Interpretation likelihood graphic I would like a clarification regards the concept of likelihood.
Example:
A random sample distributed with a density of normal probability (u, v). you get the following graph of the standardized likelihood for n = 5, $\bar{x}$ = 1 and $s^2$ = 2.

I would like to know if the graph can be interpreted in this way:
That axis goes from -1 to 3 represents the values of $\bar{x}$ = 1 (with a maximum in 1).
The side of axis that goes from 2 to 8 represents the values of $s^2$ = 2 (with maximum 2).
Is this an accurate way to interpret it?
 A: No, your interpretation is incorrect.
The likelihood is a function of the parameters, in this case, $\mu$ and $\sigma^2$; those are what the values on the axes relate to. Given the data, the values of $\bar{x}$ and $s^2$ are constants. In this case, they ($\bar{x}$ and $s^2$) are the values at the two dashed lines. When sampling from the normal distribution, they contain all the information in the data about the parameters.
It happens in this case that these particular values correspond to the peak of the likelihood function.
A: No, your intepretation is incorrect.
The function shows the likelihood of the parameter values (shown on the horizontal axes). "Likelihood" means how likely they are given the observed data. If, for example, you look at the pair of values {-1, 8}, you see that the likelihood is low, which means that it is unlikely that these parameter values are the true values given the sample you have. A more likely set of values for your parameters is {1,2}, i.e., the most likely parameter values (the mode of the plot) are 1 for the mean and 2 for the variance.
The likelihood is to be interpreted as a probability. The largest the probability is, the more the parameters corresponding to this likelihood are probable (are likely). Likelihood is a number between 0 (not probable at all) and 1 (absolutely probable); in general, it is somewhere in-between. The number can be quite small; what matter is that it is the most (i.e., the highest) likely of all the parameter values considered.
