Timeline for Is the exact value of any likelihood meaningless?
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Jun 28, 2022 at 21:10 | comment | added | Michael Lew | I now understand that I was not correct. Question and answer here: stats.stackexchange.com/questions/578944/… | |
| Jun 15, 2022 at 21:36 | comment | added | Michael Lew | @apdnu Seems unlikely to me. I'll compose a question about it and you can write an extended answer. | |
| Jun 15, 2022 at 9:00 | comment | added | apdnu | @MichaelLew Probabilities of discrete variables are dimensionless, but probabilities of continuous variables have dimensions of 1/(units of variables). Without this, the normalization condition for probability densities would make no sense, because integrals of the probability density function over the variable's domain would end up with units of (units of variable). In that case, how do you normalize to one? One in which units? | |
| Jun 15, 2022 at 7:26 | comment | added | Michael Lew | @apdnu Probability is dimensionless. | |
| Jun 15, 2022 at 6:50 | comment | added | apdnu | @MichaelLew If the data consists of continuous variables with units, then the likelihood does indeed have units of 1/(units of the data). If the data is discrete and unitless, then the likelihood is also unitless. The likelihood can be understood as the probability of the data given the model, and I think it's confusing to students to deny that that's the case, or to claim that the likelihood isn't a probability. You should explain this interpretation of the likelihood first, and after that, explain that some people have philosophical objections to it. | |
| Jun 15, 2022 at 0:22 | comment | added | Michael Lew | @apdnu Likelihood has no units. If trying to express it as something within "data space" gives it units then you have found a good reason to avoid such a way of expressing and understanding it! | |
| Jun 15, 2022 at 0:21 | comment | added | Michael Lew | @apdnu If you are restricting your considerations to continuous data then you are omitting the most common examples used to demonstrate likelihood! And, for likelihoods it is worth bearing in mind that all data are discrete because they cannot be specified with infinite precision. That leads to the numerical value of likelihood being dependent on the number of decimals attached to the data. | |
| Jun 14, 2022 at 15:08 | comment | added | apdnu | @Tim The likelihood can also be considered as a function of the data, holding the model parameters fixed. Either way of viewing the likelihood is valid. Viewing the likelihood as a function of the data gives extra intuition about what it means (i.e., it's a probability density in data space) that one wouldn't get by considering it as a function of the model parameters alone. I don't understand the resistance to viewing it both ways, especially since it's pedagogically useful and would clear up the questioner's original confusion. | |
| Jun 14, 2022 at 12:52 | comment | added | Tim | @apdnu but the likelihood is used and defined as a function of parameters with fixed data, so "in data space" is an irrelevant angle to consider it. | |
| Jun 14, 2022 at 12:35 | comment | added | apdnu | @MichaelLew There are a number of ways of seeing that the likelihood is a probability density in the data space (I'm assuming the data variables are continuous, no discrete). First, the units of the likelihood are 1/(units of data). In other words, it has the units of density in the data-space. Second, if you integrate the likelihood over the data domain, you get one. It behaves exactly like a probability density in the data-space (which is not surprising, given that it's defined as the conditional probability of the data on the model). | |
| Jun 13, 2022 at 21:18 | comment | added | Michael Lew | @apdnu Likelihood functions have the parameter(s) of interest as the x-axis scale. How is that "in the data-space"? | |
| Jun 13, 2022 at 11:59 | comment | added | apdnu | It sounds like your objections are more philosophical (Frequentist vs. Bayesian) than practical. Likelihood functions exactly like a probability (or probability density) in the data-space, and that is a perfectly valid interpretation of what it is. Indeed, that is exactly how it is generally defined, and any other definition is equivalent to this definition under a shift of your philosophical outlook (e.g., from Frequentist to Bayesian). | |
| Jun 13, 2022 at 11:53 | comment | added | Tim | @apdnu it doesn't have to be normalized, chance integrates to one, in non-Bayesian setting it is not a conditional probability, since the parameters are not random variables. You are assuming Bayesian likelihood, not the general concept. | |
| Jun 13, 2022 at 11:46 | comment | added | apdnu | The likelihood is literally defined as the probability of the data, given a model. It's a probability in data-space, integrates to one in that space, is non-negative, etc. | |
| Jun 12, 2022 at 13:20 | comment | added | Tim | @apdnu it's not, it's more complicated, see stats.stackexchange.com/q/2641/35989 Also you seem to assume a Bayesian perspective, where the term exists also outside it. | |
| Jun 12, 2022 at 12:45 | comment | added | apdnu | The likelihood is a probability (or probability density) in the data space (not the model-parameter space). Specifically, it is the probability of the data given the model parameters. The integral of the likelihood over all possible data comes to 1. In MCMC, one often works with an un-normalized version of the likelihood, and that un-normalized version is, of course, not a proper probability. | |
| Jun 11, 2022 at 6:07 | vote | accept | Alice | ||
| Jun 11, 2022 at 5:42 | history | answered | Tim | CC BY-SA 4.0 |