What is the difference between likelihood and probability

Simply put, probability points to chances while likelihood denotes a possibility. This being said, it is reasonable to infer that a probability involves the computation of chances done with formulas carefully established by mathematicians. On the other hand, a likelihood serves as an inference or forecasts that do not involve the use of a solid basis or theory. While this may be confusing, experts have come up with a system that can give clues on the proper usage of both terms.

Take, for example, a scenario with two individuals conversing. He cannot assert the probability because he has not looked into the statistics and numbers that can speak of the possible chances of the storm changing direction.

A probability follows clear parameters and computations while a likelihood is based merely on observed factors. Cite APA 7 Franscisco,. Difference Between Probability and Likelihood. That's exactly why it helped! I stopped thinking that it must have a particular meaning and instead just followed the logic.

Show 5 more comments. Joffer 3 3 bronze badges. Thylacoleo Thylacoleo 4, 5 5 gold badges 24 24 silver badges 32 32 bronze badges. Especially the three last paragraphs are very useful. How would you extend this to describe the continuous case? I don't mind math at all, but for me math is a tool ruled by what I want I don't enjoy math for its own sake, but for what it helps me do.

Only with this answer do I know the latter. Gypsy Gypsy 6 6 silver badges 2 2 bronze badges. I had to think through what it meant when I read that likelihood is not probability, but the following case occurred to me. What is the likelihood that a coin is fair, given that we see four heads in a row? We can't really say anything about probability here, but the word "trust" seems apt. Do we feel we can trust the coin? Thank you so much! I actually find it hard myself to say which "school" I am, as I have a bit of knowledge from every school.

The "Probability as extended logic" school is my favourite duh , but I don't have enough practical experience in applying it to real problems to be dogmatic about it. However, the likelihood function is proportional to the probabiilty of the observed data. Could you please address the comment that locster made? John John 21k 9 9 gold badges 47 47 silver badges 83 83 bronze badges.

Intuition and clarity above dry mathematical rigor, not to say something more derogatory. Thank you so much!! Glorfindel 1 1 gold badge 9 9 silver badges 18 18 bronze badges. Yaroslav Bulatov Yaroslav Bulatov 5, 2 2 gold badges 24 24 silver badges 38 38 bronze badges. What does it mean? So, it is often used as an objective function, but also as a performance measure to compare two models as in Bayesian model comparison.

Lenar Hoyt Lenar Hoyt 1 1 gold badge 7 7 silver badges 15 15 bronze badges. I was right all along?! English not being my mother tongue, I grew up with only one word for seemingly both of the terms or have I simply never gotten a problem where I needed to distinguish the terms? It's only now, that I know two English terms, that I begin to doubt my understanding of these things.

I wonder, why it got so few upvotes. Response Response 2 2 silver badges 4 4 bronze badges. So the real answer seems to be that the likelihood can be a probability, but is sometimes not.

Possible results are mutually exclusive and exhaustive. Suppose we ask a subject to predict the outcome of each of 10 tosses of a coin.

There are only 11 possible results 0 to 10 correct predictions. The actual result will always be one and only one of the possible results. Thus, the probabilities that attach to the possible results must sum to 1. Hypotheses, unlike results, are neither mutually exclusive nor exhaustive. Suppose that the first subject we test predicts 7 of the 10 outcomes correctly. I might hypothesize that the subject just guessed, and you might hypothesize that the subject may be somewhat clairvoyant, by which you mean that the subject may be expected to correctly predict the results at slightly greater than chance rates over the long run.

In technical terminology, my hypothesis is nested within yours. Someone else might hypothesize that the subject is strongly clairvoyant and that the observed result underestimates the probability that her next prediction will be correct. Another person could hypothesize something else altogether. There is no limit to the hypotheses one might entertain. The set of hypotheses to which we attach likelihoods is limited by our capacity to dream them up.

In practice, we can rarely be confident that we have imagined all the possible hypotheses. Our concern is to estimate the extent to which the experimental results affect the relative likelihood of the hypotheses we and others currently entertain. Because we generally do not entertain the full set of alternative hypotheses and because some are nested within others, the likelihoods that we attach to our hypotheses do not have any meaning in and of themselves; only the relative likelihoods — that is, the ratios of two likelihoods — have meaning.

Figure 1. Both panels were computed using the binopdf function. In the upper panel, I varied the possible results; in the lower, I varied the values of the p parameter.

The probability distribution function is discrete because there are only 11 possible experimental results hence, a bar plot. By contrast, the likelihood function is continuous because the probability parameter p can take on any of the infinite values between 0 and 1.

The probabilities in the top plot sum to 1, whereas the integral of the continuous likelihood function in the bottom panel is much less than 1; that is, the likelihoods do not sum to 1. The difference between probability and likelihood becomes clear when one uses the probability distribution function in general-purpose programming languages. In the present case, the function we want is the binomial distribution function.

DIST in the most common spreadsheet software and binopdf in the language I use. It has three input arguments: the number of successes, the number of tries, and the probability of a success.

When one uses it to compute probabilities, one assumes that the latter two arguments number of tries and the probability of success are given. They are the parameters of the distribution.

One varies the first argument the different possible numbers of successes in order to find the probabilities that attach to those different possible results top panel of Figure 1. Regardless of the given parameter values, the probabilities always sum to 1. By contrast, in computing a likelihood function, one is given the number of successes 7 in our example and the number of tries In other words, the given results are now treated as parameters of the function one is using.

Instead of varying the possible results, one varies the probability of success the third argument, not the first argument in order to get the binomial likelihood function bottom panel of Figure 1. One is running the function backwards, so to speak, which is why likelihood is sometimes called reverse probability. The information that the binomial likelihood function conveys is extremely intuitive. It says that given that we have observed 7 successes in 10 tries, the probability parameter of the binomial distribution from which we are drawing the distribution of successful predictions from this subject is very unlikely to be 0.

It is much more likely to be 0. The hypothesis that the long-term success rate is 0. We have a stochastic process that takes discrete values i. We also assumed implicitly that the coin tosses are independent. Alvaro Carril. The difference between probability and likelihood using coins The distinction between probability and likelihood is extremely important, though often misunderstood.

The experiment Suppose an experiment where a person has to predict the outcome of each of 10 coin tosses.