Why can the joint pmf/pdf be written as a product here?

The parameters of the Normal proposal distribution are chosen using maximum likelihood estimation. The mean $\mu$ is set to the sample mean of the retained $g^*$ samples, and the standard deviation $\sigma$ is set to the square root of the sample second central moment of those samples.

Conditions: The proposal family is restricted to Normal distributions.; The data used for fitting are the retained samples satisfying $X>\gamma$.

The likelihood function is $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$. It represents the probability of observing 7 heads and 3 tails as a function of the unknown parameter $p$, the probability of heads.

Conditions: The number of tosses $n=10$.; The observed number of heads $k=7$.; $p$ is the probability of heads on a single toss.

To compute the likelihood for an independent sample, first write the joint pmf/pdf of all observations conditional on the parameter, then use independence to factor it into a product of the individual pmf/pdf terms.

Conditions: The sample $x_1, \dots, x_n$ is independent.; Each observation has the same functional form $f(x_i \mid \theta)$.; The data values are plugged in while $\theta$ remains symbolic.

Probability fixes the parameter and varies the data, resulting in a normalized distribution where the probabilities sum to 1. Likelihood fixes the data and varies the parameter, resulting in a non-normalized curve whose integral does not equal 1.

Conditions: Both views are based on the same conditional expression $P(x \mid \theta)$.; Probability view: parameter is fixed, data varies.; Likelihood view: data is fixed, parameter varies.

The elite samples are weighted by likelihood ratios because they are drawn from the current proposal distribution, not the original target distribution. The likelihood ratio corrects for this discrepancy, ensuring that the updated proposal parameters accurately reflect the maximum likelihood estimates under the original distribution.

Conditions: Elite samples are those with values above the iteration threshold $\gamma_i$.; The likelihood ratios are computed as the original density divided by the proposal density.