In my view, the author is noting that Bayesian probabilities $p(A)$ interprets probability as quantification of our 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. light speed). 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, i.e. as" "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). In this setting "light of speed" is not a random outcome and has a fixed value that may be, up to certain level unknown level, but to which we can't apply a "frequentist" probability

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.

In my view, the author is noting that Bayesian probabilities $p(A)$ interprets probability as quantification of our 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. light speed). 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"

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, i.e. as" "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).

In my view, the author is noting that Bayesian probabilities $p(A)$ interprets probability as quantification of our 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. light speed). 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, i.e. as" "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). In this setting "light of speed" is not a random outcome and has a fixed value that may be, up to certain level unknown level, but to which we can't apply a "frequentist" probability