Introduction to Monte Carlo Methods

hi everyone today we're going to be talking about Monte Carlo methods and in particular we're going to be focusing on three key things the first is what are Monte Carlo methods how are they used what are they for the second thing that we're going to look at is applications of Monte Carlo methods and in particular we're going to look at the advantages and disadvantages of using them in our analysis a final thing that will conclude with is key issues with regards to Monte Carlo methods when we're considering incorporating them into a project this is in particular for decision-makers and for people who are not necessarily of a technical background but need to use Monte Carlo methods in their project whatever that may be this is a very non-technical overarching perspective on this method and it's important to understand that these tools even when they're complicated they have lots of low nomenclature and people from very highly educated backgrounds are very enthusiastic about them it's important to remember that they have very simple underlying concepts and for you as a decision maker it's important to say okay what is this and how does this work what is the key underlying thought behind the system and how can I use it for my project going forward a natural point to start off from is what are Monte Carlo methods Monte Carlo methods are the incorporation of probability into analysis we use this when we know that a certain set of scenarios can happen but ensure whether they will happen an example of this is my morning routine in the morning I wake up I brush my teeth I take a shower and a have breakfast those four steps are what occur in my morning routine however even if those four steps usually occur and are most likely to occur they're not the only scenario that happens in my morning sometimes if my fridge is empty I won't have breakfast that day you know even if it's unlikely for me not to have an empty fridge that that scenario could occur therefore we'll try to incorporate probability if we want to analyze Allen's morning so to see what would happen so alone in the morning we'd have to think about well okay the scenario can happen we have to incorporate it into our expectations of what does happen in Allen's daily morning routine and that's what Monte Carlo methods are for the way that Monte Carlo methods are set up is that we examine the world as a system that is we take a lens morning of waking up brushing his teeth taking a shower and having breakfast those four steps and we say this world is a system these are the four things that occur and we're going to try to model them in a comprehensive way the way we would create this is we'd formulate the world in many little steps in the case of Oh lawns morning one step would be a morning that is the four actions of waking up brushing teeth taking a shower and having breakfast those four actions constitute in this example a step that is one realization one scenario that happened within this Monte Carlo method each step that we take within this morning involves a realization of the world that is we pretend for the sake of the argument that one waking up brushing his teeth having a shower and having breakfast is a realization of the world and and some other realization of the world alone doesn't have breakfast what we do with Monte Carlos is we repeat these steps that is we see okay all on woke up brushes teeth took a shower had breakfast alone that's step number one step number two is long woke up brushes teeth took a shower and didn't have breakfast and that's another scenario that could happen and that's a different realization of the world and that will be realization number two for instance in this example and what we do is we take many of these little little steps that is we take many of these mornings and collect them together and when we collect them we can analyze it eventually what we expect that is as we collect all these little little mornings if slight variations perhaps I didn't brush my teeth because I was out of toothpaste or perhaps since I taken a shower the previous evening I don't take a shower that morning is that we expect eventually for all these collections and morning's to allow us to analyze some desirable measure of one's morning's for instance how often does Ilan eat what's the likelihood that one will eat in the morning or for instance how long does it take for one's morning to set up because sometimes if I don't take certain steps in the morning my morning may be shorter than it may be in other mornings and we would want to know let's say the average morning or the most typical morning the chart here illustrates how Monte Carlo method converges to the measure that you are looking for that is what path does the Monte Carlo system take until it gives you the desired result in the beginning since we have very few realizations or steps of the world the measure that is our desirable output is very volatile that is it changes quite substantially however we expect that as we take more and more steps our uncertainty or our lack of knowledge as to what the actual measure is gets less and less and less until it converges to a constant this is the underlying path by which all Monte Carlo methods work in a moment we'll dive deeper into what exactly a measure is and how it relates to Monte Carlo a natural question to ask is what is a measure let's imagine that there are two things there's yourself and your environment or the world around you what is the linkage between yourself and your thoughts and your environment one linkage is the direct linkage between what you know and your environment and this is called logic this is a priori information that you have that you use