View Full Version : Economic definition of rationality (and irrationality)

jaimelannister

11-08-2010, 11:33 PM

Hi all,

This is something that has been bugging me for a while. Economists and non-economists* often throw the term rationality and irrationality around when referring to subjects pertinent to economics and I often find it is defined ambiguously that it could mean whatever the user might want it to mean. If you as an economist had to define economic rationality or irrationality how would you define that? I understand the definition of economic rationality and irrationality will vary depending on context, so I was also wondering what are the different definitions rationality and irrationality used in economics?

*I'm mainly interested in economist's definitions of rationality.

kevinp123

11-08-2010, 11:48 PM

I suppose I'm not an expert, but I think about rationality as acting so as to maximize utility. This is a topic that I think a lot of people get confused about. A recent poster here claimed that he was acting rationally by not telling one of his friends about this site, and it was pretty clear from the responses that most people disagreed with him (as they should). I think the confusion comes from people associating "rationality" with "self-interest". Acting so as to maximize utility is not the same thing as acting in your best interest. It is possible (and probable) that other people's feelings also factor into your utility function. Perhaps helping your friend get into a good school is a part of your utility function, or even if you don't care what the outcome is, just helping your friend might be part of your utility function. Nothing about keeping this site from his friend was "rational," especially considering the probability of it affecting his cycle is very low.

Another example I like that illustrates this point is a story about two economists who go out to eat at a local diner while on vacation. One economist notices that the other economist is leaving the waitress a tip and remarks, "How irrational. You will never see the waitress again, so why leave her a tip?" The second economist replies, "Because not being a schmuck is part of my utility function!"

nat_resource

11-09-2010, 12:12 AM

Revealed preference. You buy an apple for lunch when you could afford an orange. Next day you purchase an orange for lunch when you could purchase the apple.

economiste

11-09-2010, 01:00 AM

Being precise, it is not irrationality. Because having an apple today is different from having one tomorrow. They are two different goods. Nicely, people don't consider it as irrationality :)

Diplomer

11-09-2010, 01:16 AM

*I'm mainly interested in economist's definitions of rationality.

For me, an agent is rational if he exhibits a goal-directed behaviour and tries to achieve this goal in the best possible way, given his constraints. ("Goal" and "best" are, of course, to be specified according to the contexts...).

On a side-note to the revealed preference argument: even if WARP does not hold, this only implies that the preference relationship of this agent is not rationalizable, i.e. we won't find a weak order so that both choices are the best ones. However, there may still be some kind of rule guiding the agent in his choices. I would not restrict rationality to the possibility to find a weak order. Otherwise, many people would actually be considered irrational. Also, probabilistic choice and simply changing preferences would immediately be considered irrational. But I don't see why this should be the case. If I buy an apple today and, suppose, it tastes like ****, I will probably buy an orange tomorrow. I do not consider this irrational. Do you?

A vague but general attempt:

An agent is rational if he/she does whatever the modeler (i.e. economist) would do in his/her position.

treblekicker

11-09-2010, 01:50 AM

in a first graduate course in microeconomics, rationality is often taken to mean that an individual satisfies WARP/has complete and transitive preferences. but seldom is this what economists mean when they invoke rationality.

dreck

11-09-2010, 02:15 AM

Hi all,

*I'm mainly interested in economist's definitions of rationality.

OK, anyone who's taken the first grad micro course should know that rational preferences are defined as being complete and transitive. Competeness: for any two items in your set X, A is (weakly) preferred to B or B is (weakly) preferred to A, or both. Completeness basically rules out indecisiveness. Transitivity: A pref. to B, and B pref to C implies A pref to C. Google the Money pump if you want a defense of transitivity.

If the set of goods is finite or countable, then these two things yield a utility function from X to R with the desired properties, in other words we know that preferences give the set a weak ordering If the set of alternatives is uncountable, we need to add continuity to guarantee the existence of a utility function. Once you have a continuous or discrete utility function, you know that we can optimize over a compact domain (like a budget set), so then you can talk about a theory of demand, for example.

When an economist says rational, that's what he means. One can show that a choice correspondence (or "choice rule") satisfies the weak axiom if/only if it can be represented by rational preferences (and the set of alternatives can be reduced to singletons...it gets a bit technical). So WARP is kind of like an equivalent characterization of rationality, though there is an important technical caveat, which becomes important when you move toward a theory of demand. The first chapter of MWG is pretty good on this, or you can look at Rubinstein.

