why cumsum the prob to render depth？

[Lazyangel]

Hello author, I'm not quite familiar with NeRF volume rendering. Could you explain why probability accumulation is summed up here to render depth? What is the corresponding mathematical formula for this process? What is the corresponding physical meaning? in my opinion，cumulative multiplication might make more sense?

def get_density(self, rays_o, rays_d, Voxel_feat, is_train, inputs):
        dtype = torch.float16 if self.opt.use_fp16 else torch.float32
        device = rays_o.device
        rays_o, rays_d, Voxel_feat = rays_o.to(dtype), rays_d.to(dtype), Voxel_feat.to(dtype)

        reg_loss = {}
        eps_time = time.time()
        with torch.no_grad():
            rays_o_i = rays_o[0, ...].flatten(0, 2)  # HXWX3
            rays_d_i = rays_d[0, ...].flatten(0, 2)  # HXWX3
            rays_pts, mask_outbbox, z_vals, rays_pts_depth = self.sample_ray(rays_o_i, rays_d_i, is_train=is_train)

        dists = rays_pts_depth[..., 1:] - rays_pts_depth[..., :-1]  # [num pixels, num points - 1]
        dists = torch.cat([dists, 1e4 * torch.ones_like(dists[..., :1])], dim=-1)  # [num pixels, num points]

        sample_ret = self.grid_sampler(rays_pts, Voxel_feat, avail_mask=~mask_outbbox)
        if self.use_semantic:
            if self.opt.semantic_sample_ratio < 1.0:
                geo_feats, mask, semantic, mask_sem, group_num, group_size = sample_ret
            else:
                geo_feats, mask, semantic = sample_ret
        else:
            geo_feats, mask = sample_ret


        if self.opt.render_type == 'prob':
            weights = torch.zeros_like(rays_pts[..., 0])
            weights[:, -1] = 1
            geo_feats = torch.sigmoid(geo_feats)
            if self.opt.last_free:
                geo_feats = 1.0 - geo_feats  # the last channel is the probability of being free
            weights[mask] = geo_feats

            # accumulate
            weights = weights.cumsum(dim=1).clamp(max=1)
            alphainv_fin = weights[..., -1]
            weights = weights.diff(dim=1, prepend=torch.zeros((rays_pts.shape[:1])).unsqueeze(1).to(device=device, dtype=dtype))
            depth = (weights * z_vals).sum(-1)
            rgb_marched = 0```