when interacting with the world example of this include math and languages there is however another linkage between the world and yourself or your thought process and that is an observation that is a piece of information or a thought that you could not have created without interacting with the world what measures do is produce your uncertainty as to the observations or uncertainty as to the information that you have that's coming from your environment that's what measures allow us to do as we are more confident in our measure we can use that to infer more about our environments this information is not a priori such as logic or math example this include speeds physics biology and many of the Natural Sciences we're now going to look at an overview of Monte Carlo methods from an overarching perspective in how does this process of analysis actually work we start off with reality which is what actually happens in our environment and from reality we infer or draw inferences and create a Monte Carlo model that is we create a system or a set of steps if you remember from the previous slide that tries to approximate that reality in this model there are inputs there's a system and the system takes those inputs and transfers them to outputs that is your Monte Carlo model however we know that the model itself doesn't always approximate reality sometimes the outputs that are driven by our system and inputs is incorrect or at least it does an approximate reality to the level that we would like to in that case we'll take information from our outputs and change the inputs that we put into the system this process of observing outputs and then changing the inputs and then re observing the outputs again is called calibration and that this occurs when you're calibrating your Monte Carlo model another thing that you can do is actually change the system itself that is observe the outputs of the model and change the entire components of the model this step is much more sophisticated and is usually left for more seasoned researchers and an less because the results of calibrating system are much more variable than some merely changing the inputs of the model this component of Monte Carlo methods is the synthesis component that is we take reality and we synthesize a Monte Carlo model and from that model we use the outputs to change the inputs and how the system translates the inputs into outputs once we have our model and we are satisfied with the results we take the results of that model and infer about reality this is the inference component of a Monte Carlo method it's important to remember that inference stage that is taking the results from your model and inferring about reality is a totally separate stage from the synthesis stage that is how one creates a Monte Carlo model and support and understand that just because you synthesize the model well does not mean you can infer from it this is a very important step to remember when considering the implications of using Monte Carlo methods we will now discuss the different types or different flavors of Monte Carlo models in a simple Monte Carlo model you have a beginning state which state N and you have resulting saved River States a States B and States C which have respective probabilities of occurring from starting in state n those probabilities are designated by the blue arrows in this case a relationship between state n and state a B and C are direct and linear that is there's a direct relationship between state n and state a state B and state C a more complicated or sophisticated manner of analyzing it is to take the simple Monte Carlo that is start from one stage and go to the next stage and expand it so that you have a two-stage Monte Carlo or even an infinitely stage Monte Carlo that is instead of going from one step to another step you can go in multiple steps or in this case you can go from state n with some probability of state C and then from Stasi with some probability of States e or states D another form of Monte Carlo models is mixture modeling that is instead of having linear or direct blue arrows the probabilities themselves are probabilistic that is there is some function or likelihood within the probabilities themselves examples of this include using two different kinds of distributions or even infinitely different kinds of distributions or a functional distribution with mixture modeling what we're doing is we're making the probabilities not linear that is they have some functional or some indirect form into transitioning that is going from state n the states a B or C or going from state n to States a B or C that is it's not always the case that the likelihood of going from state ends state a B and C is the same the final and most sophisticated Monte Carlo model is a Markov chain in this case every realization impacts the likelihood of a future realization of the model that is in state a the fact that a occurred will influence the likelihood of transitioning to state B state C and state D and in this case it's actually a recursive model that is current realizations are based upon past realizations this is the most sophisticated form modeling and is often left for those who are very experienced in Monte Carlo methods the reason for this is that because the transitions are nonlinear that is it's not a simple functional approximation to go from state a to state B to state C to state D one has to be very careful that the measure doesn't converge to something outside of the sphere of possibilities examples of this include negative volatilities or imaginary numbers and and so forth when comparing contrasting these models it is important to understand exactly how the extra sophistication education of Monte Carlo model best approximates