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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def _get_linking_probabilities(self, worlds: List[WikiTablesWorld], linking_scores: torch.FloatTensor, question_mask: torch.LongTensor, entity_type_dict: Dict[int, int]) -> torch.FloatTensor: """ Produces the probability of an entity given a question word and type. The logic below separates the entities by type since the softmax normalization term sums over entities of a single type. Parameters worlds : ``List[WikiTablesWorld]`` linking_scores : ``torch.FloatTensor`` Has shape (batch_size, num_question_tokens, num_entities). question_mask: ``torch.LongTensor`` Has shape (batch_size, num_question_tokens). entity_type_dict : ``Dict[int, int]`` This is a mapping from ((batch_index * num_entities) + entity_index) to entity type id. Returns ------- batch_probabilities : ``torch.FloatTensor`` Has shape ``(batch_size, num_question_tokens, num_entities)``. Contains all the probabilities for an entity given a question word. """

        _, num_question_tokens, num_entities = linking_scores.size()
        batch_probabilities = []

        for batch_index, world in enumerate(worlds):
            all_probabilities = []
            num_entities_in_instance = 0

            # NOTE: The way that we're doing this here relies on the fact that entities are
            # implicitly sorted by their types when we sort them by name, and that numbers come
            # before "fb:cell", and "fb:cell" comes before "fb:row".  This is not a great
            # assumption, and could easily break later, but it should work for now.
            for type_index in range(self._num_entity_types):
                # This index of 0 is for the null entity for each type, representing the case where a
                # word doesn't link to any entity.
                entity_indices = [0]
                entities = world.table_graph.entities
                for entity_index, _ in enumerate(entities):
                    if entity_type_dict[batch_index * num_entities + entity_index] == type_index:
                        entity_indices.append(entity_index)

                if len(entity_indices) == 1:
                    # No entities of this type; move along...
                    continue

                # We're subtracting one here because of the null entity we added above.
                num_entities_in_instance += len(entity_indices) - 1

                # We separate the scores by type, since normalization is done per type.  There's an
                # extra "null" entity per type, also, so we have `num_entities_per_type + 1`.  We're
                # selecting from a (num_question_tokens, num_entities) linking tensor on _dimension 1_,
                # so we get back something of shape (num_question_tokens,) for each index we're
                # selecting.  All of the selected indices together then make a tensor of shape
                # (num_question_tokens, num_entities_per_type + 1).
                indices = linking_scores.new_tensor(entity_indices, dtype=torch.long)
                entity_scores = linking_scores[batch_index].index_select(1, indices)

                # We used index 0 for the null entity, so this will actually have some values in it.
                # But we want the null entity's score to be 0, so we set that here.
                entity_scores[:, 0] = 0

                # No need for a mask here, as this is done per batch instance, with no padding.
                type_probabilities = torch.nn.functional.softmax(entity_scores, dim=1)
                all_probabilities.append(type_probabilities[:, 1:])

            # We need to add padding here if we don't have the right number of entities.
            if num_entities_in_instance != num_entities:
                zeros = linking_scores.new_zeros(num_question_tokens,
                                                 num_entities - num_entities_in_instance)
                all_probabilities.append(zeros)

            # (num_question_tokens, num_entities)
            probabilities = torch.cat(all_probabilities, dim=1)
            batch_probabilities.append(probabilities)
        batch_probabilities = torch.stack(batch_probabilities, dim=0)
        return batch_probabilities * question_mask.unsqueeze(-1).float()