On the other hand, some economists and social scientists outside the paradigm of decision theory may understand economic theory but use the word rational to mean something different. Virtually every economic model outside the realm of pure decision theory will impose additional structure on preferences, in order to generate more useful predictions (Local Nonsatiation, homotheticity, the list is very long). An important example is whether the agent is an expected utility maximizer, which means that the agent deals with uncertainty in a particular, logical way (see MWG chapter 6 I think). This assumption in particular has fallen under heavy critique, but is still the frequently used in much of applied economics, and even game theory.

And regarding Diplomer's critique: A lot of supposed counterexamples to rational preferences demonstrate how preferences may reasonably be expected to change over time. HOWEVER, the stability of preferences over time is not required for rationality. All that is required is that preferences be stable in the window in which we model the behavior. If an apple today is different than an apple tomorrow, then you can't infer much from observing choices both days. In addition, if your enjoyment of an apple is stochastic, then you're in the realm of expected utility theory (eating an apple is a lottery), specifically subjective EUT if you don't know the distribution of good to bad apples, in which case what you describe could be called Bayesian updating.

A common assumption in macro (and dynamic models), for example, is that preferences are stable over time, but decision theory makes only a weaker claim from the outset. In this sense, the stability of preferences over time is one of those additional structures imposed on preferences to make models simpler.

dreck

11-09-2010, 02:27 AM

On a side-note to the revealed preference argument: even if WARP does not hold, this only implies that the preference relationship of this agent is not rationalizable, i.e. we won't find a weak order so that both choices are the best ones. However, there may still be some kind of rule guiding the agent in his choices. I would not restrict rationality to the possibility to find a weak order. Otherwise, many people would actually be considered irrational.

There's a really cool set of choice rules in Rubinstein's lecture notes on this problem. The most famous is that in group choices, the majority rule may not satisfy transitivity (the Voting Paradox). See also Arrow's Impossibility Theorem, if you want to go deep into this...

dreck

11-09-2010, 02:46 AM

Revealed preference. You buy an apple for lunch when you could afford an orange. Next day you purchase an orange for lunch when you could purchase the apple.

1. Preferences need not be stable over time

2. Even if we assume stability, this is not a contradiction of WARP if we regard the choice over both days simultaneously, i.e. if you choose bundles on {buy nothing, orange today apple tomorrow, apple today orange tomorrow, orange today nothing tomorrow, etc}.

The intuition for WARP giving choices internal consistency is right. It's just important to be very careful in deciding if a particular behavior violates WARP. Here's another example:

If you choose vanilla ice cream when your choices are chocolate, strawberry, and vanilla, then you shouldn't choose chocolate when your choices are vanilla, chocolate, and butter pecan. Because vanilla was revealed preferred to chocolate in the first case, but after that chocolate is revealed preferred to vanilla. Which contradicts WARP.

kipfilet

11-09-2010, 03:06 AM

I guess it would just had served to direct the OP to MWG's Amazon page.

oyvey

11-09-2010, 03:33 AM

i think it is worth mentioning that there is a diffence between rational preferences and rational decisions, as decisions have to take into account a budget constraint

Diplomer

11-09-2010, 03:45 AM

OK, anyone who's taken the first grad micro course should know that rational preferences are defined as being complete and transitive.

This is how some people name such preferences. Others call it a weak order. It does, however, not necessarily reflect what people actually mean by rational agents' behaviour. The most general definition is that a person is rational if he deliberately chooses what he likes best (which is equivalent to my first description and which incorporates anything else that has been said here). This may result in a utility maximization paradigm (if your preferences are Jaffray-order separable, then there is such a function), but not necessarily so. The definition by having a weak order unnecessarily excludes much of the research done in the last two decades in decision theory and behavioural economics. People actually try to model non-weak orders so that some value function is maximized. According to the definition of rational preferences (which some people confound with rational behaviour) this would qua definition be irrational, but it can be embedded in a maximization framework and it can look quite rational to me and many other economists.

One can show that a choice correspondence (or "choice rule") satisfies the weak axiom if/only if it can be represented by rational preferences.

