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I am currently studying the textbook In All Likelihood -- Statistical Modelling and Inference Using Likelihood by Yudi Pawitan. Section Bayesians versus frequentists of chapter 1 says the following:

Once one accepts the prior $f(\theta)$ for $\theta$ and agrees it can be treated as a regular density, the way to proceed is purely deductive and (internally) consistent. Assuming that, given $\theta$, our data $x$ follows a statistical model $p_\theta(x) = f(x \vert \theta)$, then the information about $\theta$ contained in the data is given by the posterior density, using the Bayes theorem as in (1.2),

$$f(\theta \vert x) = \dfrac{f(x \vert \theta) f(\theta)}{f(x)}.$$

In Bayesian thinking there is no operational difference between a prior density $f(\theta)$, which measures belief, and $f(x \vert \theta)$, which measures an observable quantity. These two things are conceptually equal as measures of uncertainty, and they can be mixed using the Bayes theorem.

The posterior density $f(\theta \vert x)$, in principle, captures from the data all the information that is relevant for $\theta$. Hence, it is an update of the prior $f(\theta)$. In a sequence of experiments it is clear that the current posterior can function as a future prior, so the Bayesian method has a natural way of accumulating information.

My question relates to this part:

In Bayesian thinking there is no operational difference between a prior density $f(\theta)$, which measures belief, and $f(x \vert \theta)$, which measures an observable quantity. These two things are conceptually equal as measures of uncertainty, and they can be mixed using the Bayes theorem.

It is not clear to me what the author means by "there is no operational difference between a prior density $f(\theta)$, which measures belief, and $f(x \vert \theta)$, which measures an observable quantity". Furthermore, it is not clear to me how these two things are "conceptually equal as measures of uncertainty".

I would greatly appreciate it if people would please take the time to clarify this.

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In my view, the author is noting that a Bayesian interprets probability $p(A)$ as quantification of our degree uncertainty about $A$, whether $A$ represents the outcome of a random event (e.g. result of a coin flipping) or e.g. the numerical value of a physical constant (e.g. the speed of light). Both things are though and represented using a single tool (probability). So as stated "there is no operational difference between a prior density $f(\theta)$, which measures belief, and $f(x|\theta)$, which measures an observable quantity" and "the two things are conceptually equal as measures of uncertainty"

This view is often confronted to (among others) the frequentist view in which "the only sense in which probabilities have meaning is as the limiting value of the number of successes in a sequence of trials". In particular, $p($light of speed $)$ has no meaning (expect the limit case in which it equals 1 for the true value and 0 for any other values). Indeed, ng "light of speed" is not a random outcome and has a fixed value that may be, up to certain level, unknown but to which we can't apply a "frequentist" probability.

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