A distinction between belief and action is made explicit in decision theory, particularly in Bayesian decision theory (the version I'll briefly outline here). An earlier answer, referring to Pascal's wager, can be viewed as an example of the application of decision theory as a normative theory for how to choose to act in the midst of uncertainty. Indeed, Wikipedia's page on Decision theory notes that Pascal knew the key ideas of decision theory and invoked them in his eponymous wager argument.
Basically, decision theory says that how one acts should depend, not just on what one believes (and how strongly), but also on judgments about the consequences of one's actions.
(Apologies for the "bare" math notation below; it works in other StackExchange forums.)
To be especially clear, belief in (Bayesian) decision theory is distinct from feeling certain that something is true (which may be what the OP intended). There are degrees of belief, quantified via probability on a 0 to 1 scale (or equivalently with other related scales, such as percentages or odds). For a proposition (hypothesis) $H$, a probability of 0 corresponds to being certain that $H$ is false, a probability of 1 corresponds to being certain that $H$ is true, and values between these extremes correspond to degrees of belief. Probability is viewed as always contextual (i.e., based on some known or presumed information); $P(H|C)$ is the probability that $H$ is true given that the contextual information $C$ is true, the degree to which $C$ implies $H$, etc..
When faced with a choice between various actions whose consequences depend on things ("states of the world") we are not entirely certain about, decision theory requires a certain structure in order to identify an optimal decision (choice of action). The decider needs to specify the following:
- The action space (the possible decisions, $a_1, $a_2$, etc.);
- The state space (possible states the world may be in, $s_1$, $s_2$, etc., that bear on the outcome of the action);
- Enough information to assign probabilities to the states;
- A utility function, $U(a_i, s_j)$, a function of action and state that specifies the value or reward to the decider if they choose action $a_i$ when the true state is $s_j$. Equivalently (pessimistically rather than optimistically!) the decider may specify a loss function, $L(a_i, s_j)$, that specifies the cost to the decider if action $a_i$ is chosen when the true state is $s_j$.
In discrete settings (as here), the utility or loss function may be represented as a matrix.
One simple class of decision problems has as actions simply asserting that one of the possible states is in fact the true state. This class of problems arises commonly in statistics, as briefly described below. It speaks particularly clearly to the question of distinguishing action from belief.
Decision theory starts from some basic, appealing desiderata and shows that the optimal decision (choice of action among the $a_i$) is the one with the greatest expected utilitity, computed for each possible action by adding up all the utility values for that action and every possible state, weighted by the probability for each state. Equivalently, one should choose the action with the lowest expected loss.
It should be easy to see that decision theory does not in general lead one to act as if the state with the largest probability were the true state. A simple but clear example is the game of Russian roulette: You are offered some winnings, say, $1,000, to point a revolver at your head and pull the trigger, knowing that only one of its six chambers have a bullet, and after the cylinder has been spun so that the probability that the bullet is in the active chamber is $1/6$. It is five times more probable that the active chamber is empty. That is, one's belief is strong that pulling the trigger is safe. Nevertheless, one's sanity would be legitimately questioned if one agreed to play the game. It is much more likely that the active chamber is empty rather than full. But the consequences are much worse if the chamber is full and one plays (you get the winnings but lose your life), versus if the chamber is empty but one chooses not to play (you do not gain the winnings, but you keep your life).
This is what I was getting at above when I wrote that decision theory says that how one acts should depend, not just on what one believes (and how strongly), but also on judgments about the consequences of one's actions.
It's worth noting that it can be hard to specify utility or loss functions. For the Russian roulette example, in order to formulate it as a decision theory problem, you'd need to assign a dollar value to your life (though it's possible to reach a decision just by bounding that dollar value).
Decision theory is commonly used in statistics to identify the best estimate of some real-valued parameter (say, the amount of rainfall tomorrow) when the available data (and any other knowledge or assumptions) cause one to be uncertain about the parameter's value. The uncertainty can be represented by a probability density function (a so-called posterior probability density in Bayesian statistics) over the possible values of the parameter. If the utility function grants you a reward for correctly estimating the parameter's value, but nothing if you are incorrect, the best estimate the one with the largest probability density (the mode, e.g., the amount of rain for which the probability density is largest). But if there is some cost to being incorrect that depends on the size of the error you make, in general the best estimate won't be the one with the largest probability density. For example, if the loss function is quadratic with distance from the true value, then it turns out the best estimate is the posterior mean—a probability-weighted average of the possible values. When the uncertainty is symmetric about the mode, the posterior mean is equal to the mode. But in general the posterior mean is different from the mode. That is, in statistics, we quite commonly "act" (choose a parameter estimate) in a way that doesn't exactly mirror our "belief" (where the largest probability is).