It does not matter in the current context, but this is not true. Rationalizability implies WARP, but the converse is not true. Consider what happens if you have distinct choice sets.

dreck

11-09-2010, 04:26 AM

It does not matter in the current context, but this is not true. Rationalizability implies WARP, but the converse is not true. Consider what happens if you have distinct choice sets.

It is true you if you add that the family of subsets to which the correspondence applies contains all subsets of X up to three elements (MWG Prop 1D2). That's what I meant by "technical caveat" in the next sentence. And I noted that this assumption doesn't hold when we look at, say, budget sets.

I am aware that some decision theory that does not assume W.O; the professor for the course I took works on these problems.

I pointed out that some people use rationality to mean something other than W.O., but you are correct to add that even the completeness/transitivity result is sometimes replaced with some other structure on preferences that generates an intuitive rule.

jaimelannister

11-09-2010, 04:35 AM

I guess it would just had served to direct the OP to MWG's Amazon page.

Just to be clear I do know what the technical definition is (the undergraduate version anyway) but I was looking for a more in-depth discussion like the one that is currently underway. The genesis for the post was a discussion I recently had with someone who defined economic rationality to be incongruent with my own understanding, and that got me thinking about whether there were different versions.

And I was also wondering how the different schools of thought (e.g. Austrians) treat rationality. Do they have their own versions or don't use rationality at all? And what about the way behavioural economist treat the standard definition of rationality. Based on my (very) limited understanding of behavioural economics, one of things they do is look for cases of irrationality (based on the technical definition of rationality) right?

BTW if anyone is wondering this isn't for a school assessment or anything like that, but more for the benefit of my own personal understanding.

econm

11-09-2010, 04:47 AM

rational = maximizing utility

or having subgame perfect equilibria in games.

dreck

11-09-2010, 06:55 AM

There are many flavors of very behavioral economics. Most critique the traditional imposition of structure on preferences/choices; some directly critique completeness and transitivity, and still others attack stability over time or the way the agent deals with uncertainty, which do not rely on the traditional technical definition of rationality, but do depend on the more broad assumptions that constitute the general notion of rationality. There is also good work in behavioral economics which seeks to replace the weak-order assumptions with other assumptions based in psychological insights and/or experimental results, and generate different kinds of choice rules or preferences. How those things relate to rationality is a bit subjective and unclear; often the traditional theory is a nested special case of the new stuff.

The kind of behavioral economics that makes me queasy is the kind that says, "look, these choices look surprising," and concludes that they are a) irrational, and b) evidence against canonical economic models of choice. Which is what you often see in best-selling books.

Andronicus

11-09-2010, 07:20 AM

In addition to maximizing utility, rationality generally also requires that agents use all the information available to them to update their beliefs according to Bayes' rule.

Cuak2000

11-09-2010, 11:45 PM

I disagree with the idea that rationality as used in economics corresponds to the complete and transitive preferences. As someone already pointed out, at the very least we should distinguish between the rationality of preferences (as is captured by those traditional axioms) and the rationality of decisions. (Further, we could apply the rationality label to the process with which the person comes to take a decision (for which economics is usually silent, although I've read stuff from Herbert Simon & also Rubinstein that talk about it)).

Rather than being identified by particular constructs, I think rationality in economics is a pretty fundamental building block that takes different appearances in those two or three senses above (as in complete order, maximisers of utility, etc.).

Right now I don't feel like trying to characterise it though (I would probably do a bad job in any case)!

Two links which might be of interest:

Practical Reason (Stanford Encyclopedia of Philosophy) (http://plato.stanford.edu/entries/practical-reason/#MaxRat)

Oxford University Press: Decision Theory and Rationality: Jose Luis Bermudez (http://www.oup.com/us/catalog/general/subject/Philosophy/Pragmatism/?view=usa&ci=9780199548026)

Irrational

11-10-2010, 02:20 AM

Consider, that our brain might be rational in its construction, but leading to ostensibly irrational behaviour, if examined in a too small frame.

To make this clear: Imagine there is a limit in brain capacity.

We can now (in the progress of evolution) allocate brain capacities towards different tasks (like skill points in a RPG). A heuristic requires very little capacity but works pretty good and covers many tasks. Although it has weaknesses, it might possibly be an element of 'the optimal' allocation of brain capacities. Being able to behave 100% 'rational' on 1 task, on the other hand, might limit us in the variety of tasks, because it takes a lot of brain capacity. We need to cope with a variety of tasks --> it is 'rational' to use heuristics and be 'irrational' in some ways. (One extra step: Maybe it is 'rational', that we do not understand our 'rationlity', but rather spend the extra resources on reproducing/learning Danish/building rollercoasters)

Beyond this, why would we need a notion of rationality vs irrationality?