what you're analyzing often using a simple Monte Carlo model or an end stage Monte Carlo model is more than sufficient for analyzing the object that you're looking at the reason why we would want to stay with a simple Monte Carlo model or an ncage Monte Carlo is that the relationships are linear that is there's the linear relationship between my inputs and my outputs and it makes validating and understanding the model much easier when one has mixture modeling or Markov chains often the results of the system are our black box that is it's hard to see how all the interactions together create the desired results therefore I highly recommend that you should only transition to mixture modeling and Markov chains when you feel 100% comfortable with a simple Monte Carlo or end-stage Monte Carlo and you feel as though the extra sophistication will bring greater insight into your analysis now that we have a better idea of what Monte Carlo methods are next we're going to be looking at what Monte Carlo methods are actually useful for or good for one thing the Monte Carlo methods are very useful in is examining complex aggregations from simple actions an example of this is the dropping of a grain of sand from the sky to the ground that is it's very simple to model how a grain of sand drops to the ground and if we're very certain about how that process occurs we can examine how a grain of sand and dropping it contributes to a pre-existing pile of grains of sand that is a complex aggregation of dropping many little individual grains of sand Monte Carlo methods are also useful when we're trying to incorporate uncertainty into our analysis that is when we are not a hundred percent sure of whether our result will occur it is very useful to penalize our uncertainty or incorporate it into our analysis an example of this is if we're considering a business venture and we have some expectation as to how much money we'll make we can use a Monte Carlo method to help us understand how uncertain we are about that result and how that will impact the profitability of our venture another useful application of Monte Carlo models is to explore unintuitive results that is if we think we understand drivers of a certain process Monte Carlo methods can help us explore other drivers that may actually be more in and the drivers that we originally thought and that's what Monte Carlo methods are very useful for is trying to figure out which drivers could actually be contributing to our observation another use of Monte Carlo methods is to simplify a complex system such as city plumbing for instance the entire sewer systems of a city are very very complex if we want to understand a simple component of that system such as the likelihood of a breakage or how much water actually can flow through the system Monte Carlo methods are very good for simplifying those results and aggregating into key pieces of information that we can use to make our decisions however like all things in life Monte Carlo methods are not good for certain things we have to be fully aware of the flaws in using Monte Carlo analysis Monte Carlo methods are not good for examining or inferring simple actions from complex aggregations that is if there's some complex result out there such as the composition of a population within a country what ethnicities are there how are they distributed it's a very very complex aggregation that has many many underlying drivers Monte Carlo methods are not good at finding which underlying drivers are actually creating this result Monte Carlo methods are also not useful for finding realistic results that is Monte Carlo methods there are simple input and they translate them through the system into an output the actual realism or how credible that result is cannot be solved by Monte Carlo methods they are just not useful in creating reasonable results we have to use calibration and system adjustment for ourselves as researchers in order to make the Monte Carlo system provide the results that we're looking for communication is another very big issue with Monte Carlo models especially when trying to look at how we translate the models that we've created to people who are either unfamiliar with the models or may be familiar with role models but not with your model because there are so many steps and so many nuances when doing Monte Carlo method communication is a substantial barrier to incorporating them effectively robustness is another issue for Monte Carlo models many people especially those who are inexperienced with them are not aware of how sensitive Monte Carlo models are to the various inputs and the various parameters that we as researchers put into the models this can mean that slightly changing a value even if it's a credible change can totally change how you would infer the results of the model and this is a huge huge issue currently in Monte Carlo research speed is another issue though this isn't nearly as common these days as it used to be in the past but the speed of a Monte Carlo model could be very very slow depending on how the researcher is created and what environment it's in this means that when people want an immediate result Monte Carlo models are not the best template for achieving that let us start with the deeper into what we mean when we talk about complex aggregations in the context of Monte Carlo what are these methods really good for in a sense Monte Carlo methods the reasonable problems where we can easily measure the complete set of actions within the system but we're really not sure of the aggregate results examples of this include insurance market sports and demographics these three examples highlight a very important angle from which one should look at what Monte Carlo methods are