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Bayesian Philosophy

The frequentist and Bayesian frameworks disagree about what is random.

Frequentist: the parameter is fixed and unknown; the data is random. Probability statements are about the long-run behaviour of procedures.

Bayesian: the data is what it is; the parameter is uncertain, and uncertainty is described with a probability distribution.

Bayes as belief
P(θ∣data)⏟posterior=P(data∣θ)⏞likelihood  P(θ)⏞priorP(data)\underbrace{P(\theta \mid \text{data})}_{\text{posterior}} = \frac{\overbrace{P(\text{data} \mid \theta)}^{\text{likelihood}} \; \overbrace{P(\theta)}^{\text{prior}}}{P(\text{data})}

The same arithmetic as Bayes theorem, read as an update to what you believe rather than a fact about frequencies.

That difference sounds abstract until you look at what each can say.

Why traders usually want the Bayesian answer

A confidence interval cannot make a probability statement about the parameter. "95% confidence" describes the procedure's long-run hit rate, not this interval.

A Bayesian credible interval can: given the prior and the data, there is a 95% probability the parameter lies in this range. That is the statement people want when they ask "how likely is it that this strategy has an edge?", and it is the one a frequentist framework structurally cannot provide.

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