= Solution
A <proportional hazards model> has $h(t\mid g)=h_0(t)r(g)$, with a time-independent hazard ratio. In the <Cox proportional-hazards model>, $r(g)=e^{\beta g}$ and the baseline hazard $h_0$ is unspecified. At each untied event time, conditioning on the identity of the person who fails given the current <risk set> produces the factor
$$
\frac{e^{\beta g_i}}{\sum_{j\in R_i}e^{\beta g_j}}.
$$
Multiplying these factors gives the <Cox partial likelihood>. It is called partial because it uses the conditional event ordering information that identifies $\beta$ while discarding the part of the full likelihood involving the unspecified baseline hazard.
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