really useful for and all three of them the complete sphere is very easy to discern within the context of sports it's whether team or loses the rules are very well established and they do not change that much from game to game and the insurance market as well when one looks at weather insurance happens one only has to examine the incidence of a claim how large that claim is and how much the charge within the context of a person trying to buy insurance against a claim and in the case of demographics as well since demographic is a very slow-moving phenomenon and we have very good ways to measure how demographics change over time it is very well suited for Monte Carlo method on the other hand multicolor methods that have disadvantages in complex problems where we don't know the complete set of actions we really have to infer from the aggregate of outcome examples of this include market prices biological systems and the weather is important understand that what we want in a more mathematical sense is that the function is known and the input to the function is known but we're not sure what the result is or what the why is what we do not want and this is what they're bad for is we do not want to have a result that is the why and try to infer the complete set of functions and inputs that would drive that result because the fact is there's an infinite number of these combinations that you can get for this result and it doesn't lead to correct inference let's also now examine how we actually try to approximate reality or if you remember for the model synthesis stage of constructing a Monte Carlo method an example of a good use of Monte Carlo methods are one one explore what-if scenarios where we really don't know what the current realization is but we do know that it's not only possible scenario the question is what if that would happen or what if it's likely that this would happen what should we expect how should we make our decisions what risk parameters should we put in place these are really good uses of Monte Carlo methods another good use of Monte Carlo methods are the test priors and when I say priors I mean within the Bayesian context of a priori knowledge that you bring to the table and trying to examine a problem an example of this would be the likelihood that the Sun will rise tomorrow empirically even if you don't have the full capabilities of proving the Sun will rise tomorrow it's fairly likely based on your prior knowledge that the Sun is risen in the past and testing whether this is true that is whether the Sun will rise tomorrow and testing your prior to see against what is it robust and what is it sensitive to is really good use of Monte Carlo methods there are of course bad uses of these methods one is finding reasonable scenarios there's nothing within the Monte Carlo framework that tells you which scenarios are reasonable and which are not in fact it will often produce scenarios which are unrealistic or unreasonable and so for you as the person analyzed trying solves problem to understand that Monte Carlo can produce cinemas that are literally unrealistic and you have to take account of that when interpreting the results another issue that Monte Carlo methods have is the likelihood of various scenarios Monte Carlo methods will not tell you how likely each scenario is that's purely an input into the model is up to you to go and empirically measure what are the likelihoods various scenarios and then use that to construct your Monte Carlo model there's no natural part of a Monte Carlo analysis that will tell you the likelihood of the various scenarios that you're analyzing we're going to now have a slight digression into type 1 and type 2 errors this is drawn directly from Wikipedia and personally I found this very very confusing and it took me a while to understand exactly within the context of type 1 and type 2 errors what exactly this is that we're looking at and how does this relate to Monte Carlo methods here's a table that I came up with that helped me understand exactly what type 1 and type 2 errors are whenever looking at a statement they're ultimately two sides of the statement that when in combination help us create type one and type two errors one is whether the statement is in fact true that is the null hypothesis is true or it is false an example of that hypothesis is for instance the sky is blue all tables are made out of wood or apples are sweet these are merely statements these statements however in reality can be true or they could be false however there's another angle to these statements which is our judgment of these statements that is how do we perceive these statements do we reject them or do we accept them an example of how these interact is for instance is the sky blue the null hypothesis or reality says that in fact the sky is blue however we as researchers can either decide to reject that hypothesis or accept it based on criteria that we may have in the case of the sky is blue the null hypothesis is true it is true that the sky is blue however if we reject that statement we are committing a type 1 error that is we are rejecting a statement or a piece of knowledge which is in fact true that is a type 1 error a type 2 error is the opposite of that in a type 2 error we accept something which is in fact false an example of this would be mythological creatures such as unicorns and gremlins and so forth in past times people have judged that these creatures exist as true even though in reality it is false and in that case we're committing a type 2 error generally speaking people are very sensitive to type 1 errors so that is they don't want to miss out they don't want to know that there are some hypothesis that's true that they rejected however