Probability density function calculator4/19/2023 This justifies our use of the density function for continuous random variables. You can think of this approximation by visualising an integral and recalling that an integral represents the area under the curve, and so for small $\delta$ this integral can be approximated by taking the area of a rectangle which is width $\times$ height, where the width is $\delta$ and the height is $f_X(x)$. This means that we need to choose a probability distribution that is parameterised by some parameters $\boldsymbol f_X(x) dx \approx \delta p_X(x) \ $$ The idea of Maximum Likelihood is to maximising the (log-)likelihood for a given set of data. It is crucial that you understand that in general the likelihood is the probability of your observed data and not just the density or the product of densities as you may not always have iid and so it will not always boil down to taking the product of some densities. In my example below I use $p_\mu(x)$ to denote the density of the normal distribution with mean parameter $\mu$, but the density of the likelihood is the product of all the densities (because we assumed iid data). Further, as you have written $L(\theta | x) = p_\theta(x)$ I would like to clarify that if we are being precise in our definitions then this is only correct if you have one observed data point, assuming that you meant $p_\theta(x)$ is the density of $X$. Also the " $|$" may cause confusion as it looks like we are conditioning on $x$ but this is not the case - it may be better practice to use $L(\theta x)$ which is the notation used when I was learning likelihood. The probability density is used to 'measure how good' the parameters are because it is a natural way of quantifying if these parameters are good for the observed data.Īlso, as the notation often causes some confusion, $L(\theta | x)$ denotes the probability of all of your observed data, not just one value.
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