One might say, that it helps people to understand people how they should act (normative approach). To justify this, one has to convince people of his axioms.

Let me make some points on this:

a) There no such thing as inconsistent behaviour/acting/(any entity of reality). It is always the theory, that is 'wrong'.

b) Thus, any normative approach, is either positive or axiomatic/arbitrary/(unrelated to reality). If someone approved of a normative axiom, this approval is either wrong or the approach will be positive immediately (because the alleged axiom is describing reality).

c) Moral issues. I like to violate axioms (anyone you can come up with) with my behaviour. Please do not tell me how to behave. Please do not change my behaviour.

d) Decisions are context dependent. If an economist convinced an individual to behave acoording to some axioms (e.g. transitive choice) for the rest of his lifetime, the individual's context has changed for the rest of lifetime. To justify axioms, one has to justify this shift. But this impossible without argumenting either arbitrary (= more axioms) or using individual decision (circlular reasoning) or using descriptive arguments (which themselves can hardly serve as normative, see naturalistic fallacy)

Brownie Point argument for Gödel friends: I like to behave, such that my behaviour cannot be formalized. Neither in the notion of rational/irrational nor in any formal system (nor in a sentence).

So the point, I want to make with this post, is that a notion of "rational vs. irrational" is of illusional utility. Of course, it seems interesting to find out what an indidual would do, if he/she behaves according to a set of axioms, that seem desirable at first sight. But for the comparison with actual behaviour at the end of the day, it is of no more worth than the behaviour of dinosaurs in post-war's society.

Because my points are formulated extremely, let me talk some positives: I believe, it is great, fun and important to examine our individual behaviour and despite Gödel's results try to come up with good models for it. And this is actually the reason, why we should get over the unfortunately very prevalent comparison of "irrational" and "rational" behaviour and finally start focussing on important descriptive research.

I am not yet fully convinced of everything up there. And I would love to get any feedback!

Elliephant

11-10-2010, 02:40 AM

And I would love to get any feedback!

Who are you to define and defame rationality? Surely we can't take you seriously since you are Irrational ;)

dreck

11-10-2010, 03:39 AM

we should get over the unfortunately very prevalent comparison of "irrational" and "rational" behaviour and finally start focussing on important descriptive research.

Well, many decision theorists are doing just that.

The reason many people seek to create a distinction between rational and irrational behavior is to critique economic theory, and its predictions on behavior. However, in its full-blown form, economic theory makes weaker assumptions and more general predictions than one might think. That's what I was trying to point out in the first post, which was probably too long in retrospect.

We can often improve theoretic prediction, however, if we incorporate more information on how individuals make decisions and change to more realistic assumptions, and thats where the research is headed now. Trying to get some individual to conform to another's idea of rationality is virtually never a goal of economic research (des gustibus non est disputandum, though that's not the golden rule it once was, see Stigler and Becker's critique).

Furthermore, in the case that preferences change unexpectedly over time, ideas like pre-commitment mechanisms can lead to welfare improvements. Some cast this in the light of rationality, but one can reduce the idea to something involving preferences and/or choices. Come to think of it, most concepts that are supposedly about rationality are really about preferences and choices, which have a much more specific language. So if you want to be an economist, that's the language you should use.

Irrational

11-10-2010, 12:34 PM

Okay, sorry for the last sentence, it was probably too dramatic.

The reason many people seek to create a distinction between rational and irrational behavior is to critique economic theory, and its predictions on behavior. However, in its full-blown form, economic theory makes weaker assumptions and more general predictions than one might think.Yes. Of course part of the epistemology is comparing predictions of our theory with actual behaviour. And then enrichen our theory (Kuhn?). And yes, decision theorist are doing good descriptive research (afaik). But many did not get over "rationality".

And it's just the "rational vs. irrational" thing, that I "define and defame". Using the notion of 'rationality' is arbitrary and irritating ("This is something that has been bugging me for a while."). And in the end, it doesn't help us with our problems.