we should be equally as cautious to committing type 2 errors that is resetting hypotheses that are in fact false and Monte Carlo methods are very useful way to account for type 2 errors because within the framework of a Monte Carlo method you can test if your hypothesis was true what are the results and would those results be coinciding with reality you remember from a previous slide that's where the power of Monte Carlo methods really come about is where they allow us to actually judge hypotheses and see if they are true what they would in fact imply about our observations within a Monte Carlo world and if our multicolored world is very very different in the world we actually see we can actually use that to infer about our original hypotheses there are other applications monte carlo methods these applications have both advantages and disadvantages the first application is expanding the possible scenarios monte carlo provides us with a range of standards that is really hard for the human mind to replicate and this opens our minds up to opportunities that we wouldn't have foreseen when we just had to sit there and actually think up of possible scenarios disadvantage of this method is that the scenario is often very hard to justify and communicate and especially for those outside of a quantitative mindset in the sense that when we're talking about potential scenarios since they haven't happened in the past it's very hard to communicate with you who aren't familiar with using Monte Carlo methods the scenarios that haven't happened could potentially happen and this is a very difficult issue of communication another application of Monte Carlo methods is sensitivity trained and this allows us to easily incorporate conditional changes to a system that is if we want to examine how if we keep everything else constant and we change one parameter or one input or one element of the system we can easily test what's going to happen to that system as we change that parameter a disadvantage with using this method of analysis is that it doesn't eliminate the issues with conditional analysis that is is it realistic to expect that one element of the system or one parameter should change while holding everything else constant most likely it is not because in our world all parameters and the entire system change at once in a continuous basis and conditional analysis can often be very misleading in that case another application of multicolor methods is the reduction in complexity this permits us to create a comprehensive measurement of the complex systems that is even if we can go out empirically and measure everything that's in the world if we create a Monte Carlo model that approximates that world sufficiently we can gather for vastly richer information than we would if we had to actually go out and retrieve it simply because it's much more costly to go out and retrieve all this information than to create it within this context of a simulation a disadvantage of this is the measurements are only as credible as models being simulated and the ability to reduce complexity will really depend on the assumptions that you use when you're creating the model but it's finished with some key lessons that decision-makers can take from its presentation the first thing is do not trust Monte Carlo methods blindly regardless of how well they've worked in the past or the credentials behind the person who's created them it is really important not to trust them blindly and there are things that you can do as a non-technical person in order to better establish how useful this models actually are it's important to remember that every monte carlo model is as such there are outputs a system and inputs and we use the inputs filter them through the system and create outputs we then take those outputs and look at the inputs in turn and also try to change the system all Montecarlo methods work this way and is important for you as a decision-maker to ask are these inputs reasonable is this system enough to explain what it is trying to explain are the outputs within reason of what we observe in the world these are the key questions that you should really keep to yourself before you decide to incorporate Monte Carlo analysis the first is well these methods contribute and this is a really big question before we decide to go to the extra complexity will this actually help us answer the decision we're trying to answer another is is the architect of this model capable of managing that and this is a big issue is the person who's creating the system really capable making sure that the system is congruent with the other processes that are going on in this project often you'll find that people who are very talented at creating these models are not capable of either communicating them to stakeholders or managing the implementation of this model within the work environment another is are the assumptions of these models reasonable and this is a really big issue is it okay if you have a model that has totally unrealistic assumptions that would never ever ever happen in my opinion it is not and for you as a decision maker you should be very careful about that another is all the outputs of this model realistic that is are these outputs validated and are they congruent with what we see in the real world and a final question which is the most important question is is the model properly validated that is do we know that we can validate the results of this model and is that validation credible and that's the most important thing that you can do for any Monte Carlo model

Channel: Alon Honig

Views: 170,978

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Id: t0F3S-46bIQ

Length: 30min 29sec (1829 seconds)

Published: Sat Feb 23 2013