Probability vs. Likelihood ... MADE EASY!!!

today we're going to learn about probability versus likelihood so what is probability well probability means exactly what you think it means it is the longrun frequency of how often something occurs probability is the word we are most accustomed to probability exists in one Universe what I mean by this is when we talk about Pro the probability of an event we talk about it in relation to other events that may occur in the same universe this means that probabilities are related in an important way they must add up to one something must happen so for example if I flip one coin a Fair coin that lands 50% heads and 50% tails the probability of heads is 0.5 and the probability of tails is 0.5 these two numbers add up to one because they exist in the same universe and one of the two things must happen probabilities must add up to one so by the same universe I mean the universe in which we have a fair coin if a coin is fair the coin must be heads or tails the probabilities must add up to one we can discuss probabilities when we understand which Universe we are in in probability we know the parameters which exactly describe the situation and how often things occur but in statistics we often do not know which Universe we are in we only observe data but we don't understand the process that created the data so when we talk about a coin flip suppose we flip a coin with an unknown probability of heads and I observe heads I can't say that the probability of heads is 0.5 because I don't know what type of coin this is if the coin is fair then the probability would be 0.5 like we see over here but if the coin is not fair then the probability could be 0.7 or it could be 0.9 or any other number it could be 0.1 probability of heads I do not know which Universe I'm in so I do not know the probability of heads I just know that I saw heads I observe heads I don't know what the probability of heads is but I can talk about what the probability of heads would be with different types of coins all of the numbers below are probabilities in a sense but they exist in different universes and don't add up to one so we don't use them in the same way that we use probabilities they are likelihoods of our observed data under different scenarios likelihoods in some scenarios are the exact same numbers as probabilities calling them a likelihood comes from the context that we're using them in probabilties add up to one because we're considering things in one Universe I.E for a single value of a parameter something must happen in probability world but likelihoods are a probability of The observed data under a hypothetical scenario since there are many different hypothetical scenarios and in statistics we don't know which one there are many likelihoods that do not add up to one and thus cannot be interpreted as probabilities uh let's look at another example this time we're flipping two coins suppose we have a fair coin then I can talk about the probability of zero heads being 1/4 the probability of one head being 1/2 and the probability of two heads being 1/4 these are probabilities these numbers add up to one and in the universe where we have a fair coin that we flip twice one of these three things will happen which is why the probabilities add up to one but if I had another coin if the coin is heads 90% of the time that probability of zero heads is 01 the probability of one head is18 and the probability of two heads is 81 in the universe where we have an unfair coin that we flipped twice one of these three things will happen so the probabilities add up to one but let's flip the situation around suppose we just observe two heads now we know if it was a Fair coin the likelihood of two heads is 25% but if it's a 90% heads coin the likelihood of two heads is 81 if it's a coin that lands on heads 10% of the time the likelihood of two heads is only 1% these numbers don't add up to one because they occur in different universes it doesn't make sense in this context to talk about them as real probabilities they are one of many possible probabilities of what we observed again these numbers do not add up to one so in summary in the context of these discret probability distributions where we're talking about coin flips probability and likelihood are really the same thing but from different perspectives probability means we know which Universe we're in and the probabilities all add up to one where it's likelihood we know what we observed and we consider the probability of what we observed in any past POS universe but what about continuous distributions like a normal distribution in a normal distribution the probability is represented by the area under the curve of a density function so the normal bell curve with standard deviation one is described by this density function now this function does not tell us probabilities uh they are used to compute probabilities we integrate the function and find the area under the curve to compute a probability so here we have the density of a normal distribution at one and this number is not a probability and is also not a likelihood in this context either it is a probability density it's related to probability and it's used to compute probabilities uh here the value zero uh has a higher likelihood of 39 now both 0 and one occur with probability zero because it's a continuous distribution but values near zero are still more likely to occur than values near one now suppose we observe data instead we observe xals 1 but we don't know what Universe we're in well what is the probability that x equals 1 now that's not even a meaningful question for a normal distribution because every outcome has probability zero well what is the density at xals 1 again this depends on what Universe we're in and which normal distribution we have so for a continuous distribution the relevant question is not what is a likelihood versus a probability rather it is a density versus a likelihood so here we see that we observe our data point xal 1 and we say what is the density at that point well I don't know because it depends what Universe we're in for the green distribution the likelihood is very high for the red in Black distributions the d uh the likelihood is in the middle and for the blue distribution the likelihood is very low so this same function's density function can be viewed in two ways it is a probability density function if we View it as a function of X where mu is known so if we know the parameter and we know which Universe we're in then it is a density function but if we fix X if we observe our data and view as a function of mu the unknown parameter then we call it a likelihood function we we can see this illustrated by talking about the idea of maximizing the probability versus maximizing the likelihood so a major use of likelihood functions is maximum likelihood estimation and this can help us understand the differ between the two so here is an illustration of probability what is the maximum probability well the most likely outcome with a Fair coin is one head however likelihood takes place in any universe so if we observe our data we observe two heads well we have different likelihoods depending on which Universe we chose so we're going to chose the universe in which our data was most likely where the probability of heads was very high that's the end uh please like And subscribe for more statistics itics content

Channel: Brian Greco - Learn Statistics!

Views: 25,520

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Id: bXGjQnpGGIo

Length: 7min 31sec (451 seconds)

Published: Wed Mar 27 2024