tips_and_updatesbefore the data

What a prior actually is

In a frequentist workflow you usually start with nothing but the data. In a Bayesian workflow you start by writing down, as a probability distribution, what you already believed about your audience before this study: from theory, from a pilot, from a decade of cultivation-theory literature, or from genuine uncertainty.

spa θ (theta) is whatever you're estimating

θ stands for the proportion (or mean, or rate, or correlation) you care about: the share of viewers whose attitude shifted, the share of heavy viewers who overestimate crime, the mean score on a parasocial-interaction scale, the number of exposures per week. The prior is a probability distribution that says how plausible you think each possible value of θ is, before a single respondent answers.

Try on three researcher postures

Each is a genuine Beta(α,β) distribution: the natural prior for a proportion. Pick one and see what it claims about θ.

sliders Feel how a prior encodes belief

Drag the two sliders that define a Beta prior and watch it change shape. α acts like "prior successes," β like "prior failures," and α+β is the prior's effective sample size, how many hypothetical respondents the prior is worth.

α (pseudo-successes) 2

β (pseudo-failures) 2

Prior mean

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Prior SD

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Effective prior n (α+β)

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"I'll just let the data speak, no prior." There is no such analysis. A flat Beta(1,1) that treats every θ as equally plausible is itself a prior, a genuine, often ill-advised, commitment. Flat is a choice, not a vacuum. This is one of the most-cited warnings in the applied Bayesian literature (see Gill, 2007; Gelman et al., 2013, §2.9).

quizcheck yourself

Which of these is a defensible justification for choosing a particular prior?

It came from a careful reading of prior theory and empirical estimates, described transparently before analysis. It makes your find more publishable. It's the smallest number you could think of.

functionsthe machine

Bayes' theorem, in research vocabulary

calculatebayes' theorem

P(θ | data) = P(data | θ) · P(θ)P(data)

P(θ)

Prior. What theory and prior work told you before this dataset.

P(data | θ)

Likelihood. How compatible your responses are with each possible θ.

P(data)

Marginal likelihood. Normalizing constant: total probability of the data over every θ.

P(θ | data)

Posterior. What you're entitled to believe now, theory and evidence combined.

calculatethe beta density: the shape of θ itself

f(θ; α, β) = θ^α−1 (1−θ)^β−1B(α, β)

B(α,β) is the beta function, the normalizing constant that makes the area under the curve equal 1. Because a Beta prior and a binomial likelihood multiply into another Beta distribution (a conjugate pair), the whole update has an exact closed-form answer. No simulation required.

device_hub Conjugate priors: why the math stays exact

A conjugate prior is chosen so that, combined with a particular likelihood, the posterior stays in the same distributional family. The table below is the quiet superpower behind most of this site, it's why the "lab" pages compute exact posteriors rather than running simulations.

Likelihood (your data)	Conjugate prior	Posterior params	Typical media use
Bernoulli / Binomial	Beta(α,β)	Beta(α+x, β+n−x)	yes/no attitude items
Normal (known var)	Normal(μ₀, σ₀)	Normal (precision add)	Likert composites, means
Poisson	Gamma(α,β)	Gamma(α+Σx, β+n)	exposure counts, frequencies
Multinomial	Dirichlet(α)	Dirichlet(α+counts)	genre/source preference
Multivariate Normal	Normal-Inverse-Wishart	conjugate Normal-IW	relationships between measures

Conjugacy is discussed throughout Gill (2007) and Gelman et al. (2013, §2.4–2.8). When no conjugate prior fits, one uses numerical methods, which this site does too, on the Compare and Meta pages.