distributions.
- Separate probability density from integrated probability and estimator summaries.
A likelihood is a response to a parameter choice
A sampling model assigns probabilities to possible observations for a fixed parameter. Once an observation is supplied, its likelihood is that same expression considered as a function of the parameter.
The likelihood need not integrate to one over the parameter. Its normalization as a sampling distribution concerns possible data at each fixed parameter value.
A Bayesian posterior also uses a prior distribution over the declared parameter space. The posterior is proportional to likelihood times prior, with the normalizing evidence obtained by integrating that product.
These definitions are useful in cosmology, where a probability model connects physical predictions and observational responses. The small count below is hypothetical and supplies no astronomical observations.
The observed event determines the combinatorial factor
Let θ lie in [0,1] and denote the success probability in three independent, identically distributed Bernoulli trials. Assume fixed perfect selection and supply the count K = 2 successes.
The count can arise from three ordered sequences. Its likelihood is L(θ;K=2) = 3θ²(1−θ), where the factor 3 counts those possibilities.
This is a probability of the specified count for each θ. It differs from the likelihood of one particular ordered sequence with two successes, which lacks that factor.
Compare two explicitly normalized priors: a uniform density π₁(θ) = 1, and π₂(θ) = 6θ(1−θ), both on [0,1]. The latter is often named Beta(2,2); the polynomial defines it completely.
The same data can update different priors
For the uniform prior, the evidence is Z₁ = ∫₀¹3θ²(1−θ)dθ = 1/4. Dividing by it gives p₁(θ|K=2) = 12θ²(1−θ).
For the second prior, the likelihood-prior product is 18θ³(1−θ)². Its integral is Z₂ = 3/10, giving p₂(θ|K=2) = 60θ³(1−θ)².
Each posterior density integrates to one. Their compact names are Beta(3,2) and Beta(4,3), but no special-function notation is needed to use the explicit formulas.
The evidences are prior-predictive probabilities of the count K = 2 under the two assumptions. They are not posterior probabilities that either prior is true.
| θ | L(θ) | Uniform-prior posterior | Beta(2,2)-prior posterior |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0.25 | 0.140625 | 0.5625 | 0.527344 |
| 0.5 | 0.375 | 1.5 | 1.875 |
| 0.75 | 0.421875 | 1.6875 | 1.582031 |
| 1 | 0 | 0 | 0 |
Density height is different from probability mass
For a continuous parameter, the posterior probability of a range is the area under its density over that range. The probability of one exact point is zero under these continuous posteriors.
The first posterior has cumulative distribution F₁(θ) = 4θ³−3θ⁴. Thus P₁(θ>1/2|K=2) = 1−F₁(1/2) = 11/16 = 0.6875.
Its density at θ = 1/2 is 1.5 per unit θ. A density may exceed one because a density height is not itself a probability; a properly integrated probability remains between zero and one.
The second posterior gives P₂(θ>1/2|K=2) = 21/32 = 0.65625. The difference traces an explicit prior choice rather than a change in the supplied count.
A single best value does not describe the whole inference
The first posterior mean is 3/5 and its mode is 2/3. For the second, the mean is 4/7 and the mode is 3/5. A mean and a density maximum answer different questions.
A posterior range or summary remains conditional on the declared trial model, prior, selection and observation. More complicated cosmological likelihoods also require physical predictions and observational nuisance models.
The count normalization can be checked separately: the four prior-predictive probabilities for K = 0,1,2,3 are (1/4,1/4,1/4,1/4) under the uniform prior and (1/5,3/10,3/10,1/5) under the second. Each list sums to one.
Posterior content is not automatically the repeated-sampling coverage of a confidence procedure. Keep the data model, prior, normalization and interpretation visible when reporting a parameter estimate.
Worked reasoning
Compute a range probability
Under the uniform prior, is the posterior probability θ>1/2 equal to its density at 1/2?
- Normalize the count likelihood by Z₁ = 1/4 to obtain 12θ²(1−θ).
- Integrate with cumulative distribution 4θ³−3θ⁴ from 1/2 to 1.
- Obtain 11/16 = 0.6875, distinct from density 1.5 per unit θ at 1/2.
No. Probability is integrated density over the stated range.
Try the reasoning
Why does the likelihood for K = 2 contain the factor 3?
Show the answer and reasoning
Three ordered trial sequences have two successes.
The observation is the count, so their probabilities are added.
What normalizes likelihood times prior?
Show the answer and reasoning
The evidence, integrated over the declared parameter space.
For the uniform prior here it is 1/4; for 6θ(1−θ) it is 3/10.
Does a posterior density of 1.875 violate probability normalization?
Show the answer and reasoning
No.
The value is a density per unit θ; its integral over the full allowed range is one.
What to carry forward
- A likelihood fixes the observed event and is not a normalized parameter density.
- Posterior normalization and shape depend on the declared prior.
- Range probabilities, density maxima and posterior means carry different meanings.
References
Sources supporting the explanations, physical distinctions and examples in this lesson.