For instance, what can someone possibly mean by talking about 'rational behaviour'? He could talk about a set of axioms defining/predicting behaviour. But there is no unique set of axioms, that one can assign the label 'rational' to. And even if all economist could agree "yes these axioms describe rational behaviour", what would that label help us with our behavioural questions? (The answer is: nothing without being normative, which I have discussed in my previous post.)

Let's forget about "rationality". We can do well with "predictions vs. reality" and the "language of preferences and choice", as dreck has pointed out.

Unidentified

11-10-2010, 05:51 PM

I think the definition of rationality which most economists do have in mind can be summarized under the term Bayesian. Which does not only mean that information is updated according to Bayes rule but also that we can assign probabilities to events and can assign preferences over acts.

All papers (I have read) in decision theory fall into this category, even papers which try to model 'irrational' behavior like prospect theory.

The question of what is rationality becomes even more complex if we move from static decisions to dynamic decisions. In economics, decision theorists demand that behavior should be 'dynamically consistent', i.e. they don't deviate from a pre-set plan. Even if the behavior would not be dynamically consistent (e.g. Hyperbolic discounting) economist try to make the behavior dynamically consistent.

It is quite ironic that economists use the term 'rationality' in a very informal way.

Andronicus

11-11-2010, 12:40 AM

I think the definition of rationality which most economists do have in mind can be summarized under the term Bayesian. Which does not only mean that information is updated according to Bayes rule but also that we can assign probabilities to events and can assign preferences over acts.

All papers (I have read) in decision theory fall into this category, even papers which try to model 'irrational' behavior like prospect theory.

There's lots of decision theory papers that deal with non-Bayesian decisions (Knightian uncertainty) such as the work of Schmeidler and Gilboa on Choquet expected utility and maxmin expected utility.

azzobrother

11-11-2010, 04:57 AM

Your rational if you can be expressed as the ratio of two integers. Your irrational if and only if your not rational.

Joke aside, my understanding of rationality is based on whether or not a decision made by a person follows logically from their previous action. Or the decision that they make is logical given the circumstance that they are in. From this, it's just a matter of representing the idea in symbols (binary relations, X={set of goods x1, x2, x3,...}, etc.). For example, suppose you get into a terrible car accident, the logical thing to do is to call 911 and get the other guy's insurance information. It would be illogical of you if instead you decide to order a medium pizza. This is an extreme example, but I hope it demonstrates my point.

Unidentified

11-11-2010, 08:56 AM

There's lots of decision theory papers that deal with non-Bayesian decisions (Knightian uncertainty) such as the work of Schmeidler and Gilboa on Choquet expected utility and maxmin expected utility.

We still have some states of the world, assign objects called probabilities (or sets of probabilities, or generalized probabilities) with these states and have preferences over acts. Therefore these models are still Bayesian.

Correction:

Maybe I should have been more precise as probabilities are not primitives. In Bayesian decision theory we have states of the world, acts, consequences, and preferences over acts. The mentioned models are still bayesian according to this more precise definition.

Unidentified

11-11-2010, 09:46 AM

Your rational if you can be expressed as the ratio of two integers. Your irrational if and only if your not rational.

Joke aside, my understanding of rationality is based on whether or not a decision made by a person follows logically from their previous action. Or the decision that they make is logical given the circumstance that they are in. From this, it's just a matter of representing the idea in symbols (binary relations, X={set of goods x1, x2, x3,...}, etc.). For example, suppose you get into a terrible car accident, the logical thing to do is to call 911 and get the other guy's insurance information. It would be illogical of you if instead you decide to order a medium pizza. This is an extreme example, but I hope it demonstrates my point.

Your decision of rationality has the problem that you just replace the word 'rational' with 'logical'. But you didn't define what a 'logical' decision. If you jump from the roof of a 50m building and survive then it can be a logical decision to jump of the roof from a 60m building (at least for some people). It passes your test of rationality, but most people would say that this behavior is 'irrational'.

Irrational

11-11-2010, 01:34 PM

We still have some states of the world, assign objects called probabilities (or sets of probabilities, or generalized probabilities) with these states and have preferences over acts. Therefore these models are still Bayesian.

Correction:

Maybe I should have been more precise as probabilities are not primitives. In Bayesian decision theory we have states of the world, acts, consequences, and preferences over acts. The mentioned models are still bayesian according to this more precise definition.

This sounds very interesting. I am not quite sure, whether I understand it correctly. Do you say that, even if I a develop a model of behaviour that is completely random in its decision making, it would be Bayesian? Although people do _not_ assign probabilites (their decision is just randomly determined).

So is every model that works with probabilty Bayesian (according to your sense) or is it only the models that assume individuals work with probabilities?

And I second your critique on shifting the 'rational' problem to the term 'logical'.

Unidentified

11-11-2010, 02:26 PM

This sounds very interesting. I am not quite sure, whether I understand it correctly. Do you say that, even if I a develop a model of behaviour that is completely random in its decision making, it would be Bayesian? Although people do _not_ assign probabilites (their decision is just randomly determined).

So is every model that works with probabilty Bayesian (according to your sense) or is it only the models that assume individuals work with probabilities?

And I second your critique on shifting the 'rational' problem to the term 'logical'.

I think I didn't express myself clearly enough. Probabilities are not primitives in the bayesian approach. In Bayesian decision theory you have as primitives: states of the world, acts, consequences and preferences as primitives. Any model of behavior which has these 4 primitives can be considered as Bayesian.

And yes, if you could model such behavior with these 4 primitives then it can be considered Bayesian. But I doubt that someone will be able to find preferences which represent a total random behavior.

Irrational

11-11-2010, 02:52 PM

Okay, so Bayesian is everything that uses the primitives states of the world, acts, consequences and preferences. And you haven't seen anything in decision theory that does not rely on those primitives, because even stuff like Knightian Uncertainty relies on those primitives. Have I understood correctly? Assuming so, I'd like to make 2 points:

1) "I think the definition of rationality which most economists do have in mind can be summarized under the term Bayesian." I don't get this. If any known model can be considered Bayesian, what do you think is use of a distinction of rationality (for most economist)?

(I put this in italics, because it does no longer really relate to the topic of 'rationality')

2) "But I doubt that someone will be able to find preferences which represent a total random behavior." If you drop assumptions of determinism, it's not that hard any more :) Consider that total random behaviour does not mean euqal probabilities and that there are some physical theories of randomness. This randomness is in our brain, which is "probably" related to our decisions. Surely, this is not too practical. However, it demonstrates that the Bayesian approach is neither perfectly firm nor essential.

Unidentified

11-11-2010, 03:05 PM

Okay, so Bayesian is everything that uses the primitives states of the world, acts, consequences and preferences. And you haven't seen anything in decision theory that does not rely on those primitives, because even stuff like Knightian Uncertainty relies on those primitives. Have I understood correctly? Assuming so, I'd like to make 2 points:

1) "I think the definition of rationality which most economists do have in mind can be summarized under the term Bayesian." I don't get this. If any known model can be considered Bayesian, what do you think is use of a distinction of rationality (for most economist)?

(I put this in italics, because it does no longer really relate to the topic of 'rationality')

2) "But I doubt that someone will be able to find preferences which represent a total random behavior." If you drop assumptions of determinism, it's not that hard any more :) Consider that total random behaviour does not mean euqal probabilities and that there are some physical theories of randomness. This randomness is in our brain, which is "probably" related to our decisions. Surely, this is not too practical. However, it demonstrates that the Bayesian approach is neither perfectly firm nor essential.

1) bayesian decision theory is the predominant approach in economics but not all decision models can be considered as bayesian. If you look outside economics, there is a lot of work and research which can be considered non-bayesian, e.g. in computer science or philosophy. A good reference to this work "Non-Bayesian Decision Theory" by Peterson (a Philosopher), of course he pushes his own theory of how people should act in this book.

2)I think we have two different interpretations of the words preferences. With mean preferences I mean the binary relation we have over the acts. And I doubt that such a binary relation can be found.

Irrational

11-11-2010, 03:36 PM

1) Thank you for the reference. What do you think is the use of a term 'rational'? Is it merely another word for 'Bayesian'?

2) a) It might also be a discrepancy of understanding the term "random behaviour". Consider someone whose preference over the acts "eating apple" and "eating orange" change every year due to initially random events. Would you say he inhibits random preferences?

(One could put this in a more specific frame and say that the particles determining his preference are randomly either acitivated or not after an individual sees an apple juice commercial.)

b) How is a natural uncertainty modeled in Bayesian theory with only those 4 primitives?